<rss xmlns:itunes="http://www.itunes.com/dtds/podcast-1.0.dtd" version="2.0">
  <channel>
    <title>MaplePrimes - Newest Questions</title>
    <link>http://www.mapleprimes.com/questions</link>
    <language>en-us</language>
    <copyright>2026 Maplesoft, A Division of Waterloo Maple Inc.</copyright>
    <generator>Maplesoft Document System</generator>
    <lastBuildDate>Sat, 04 Jul 2026 05:15:28 GMT</lastBuildDate>
    <pubDate>Sat, 04 Jul 2026 05:15:28 GMT</pubDate>
    <itunes:subtitle />
    <itunes:summary />
    <description>The questions most recently asked on MaplePrimes</description>
    <image>
      <url>http://www.mapleprimes.com/images/mapleprimeswhite.jpg</url>
      <title>MaplePrimes - Newest Questions</title>
      <link>http://www.mapleprimes.com/questions</link>
    </image>
    <item>
      <title>How to make ChatGPT use Maple via MCP</title>
      <link>http://www.mapleprimes.com/questions/243664-How-To-Make-ChatGPT-Use-Maple-Via-MCP?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;Dear Maple users&lt;/p&gt;

&lt;p&gt;I am testing Maple 2026, which will be used at our school after the summer holiday. I see that AI have now found its way to Maple on a new level. AI can be used internally via the AI Assistant, but as I understand it will also be possible to let ChatGPT use Maple in order to provide an answer, instead of using it&amp;#39;s own way to do math. I just cannot figure out how it is done. Having a school license for Maple means we take part in the Maplesoft Elite Maintenance Program (EMP). In Maple 2026 i have found Maple MCP on the &amp;quot;My Maple&amp;quot; b&amp;aring;ndet. When I click it I am however just referred to the main page of Maplesoft. My question: How can I make Maple MCP work for me?&lt;/p&gt;

&lt;p&gt;Kind regards,&lt;/p&gt;

&lt;p&gt;Erik V.&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;Dear Maple users&lt;/p&gt;

&lt;p&gt;I am testing Maple 2026, which will be used at our school after the summer holiday. I see that AI have now found its way to Maple on a new level. AI can be used internally via the AI Assistant, but as I understand it will also be possible to let ChatGPT use Maple in order to provide an answer, instead of using it&amp;#39;s own way to do math. I just cannot figure out how it is done. Having a school license for Maple means we take part in the Maplesoft Elite Maintenance Program (EMP). In Maple 2026 i have found Maple MCP on the &amp;quot;My Maple&amp;quot; b&amp;aring;ndet. When I click it I am however just referred to the main page of Maplesoft. My question: How can I make Maple MCP work for me?&lt;/p&gt;

&lt;p&gt;Kind regards,&lt;/p&gt;

&lt;p&gt;Erik V.&lt;/p&gt;
</description>
      <guid>243664</guid>
      <pubDate>Fri, 03 Jul 2026 23:00:04 Z</pubDate>
      <itunes:author>erik10</itunes:author>
      <author>erik10</author>
    </item>
    <item>
      <title>plotting piecewise function</title>
      <link>http://www.mapleprimes.com/questions/243663-Plotting-Piecewise-Function?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;Why doesn&amp;#39;t a piecewise function plot correctly?&lt;/p&gt;

&lt;p&gt;(1)&amp;nbsp; &amp;nbsp;plot(sin((2*Pi)*100*t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; Plots correctly&lt;/p&gt;

&lt;p&gt;(2)&amp;nbsp; &amp;nbsp;s1 := t -&amp;gt; sin(200*t*Pi);&lt;br&gt;
&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;plot(s1(t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;Plots correctly&lt;/p&gt;

&lt;p&gt;(3)&amp;nbsp; &amp;nbsp;s2 := t -&amp;gt; piecewise(0 &amp;lt; t, 0, t &amp;lt; 10, sin(200*t*Pi), 0);&lt;br&gt;
&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;s2 := proc (t) options operator, arrow; piecewise(0 &amp;lt; t, 0, t &amp;lt;&amp;nbsp;10, sin(200*t*Pi), 0) end proc&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;plot(s2(t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; Does not plot correctly. Only a blank plot is displayed.&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;Why doesn&amp;#39;t a piecewise function plot correctly?&lt;/p&gt;

&lt;p&gt;(1)&amp;nbsp; &amp;nbsp;plot(sin((2*Pi)*100*t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; Plots correctly&lt;/p&gt;

&lt;p&gt;(2)&amp;nbsp; &amp;nbsp;s1 := t -&amp;gt; sin(200*t*Pi);&lt;br /&gt;
&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;plot(s1(t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;Plots correctly&lt;/p&gt;

&lt;p&gt;(3)&amp;nbsp; &amp;nbsp;s2 := t -&amp;gt; piecewise(0 &amp;lt; t, 0, t &amp;lt; 10, sin(200*t*Pi), 0);&lt;br /&gt;
&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;s2 := proc (t) options operator, arrow; piecewise(0 &amp;lt; t, 0, t &amp;lt;&amp;nbsp;10, sin(200*t*Pi), 0) end proc&lt;/p&gt;

&lt;p&gt;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp;plot(s2(t), t = 0 .. 1);&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; Does not plot correctly. Only a blank plot is displayed.&lt;/p&gt;
</description>
      <guid>243663</guid>
      <pubDate>Fri, 03 Jul 2026 17:52:49 Z</pubDate>
      <itunes:author>anthonymlinar</itunes:author>
      <author>anthonymlinar</author>
    </item>
    <item>
      <title>Upgrading from Maple 2016.2  to Maple 2026</title>
      <link>http://www.mapleprimes.com/questions/243662-Upgrading-From-Maple-20162--To-Maple-2026?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;A very stupid question :I want my one-dimensional output from Maple 2016 and also Maple 2026, use as one-dimensional input in my Maple 2026&amp;nbsp;, which I just bought. I&amp;#39;am very inpatient and think it must be very simple but too complicated for me. Before making a very extensive and timeconsuming study somebody should be able to tell me to perform this&amp;nbsp; by a few clicks .&lt;br&gt;
Thanking you beforehand and I&amp;#39;ll be very gratefull for your help.&lt;/p&gt;

&lt;p&gt;Bartele de Jong The netherlands&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;A very stupid question :I want my one-dimensional output from Maple 2016 and also Maple 2026, use as one-dimensional input in my Maple 2026&amp;nbsp;, which I just bought. I&amp;#39;am very inpatient and think it must be very simple but too complicated for me. Before making a very extensive and timeconsuming study somebody should be able to tell me to perform this&amp;nbsp; by a few clicks .&lt;br /&gt;
Thanking you beforehand and I&amp;#39;ll be very gratefull for your help.&lt;/p&gt;

&lt;p&gt;Bartele de Jong The netherlands&lt;/p&gt;
</description>
      <guid>243662</guid>
      <pubDate>Wed, 01 Jul 2026 13:09:05 Z</pubDate>
      <itunes:author>Bart_de_Jong</itunes:author>
      <author>Bart_de_Jong</author>
    </item>
    <item>
      <title>Missing operation (multiplication) in 2D-Math</title>
      <link>http://www.mapleprimes.com/questions/243658-Missing-Operation-multiplication-In-2DMath?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;In 2-D Math input:&lt;br&gt;
In a product of more than two factors space is not allways sufficient to delimite factors when one of the factors is of type numeric.&lt;/p&gt;

&lt;p&gt;Just for my interest: Is there a reason or a rule for that?&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="F5424676DE4C50DDC4DA7F95D02FEDD7"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;2-D Math: space interpreted as multiplication&lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="a*b*c" height="23" src="/view.aspx?sf=243658_question/1edf683c78bda9d7f706d5a9dfa6a213.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="a*b*c" height="23" src="/view.aspx?sf=243658_question/025b40f965b9b241d299986445e13a9b.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*b*c" height="23" src="/view.aspx?sf=243658_question/80739ee7352b6a06a85049525afaab9c.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*b*c" height="23" src="/view.aspx?sf=243658_question/6f34bbb07e3f2e212268c7e0f3d1b753.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;With&amp;nbsp;numbers this does not work in these cases &lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a 2 c&amp;quot;" height="23" src="/view.aspx?sf=243658_question/c815e836509e54138adf5bb29d68a9c4.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;a href="http://www.maplesoft.com/support/help/errors/view.aspx?path=Error,%20missing%20operation"&gt;&lt;span style="color:#ff00ff;font-size: 100%;font-family: Courier New,monospace;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Error, missing operation&lt;/u&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;

						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a 2 c&amp;quot;" height="24" src="/view.aspx?sf=243658_question/92966489ba41ca70dfe0f1a970497f71.gif" style="vertical-align:-7px" width="44"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a b 2&amp;quot;" height="6" src="/view.aspx?sf=243658_question/264c009f5a7c94d8a99f371500830520.gif" style="vertical-align:-6px" width="768"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;a href="http://www.maplesoft.com/support/help/errors/view.aspx?path=Error,%20missing%20operation"&gt;&lt;span style="color:#ff00ff;font-size: 100%;font-family: Courier New,monospace;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Error, missing operation&lt;/u&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;

						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a b 2&amp;quot;" height="24" src="/view.aspx?sf=243658_question/35dd344cf82825f1bd0168aab6e53aca.gif" style="vertical-align:-7px" width="45"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;Multiplication operators are required&lt;/span&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;2 2 c&amp;quot;" height="24" src="/view.aspx?sf=243658_question/c2255a3bd85ee4c8b091929be863c4f7.gif" style="vertical-align:-7px" width="44"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*c" height="23" src="/view.aspx?sf=243658_question/f8bf7968a5265a06d9899ea18c4d2257.gif" style="vertical-align:-6px" width="50"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*c" height="23" src="/view.aspx?sf=243658_question/facc9ebf6827b9a1dde97e46c0a9d838.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*b" height="23" src="/view.aspx?sf=243658_question/b9dac3286f2c77bbc6ebbe0277d67cb8.gif" style="vertical-align:-6px" width="42"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*b" height="23" src="/view.aspx?sf=243658_question/5919110ec26e960771d282c734e6d755.gif" style="vertical-align:-6px" width="33"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="NULL" height="23" src="/view.aspx?sf=243658_question/a6cede8a69d182df01d05b01d29373a1.gif" style="vertical-align:-6px" width="9"&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243658_question/Missing_operation.mw"&gt;Download Missing_operation.mw&lt;/a&gt;&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;In 2-D Math input:&lt;br&gt;
In a product of more than two factors space is not allways sufficient to delimite factors when one of the factors is of type numeric.&lt;/p&gt;

&lt;p&gt;Just for my interest: Is there a reason or a rule for that?&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="F5424676DE4C50DDC4DA7F95D02FEDD7"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;2-D Math: space interpreted as multiplication&lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="a*b*c" height="23" src="/view.aspx?sf=243658_question/1edf683c78bda9d7f706d5a9dfa6a213.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="a*b*c" height="23" src="/view.aspx?sf=243658_question/025b40f965b9b241d299986445e13a9b.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*b*c" height="23" src="/view.aspx?sf=243658_question/80739ee7352b6a06a85049525afaab9c.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*b*c" height="23" src="/view.aspx?sf=243658_question/6f34bbb07e3f2e212268c7e0f3d1b753.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;With&amp;nbsp;numbers this does not work in these cases &lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a 2 c&amp;quot;" height="23" src="/view.aspx?sf=243658_question/c815e836509e54138adf5bb29d68a9c4.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;a href="http://www.maplesoft.com/support/help/errors/view.aspx?path=Error,%20missing%20operation"&gt;&lt;span style="color:#ff00ff;font-size: 100%;font-family: Courier New,monospace;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Error, missing operation&lt;/u&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;

						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a 2 c&amp;quot;" height="24" src="/view.aspx?sf=243658_question/92966489ba41ca70dfe0f1a970497f71.gif" style="vertical-align:-7px" width="44"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a b 2&amp;quot;" height="6" src="/view.aspx?sf=243658_question/264c009f5a7c94d8a99f371500830520.gif" style="vertical-align:-6px" width="768"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;a href="http://www.maplesoft.com/support/help/errors/view.aspx?path=Error,%20missing%20operation"&gt;&lt;span style="color:#ff00ff;font-size: 100%;font-family: Courier New,monospace;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Error, missing operation&lt;/u&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;

						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;a b 2&amp;quot;" height="24" src="/view.aspx?sf=243658_question/35dd344cf82825f1bd0168aab6e53aca.gif" style="vertical-align:-7px" width="45"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Trebuchet MS,sans-serif;font-weight:normal;font-style:normal;"&gt;Multiplication operators are required&lt;/span&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="&amp;quot;2 2 c&amp;quot;" height="24" src="/view.aspx?sf=243658_question/c2255a3bd85ee4c8b091929be863c4f7.gif" style="vertical-align:-7px" width="44"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*c" height="23" src="/view.aspx?sf=243658_question/f8bf7968a5265a06d9899ea18c4d2257.gif" style="vertical-align:-6px" width="50"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*c" height="23" src="/view.aspx?sf=243658_question/facc9ebf6827b9a1dde97e46c0a9d838.gif" style="vertical-align:-6px" width="32"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*b" height="23" src="/view.aspx?sf=243658_question/b9dac3286f2c77bbc6ebbe0277d67cb8.gif" style="vertical-align:-6px" width="42"&gt;&lt;/p&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="2*a*b" height="23" src="/view.aspx?sf=243658_question/5919110ec26e960771d282c734e6d755.gif" style="vertical-align:-6px" width="33"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="NULL" height="23" src="/view.aspx?sf=243658_question/a6cede8a69d182df01d05b01d29373a1.gif" style="vertical-align:-6px" width="9"&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243658_question/Missing_operation.mw"&gt;Download Missing_operation.mw&lt;/a&gt;&lt;/p&gt;
</description>
      <guid>243658</guid>
      <pubDate>Sat, 27 Jun 2026 11:34:01 Z</pubDate>
      <itunes:author>C_R</itunes:author>
      <author>C_R</author>
    </item>
    <item>
      <title>How can I get the correct inverse metric with Physics? </title>
      <link>http://www.mapleprimes.com/questions/243656-How-Can-I-Get-The-Correct-Inverse-Metric?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;I am looking to do some gravitational perturbations around a generic background spacetime. But before doing that, I wanted to look at just linearized gravity, and make sure all the standard calculations work with Physics before throwing something a little more complicated at it. I went to&amp;nbsp;&lt;strong&gt;?Physics,Library&amp;nbsp;&lt;/strong&gt;and found the&amp;nbsp;&lt;strong&gt;Linearize&amp;nbsp;&lt;/strong&gt;command, and I thought this was great! When I was reading through it however, I found that the sign infront of the perturbation&amp;nbsp;&lt;strong&gt;h&amp;nbsp;&lt;/strong&gt;in the inverse metric is incorrect. Now, this does not give any invalid results for the Ricci tensor as displayed in the worksheet, since we are only going to linear order, but if we want to go beyond linear order, this will start to cause issues.&amp;nbsp;&lt;/p&gt;

&lt;p&gt;Is there a way that Maple can handle this? Or do I have to do some sort of double Define for the metric: one with all downstairs indices, and one with all upstairs indices? If so, how do I do that?&amp;nbsp;&lt;/p&gt;

&lt;p&gt;Any help would be greatly appreciated!&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="40A88B63DE8575A8676A0BEEF4C00FA4"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;restart: with(Physics): with(Library):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Setup(coordinates = cartesian,signature=`-+++`):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Systems of spacetime coordinates are:`*{X = (t, x, y, z)}" height="23" src="/view.aspx?sf=243656_question/cf3840e0f3110ef5df01087864446e5a.gif" style="vertical-align:-6px" width="338"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/b82c5bde0eb33ae685678d3412cae215.gif" style="vertical-align:-6px" width="418"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="g_[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/58ead79797e3f31eada232d93f5c86fc.gif" style="vertical-align:-46px" width="134"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(h[mu, nu],symmetric)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/3174a0aedcda8a01353b389ecc58997c.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/574d6e020c66ba75388113dc2700ebaa.gif" style="vertical-align:-16px" width="217"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(eta[mu,nu]=rhs(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(2)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/bc7fac4cd9237da2724f9b528ebec53f.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/ad79cbcfc2ff9048883fd373c653ea7c.gif" style="vertical-align:-16px" width="251"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu,nu]=eta[mu,nu]+epsilon*h[mu,nu]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = epsilon*h[mu, nu]+eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/94dc6b57129defda2649c0f7807cc31e.gif" style="vertical-align:-15px" width="128"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(5)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Lets &amp;quot;define&amp;quot; the inverse metric as it appears from the Library:-Linearize worksheet. &lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[~mu,~alpha]=eta[~mu,~alpha]+epsilon*h[~mu,~alpha]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[`~alpha`, `~mu`] = epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/f739cb3156c9904f96b0879c3f696310.gif" style="vertical-align:-15px" width="140"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(6)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;If we multiply the metric and its inverse together, we should expact that we return the KroneckerDelta by definition -- if we consider only to linear order. &amp;nbsp;&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;*&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(6)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = (epsilon*h[mu, nu]+eta[mu, nu])*(epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`])" height="41" src="/view.aspx?sf=243656_question/6d05a3710022cc2f135536e53a81f2e2.gif" style="vertical-align:-15px" width="290"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(7)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;expand(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon^2*h[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[`~alpha`, `~mu`]*h[mu, nu]+eta[mu, nu]*eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/a1b152cc9133da69ef28ec149d93bcad.gif" style="vertical-align:-15px" width="408"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(8)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(8)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon^2*h[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[`~alpha`, `~mu`]*h[mu, nu]+Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`]" height="38" src="/view.aspx?sf=243656_question/d248903f93d99140aef7e546c4bf62d7.gif" style="vertical-align:-14px" width="400"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(9)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(epsilon^2=0,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(9)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon*Physics:-g_[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[`~alpha`, `~mu`]*h[mu, nu]+Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`]" height="38" src="/view.aspx?sf=243656_question/880fddc8cda45ed4215882990ec8c4a1.gif" style="vertical-align:-14px" width="308"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(10)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(%)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = 2*epsilon*h[nu, `~alpha`]+Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/ce33a06c18f536837542eb8eb1b5951e.gif" style="vertical-align:-15px" width="121"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(11)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;As we can see, we do not get delta alone on the right-hand-side, but instead we still have the perturbation still. &lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;If we instead, use the proper way the inverse should look, which of course comes from the definition of the inverse, it should have minus sign. &lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[~mu,~alpha]=eta[~mu,~alpha]-epsilon*h[~mu,~alpha]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[`~alpha`, `~mu`] = -epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/8956740c2d53528e6a0e4346fc19456b.gif" style="vertical-align:-15px" width="150"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(12)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(epsilon^2=0,expand(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;*&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(12)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = -epsilon*eta[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[`~alpha`, `~mu`]*h[mu, nu]+eta[mu, nu]*eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/0636387c09a95099df08020fba15bd84.gif" style="vertical-align:-15px" width="326"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(13)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(Substitute(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(13)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/9d9e0a6952574aba270523a25acf92eb.gif" style="vertical-align:-15px" width="62"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(14)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Which is the desired result we want. So, my question: is there a way that Maple can produce the correct inverse metric not only to linear order, but to say quadratic, without explicitly deriving it ourselves? &lt;/span&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;Here is the Physics:-Library(Linearize) Worksheet/Example with some comments&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="5FBF0E7F102D25877EA91E150DC7DE6C"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;restart: with(Physics): with(Library):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Setup(coordinates = cartesian);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Systems of spacetime coordinates are:`*{X = (x, y, z, t)}" height="23" src="/view.aspx?sf=243656_question/878b704bd74801b33ae2bb714eba23b4.gif" style="vertical-align:-6px" width="338"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/018d3887927dd3d5291e51b672d2de60.gif" style="vertical-align:-6px" width="418"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="[coordinatesystems = {X}]" height="23" src="/view.aspx?sf=243656_question/7283a8af3423657b797fa502f1c22e1c.gif" style="vertical-align:-6px" width="172"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The default metric when Physics is loaded is the Minkowski metric, representing a flat (no curvature) spacetime&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="g_[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/5ef096b89530302530c2ae5904589b7d.gif" style="vertical-align:-46px" width="162"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Suppose you want to define a small perturbation around this metric. For that purpose, define a perturbation tensor &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/88d43a56eaede21235c2e4725f734215.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;, that in the general case depends on the coordinates and is not diagonal, the only requirement is that it is symmetric (to have it diagonal, change symmetric by diagonal; to have it constant, change &lt;/span&gt;&lt;img alt="delta[i, j](X)" height="31" src="/view.aspx?sf=243656_question/faf9236d814da6be25c86a31f86455a4.gif" style="vertical-align:-12px" width="50"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;by &lt;/span&gt;&lt;img alt="delta[i, j]" height="31" src="/view.aspx?sf=243656_question/db07e4d487f0635ea03bc39af520fdc4.gif" style="vertical-align:-12px" width="25"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;h[mu, nu] = Matrix(4, (i, j) -&amp;gt; delta[i, j](X), shape = symmetric);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="h[mu, nu] = (Matrix(4, 4, {(1, 1) = delta[1, 1](x, y, z, t), (1, 2) = delta[1, 2](x, y, z, t), (1, 3) = delta[1, 3](x, y, z, t), (1, 4) = delta[1, 4](x, y, z, t), (2, 1) = delta[1, 2](x, y, z, t), (2, 2) = delta[2, 2](x, y, z, t), (2, 3) = delta[2, 3](x, y, z, t), (2, 4) = delta[2, 4](x, y, z, t), (3, 1) = delta[1, 3](x, y, z, t), (3, 2) = delta[2, 3](x, y, z, t), (3, 3) = delta[3, 3](x, y, z, t), (3, 4) = delta[3, 4](x, y, z, t), (4, 1) = delta[1, 4](x, y, z, t), (4, 2) = delta[2, 4](x, y, z, t), (4, 3) = delta[3, 4](x, y, z, t), (4, 4) = delta[4, 4](x, y, z, t)}))" height="135" src="/view.aspx?sf=243656_question/d3a384982e56dc179292fa75137073bf.gif" style="vertical-align:-62px" width="292"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;In the above it is understood that &lt;/span&gt;&lt;img alt="delta[i, j]" height="31" src="/view.aspx?sf=243656_question/8f1f919532ec4e69bbc21cb059453f55.gif" style="vertical-align:-12px" width="25"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;are small quantities, so that quadratic or higher powers of it can be approximated to 0 (i.e., discarded). Define the components of &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/e5830ba45cafcf7c8f060fc572475409.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;accordingly&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(3)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/109619a20335d25c9dcd231bca961162.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/713ccba30a16b2408b45fff4b0a4cd7e.gif" style="vertical-align:-16px" width="217"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Define also a tensor &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/411d04182a0db120bece5de6552912f4.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;representing the unperturbed Minkowski metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;eta[mu, nu] = rhs(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(2)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="eta[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/72b8f3f6b0cbb13a2fe4058310aa4f4d.gif" style="vertical-align:-46px" width="164"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(5)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/48ba24bdc6e6805159b7bcbe9970eee9.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/f14dcad620490a352a44d4d5c0d93866.gif" style="vertical-align:-16px" width="251"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(6)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The weakly perturbed metric is given by&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu, nu] = eta[mu, nu] + h[mu, nu];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = eta[mu, nu]+h[mu, nu]" height="34" src="/view.aspx?sf=243656_question/9c69391c3b31468de474dd218babec60.gif" style="vertical-align:-15px" width="119"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(7)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Make this be the definition of the metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/4b0a7119fbf07258703361d0a1818841.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Coordinates: `[x, y, z, t]*`. Signature: `(`- - - +`)" height="23" src="/view.aspx?sf=243656_question/81c7d230652109120279afef36712980.gif" style="vertical-align:-6px" width="271"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/9c8682d4cb3c1a19e6f3bb24c78f52c1.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = Matrix(%id = 36893488152142178892)" height="135" src="/view.aspx?sf=243656_question/a752436901d582200090a689f254b260.gif" style="vertical-align:-62px" width="430"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/a37a7ef8626bc9dc6340bbbba1db6beb.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Setting `*lowercaselatin_is*` letters to represent `*space*` indices`" height="23" src="/view.aspx?sf=243656_question/282f14cc4c6cff23eb601c7b98d9ba62.gif" style="vertical-align:-6px" width="356"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/d917bc089756c2ecbac6d5fa5b6547e9.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], h[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/8d5812b423fc420b5d7d418da5aa0158.gif" style="vertical-align:-16px" width="538"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(8)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The linearized form of the Ricci tensor is computed by introducing this weakly perturbed metric in the expression of the &lt;/span&gt;&lt;!-- HelpHyperlink topic=Ricci --&gt; &lt;span style="color:#008080;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Ricci&lt;/u&gt;&lt;/span&gt; &lt;!-- /HelpHyperlink --&gt; &lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;tensor as a function of the metric. This can be accomplished in different ways, the simpler being to use the conversion network between tensors, but for illustration purposes, showing steps one at time, a substitution of definitions one into the other one is used&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Ricci[definition];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-Ricci[mu, nu] = Physics:-d_[alpha](Physics:-Christoffel[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~alpha`, mu, alpha], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, alpha]-Physics:-Christoffel[`~beta`, mu, alpha]*Physics:-Christoffel[`~alpha`, nu, beta]" height="42" src="/view.aspx?sf=243656_question/eb3df9d8664a0ba8828f4a5c95ba113c.gif" style="vertical-align:-16px" width="403"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(9)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Christoffel[~alpha, mu, nu, definition];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-Christoffel[`~alpha`, mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~beta`]*(Physics:-d_[nu](Physics:-g_[beta, mu], [X])+Physics:-d_[mu](Physics:-g_[beta, nu], [X])-Physics:-d_[beta](Physics:-g_[mu, nu], [X]))" height="59" src="/view.aspx?sf=243656_question/39d599bc82ae28128c982c2c47a0bc4a.gif" style="vertical-align:-16px" width="319"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(10)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(10)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(9)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="Physics:-Ricci[mu, nu] = Physics:-d_[alpha]((1/2)*Physics:-g_[`~alpha`, `~kappa`]*(Physics:-d_[nu](Physics:-g_[kappa, mu], [X])+Physics:-d_[mu](Physics:-g_[kappa, nu], [X])-Physics:-d_[kappa](Physics:-g_[mu, nu], [X])), [X])-Physics:-d_[nu]((1/2)*Physics:-g_[`~alpha`, `~tau`]*(Physics:-d_[mu](Physics:-g_[tau, alpha], [X])+Physics:-d_[alpha](Physics:-g_[tau, mu], [X])-Physics:-d_[tau](Physics:-g_[alpha, mu], [X])), [X])+(1/4)*Physics:-g_[`~beta`, `~iota`]*(Physics:-d_[nu](Physics:-g_[iota, mu], [X])+Physics:-d_[mu](Physics:-g_[iota, nu], [X])-Physics:-d_[iota](Physics:-g_[mu, nu], [X]))*Physics:-g_[`~alpha`, `~lambda`]*(Physics:-d_[beta](Physics:-g_[lambda, alpha], [X])+Physics:-d_[alpha](Physics:-g_[lambda, beta], [X])-Physics:-d_[lambda](Physics:-g_[alpha, beta], [X]))-(1/4)*Physics:-g_[`~beta`, `~omega`]*(Physics:-d_[mu](Physics:-g_[omega, alpha], [X])+Physics:-d_[alpha](Physics:-g_[omega, mu], [X])-Physics:-d_[omega](Physics:-g_[alpha, mu], [X]))*Physics:-g_[`~alpha`, `~chi`]*(Physics:-d_[nu](Physics:-g_[chi, beta], [X])+Physics:-d_[beta](Physics:-g_[chi, nu], [X])-Physics:-d_[chi](Physics:-g_[beta, nu], [X]))" height="165" src="/view.aspx?sf=243656_question/f65889bbff45c4cccce7308270597a12.gif" style="vertical-align:-122px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(11)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Introducing the perturbed metric, and the inert form of Ricci for simplification purposes&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, Ricci = %Ricci, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(11)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*Physics:-d_[alpha](eta[`~alpha`, `~kappa`]+h[`~alpha`, `~kappa`], [X])*(Physics:-d_[nu](eta[kappa, mu]+h[kappa, mu], [X])+Physics:-d_[mu](eta[kappa, nu]+h[kappa, nu], [X])-Physics:-d_[kappa](eta[mu, nu]+h[mu, nu], [X]))+(1/2)*(eta[`~alpha`, `~kappa`]+h[`~alpha`, `~kappa`])*(Physics:-d_[alpha](Physics:-d_[nu](eta[kappa, mu]+h[kappa, mu], [X]), [X])+Physics:-d_[alpha](Physics:-d_[mu](eta[kappa, nu]+h[kappa, nu], [X]), [X])-Physics:-d_[alpha](Physics:-d_[kappa](eta[mu, nu]+h[mu, nu], [X]), [X]))-(1/2)*Physics:-d_[nu](eta[`~alpha`, `~tau`]+h[`~alpha`, `~tau`], [X])*(Physics:-d_[mu](eta[alpha, tau]+h[alpha, tau], [X])+Physics:-d_[alpha](eta[mu, tau]+h[mu, tau], [X])-Physics:-d_[tau](eta[alpha, mu]+h[alpha, mu], [X]))-(1/2)*(eta[`~alpha`, `~tau`]+h[`~alpha`, `~tau`])*(Physics:-d_[mu](Physics:-d_[nu](eta[alpha, tau]+h[alpha, tau], [X]), [X])+Physics:-d_[alpha](Physics:-d_[nu](eta[mu, tau]+h[mu, tau], [X]), [X])-Physics:-d_[nu](Physics:-d_[tau](eta[alpha, mu]+h[alpha, mu], [X]), [X]))+(1/4)*(eta[`~beta`, `~iota`]+h[`~beta`, `~iota`])*(Physics:-d_[nu](eta[iota, mu]+h[iota, mu], [X])+Physics:-d_[mu](eta[iota, nu]+h[iota, nu], [X])-Physics:-d_[iota](eta[mu, nu]+h[mu, nu], [X]))*(eta[`~alpha`, `~lambda`]+h[`~alpha`, `~lambda`])*(Physics:-d_[beta](eta[alpha, lambda]+h[alpha, lambda], [X])+Physics:-d_[alpha](eta[beta, lambda]+h[beta, lambda], [X])-Physics:-d_[lambda](eta[alpha, beta]+h[alpha, beta], [X]))-(1/4)*(eta[`~beta`, `~omega`]+h[`~beta`, `~omega`])*(Physics:-d_[mu](eta[alpha, omega]+h[alpha, omega], [X])+Physics:-d_[alpha](eta[mu, omega]+h[mu, omega], [X])-Physics:-d_[omega](eta[alpha, mu]+h[alpha, mu], [X]))*(eta[`~alpha`, `~chi`]+h[`~alpha`, `~chi`])*(Physics:-d_[nu](eta[beta, chi]+h[beta, chi], [X])+Physics:-d_[beta](eta[chi, nu]+h[chi, nu], [X])-Physics:-d_[chi](eta[beta, nu]+h[beta, nu], [X]))" height="334" src="/view.aspx?sf=243656_question/ee96858b01f691d0375584bdf698017e.gif" style="vertical-align:-289px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(12)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;The sign infront of the perturbation in the inverse metric is wrong, it should be minus. &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;This expression contains several terms quadratic in the small perturbation &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/a983bc035c44215bd1a5c39435625910.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;. The routine to filter out those terms is &lt;/span&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:italic;"&gt;Linearize&lt;/span&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;, that takes as second argument the symbol representing the small quantities (perturbation)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;Lets look at the metric times inverse in this setup&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu,nu,definition]*g_[~mu,~alpha,definition]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~mu`, `~alpha`] = (eta[mu, nu]+h[mu, nu])*(eta[`~mu`, `~alpha`]+h[`~mu`, `~alpha`])" height="41" src="/view.aspx?sf=243656_question/fddc2df8d0afa5e197ee55322fbb4afb.gif" style="vertical-align:-15px" width="272"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(13)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Linearize(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(13)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;,h)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = eta[mu, nu]*eta[`~alpha`, `~mu`]+eta[mu, nu]*h[`~alpha`, `~mu`]+eta[`~alpha`, `~mu`]*h[mu, nu]" height="41" src="/view.aspx?sf=243656_question/39f753644af5069020e2d3989f4dc70b.gif" style="vertical-align:-15px" width="298"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(14)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(subs(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(14)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]+2*h[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/c9dff1e3442c26c3e7e53bb7ddef6d0d.gif" style="vertical-align:-15px" width="112"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(15)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;The result is not correct, left-hand-side does not match right-hand-side, this is because the inverse metric has the wrong. If it were a minus, we would get:&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu, nu]*g_[~alpha, ~mu] = eta[mu, nu]*eta[~alpha, ~mu] - eta[mu, nu]*h[~alpha, ~mu] + eta[~alpha, ~mu]*h[mu, nu]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = eta[mu, nu]*eta[`~alpha`, `~mu`]-eta[mu, nu]*h[`~alpha`, `~mu`]+eta[`~alpha`, `~mu`]*h[mu, nu]" height="41" src="/view.aspx?sf=243656_question/3780cec4644972a2ab9aaf5dfdd6cffc.gif" style="vertical-align:-15px" width="298"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(16)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(subs(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(16)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/36728c4f2f9fab825be9f4d947a209a2.gif" style="vertical-align:-15px" width="62"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(17)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;Which is correct. The continued calculation from the Help page is below. &lt;/span&gt;&lt;/p&gt;
			&amp;nbsp;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Linearize(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(12)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, h);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[nu](Physics:-d_[tau](h[alpha, mu], [X]), [X])-(1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[mu](Physics:-d_[nu](h[alpha, tau], [X]), [X])-(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[kappa](h[mu, nu], [X]), [X])+(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[nu](h[kappa, mu], [X]), [X])+(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[mu](h[kappa, nu], [X]), [X])-(1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[alpha](Physics:-d_[nu](h[mu, tau], [X]), [X])" height="116" src="/view.aspx?sf=243656_question/161b36d55a53a47f0d0114712b250360.gif" style="vertical-align:-71px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(18)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;
			&amp;nbsp;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;In this result, &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/d343e2294efb91788cb58aa443ffe126.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;is the flat Minkowski metric. To further simplify this expression using the internal algorithms for a flat metric it is practical to reintroduce &lt;/span&gt;&lt;img alt="g[mu, nu]" height="31" src="/view.aspx?sf=243656_question/4620c519c121dd9a6f413f1991adb15a.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;representing that Minkowski metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[min];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/c7f34f870c0debd7898318d0a9871c18.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`The Minkowski metric in coordinates `*[x, y, z, t]" height="23" src="/view.aspx?sf=243656_question/15bee490de5359843d002ba72da94e9e.gif" style="vertical-align:-6px" width="294"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Signature: `(`- - - +`)" height="23" src="/view.aspx?sf=243656_question/faacf14c02220772a09fb186eb7f3dfe.gif" style="vertical-align:-6px" width="127"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/d630f95755e537c67cfd5567963fcc23.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = Matrix(%id = 36893488152069364060)" height="103" src="/view.aspx?sf=243656_question/d808bef8d776fe521927d2a268bcb52b.gif" style="vertical-align:-46px" width="162"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(19)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Replace in the expression for the Ricci tensor the intermediate Minkowski &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/bc5114140920505409534b9e719710b6.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;by &lt;/span&gt;&lt;img alt="g[mu, nu]" height="31" src="/view.aspx?sf=243656_question/2dafbc0ed5f2199727add3c84366ccd2.gif" style="vertical-align:-14px" width="31"&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(eta = g_, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(18)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[nu](Physics:-d_[tau](h[alpha, mu], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[mu](Physics:-d_[nu](h[alpha, tau], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[kappa](h[mu, nu], [X]), [X])+(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[nu](h[kappa, mu], [X]), [X])+(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[mu](h[kappa, nu], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[alpha](Physics:-d_[nu](h[mu, tau], [X]), [X])" height="112" src="/view.aspx?sf=243656_question/247ace3da0af4e444b7180f89774c547.gif" style="vertical-align:-69px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(20)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Simplifying, results in the linearized form of the Ricci tensor&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(20)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="%Ricci[mu, nu] = -(1/2)*Physics:-d_[mu](Physics:-d_[nu](h[tau, `~tau`], [X]), [X])-(1/2)*Physics:-dAlembertian(h[mu, nu], [X])+(1/2)*Physics:-d_[nu](Physics:-d_[tau](h[mu, `~tau`], [X]), [X])+(1/2)*Physics:-d_[mu](Physics:-d_[tau](h[nu, `~tau`], [X]), [X])" height="59" src="/view.aspx?sf=243656_question/682cfa427f7b91c481419818d27e7bff.gif" style="vertical-align:-16px" width="457"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(21)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;This is correct result, because we are going to linear order only the +/- does not have an effect on the end result. &lt;/span&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243656_question/LinearizedWorksheet-Comments.mw"&gt;Download LinearizedWorksheet-Comments.mw&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243656_question/LinearQuestion.mw"&gt;Download LinearQuestion.mw&lt;/a&gt;&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;I am looking to do some gravitational perturbations around a generic background spacetime. But before doing that, I wanted to look at just linearized gravity, and make sure all the standard calculations work with Physics before throwing something a little more complicated at it. I went to&amp;nbsp;&lt;strong&gt;?Physics,Library&amp;nbsp;&lt;/strong&gt;and found the&amp;nbsp;&lt;strong&gt;Linearize&amp;nbsp;&lt;/strong&gt;command, and I thought this was great! When I was reading through it however, I found that the sign infront of the perturbation&amp;nbsp;&lt;strong&gt;h&amp;nbsp;&lt;/strong&gt;in the inverse metric is incorrect. Now, this does not give any invalid results for the Ricci tensor as displayed in the worksheet, since we are only going to linear order, but if we want to go beyond linear order, this will start to cause issues.&amp;nbsp;&lt;/p&gt;

&lt;p&gt;Is there a way that Maple can handle this? Or do I have to do some sort of double Define for the metric: one with all downstairs indices, and one with all upstairs indices? If so, how do I do that?&amp;nbsp;&lt;/p&gt;

&lt;p&gt;Any help would be greatly appreciated!&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="40A88B63DE8575A8676A0BEEF4C00FA4"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;restart: with(Physics): with(Library):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Setup(coordinates = cartesian,signature=`-+++`):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Systems of spacetime coordinates are:`*{X = (t, x, y, z)}" height="23" src="/view.aspx?sf=243656_question/cf3840e0f3110ef5df01087864446e5a.gif" style="vertical-align:-6px" width="338"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/b82c5bde0eb33ae685678d3412cae215.gif" style="vertical-align:-6px" width="418"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="g_[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/58ead79797e3f31eada232d93f5c86fc.gif" style="vertical-align:-46px" width="134"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(h[mu, nu],symmetric)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/3174a0aedcda8a01353b389ecc58997c.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/574d6e020c66ba75388113dc2700ebaa.gif" style="vertical-align:-16px" width="217"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(eta[mu,nu]=rhs(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(2)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/bc7fac4cd9237da2724f9b528ebec53f.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/ad79cbcfc2ff9048883fd373c653ea7c.gif" style="vertical-align:-16px" width="251"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu,nu]=eta[mu,nu]+epsilon*h[mu,nu]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = epsilon*h[mu, nu]+eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/94dc6b57129defda2649c0f7807cc31e.gif" style="vertical-align:-15px" width="128"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(5)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Lets &amp;quot;define&amp;quot; the inverse metric as it appears from the Library:-Linearize worksheet. &lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[~mu,~alpha]=eta[~mu,~alpha]+epsilon*h[~mu,~alpha]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[`~alpha`, `~mu`] = epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/f739cb3156c9904f96b0879c3f696310.gif" style="vertical-align:-15px" width="140"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(6)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;If we multiply the metric and its inverse together, we should expact that we return the KroneckerDelta by definition -- if we consider only to linear order. &amp;nbsp;&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;*&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(6)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = (epsilon*h[mu, nu]+eta[mu, nu])*(epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`])" height="41" src="/view.aspx?sf=243656_question/6d05a3710022cc2f135536e53a81f2e2.gif" style="vertical-align:-15px" width="290"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(7)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;expand(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon^2*h[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[`~alpha`, `~mu`]*h[mu, nu]+eta[mu, nu]*eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/a1b152cc9133da69ef28ec149d93bcad.gif" style="vertical-align:-15px" width="408"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(8)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(8)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon^2*h[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[`~alpha`, `~mu`]*h[mu, nu]+Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`]" height="38" src="/view.aspx?sf=243656_question/d248903f93d99140aef7e546c4bf62d7.gif" style="vertical-align:-14px" width="400"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(9)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(epsilon^2=0,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(9)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = epsilon*Physics:-g_[mu, nu]*h[`~mu`, `~alpha`]+epsilon*Physics:-g_[`~alpha`, `~mu`]*h[mu, nu]+Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`]" height="38" src="/view.aspx?sf=243656_question/880fddc8cda45ed4215882990ec8c4a1.gif" style="vertical-align:-14px" width="308"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(10)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(%)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = 2*epsilon*h[nu, `~alpha`]+Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/ce33a06c18f536837542eb8eb1b5951e.gif" style="vertical-align:-15px" width="121"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(11)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;As we can see, we do not get delta alone on the right-hand-side, but instead we still have the perturbation still. &lt;/span&gt;&lt;/p&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;If we instead, use the proper way the inverse should look, which of course comes from the definition of the inverse, it should have minus sign. &lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[~mu,~alpha]=eta[~mu,~alpha]-epsilon*h[~mu,~alpha]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[`~alpha`, `~mu`] = -epsilon*h[`~mu`, `~alpha`]+eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/8956740c2d53528e6a0e4346fc19456b.gif" style="vertical-align:-15px" width="150"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(12)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(epsilon^2=0,expand(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;*&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(12)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = -epsilon*eta[mu, nu]*h[`~mu`, `~alpha`]+epsilon*eta[`~alpha`, `~mu`]*h[mu, nu]+eta[mu, nu]*eta[`~alpha`, `~mu`]" height="41" src="/view.aspx?sf=243656_question/0636387c09a95099df08020fba15bd84.gif" style="vertical-align:-15px" width="326"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(13)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(Substitute(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(13)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/9d9e0a6952574aba270523a25acf92eb.gif" style="vertical-align:-15px" width="62"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(14)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Which is the desired result we want. So, my question: is there a way that Maple can produce the correct inverse metric not only to linear order, but to say quadratic, without explicitly deriving it ourselves? &lt;/span&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;Here is the Physics:-Library(Linearize) Worksheet/Example with some comments&lt;/p&gt;

&lt;form name="worksheet_form"&gt;&lt;input name="md.ref" type="hidden" value="5FBF0E7F102D25877EA91E150DC7DE6C"&gt;
&lt;table align="center" width="768"&gt;
	&lt;tbody&gt;
		&lt;tr&gt;
			&lt;td&gt;
			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;restart: with(Physics): with(Library):&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Setup(coordinates = cartesian);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Systems of spacetime coordinates are:`*{X = (x, y, z, t)}" height="23" src="/view.aspx?sf=243656_question/878b704bd74801b33ae2bb714eba23b4.gif" style="vertical-align:-6px" width="338"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/018d3887927dd3d5291e51b672d2de60.gif" style="vertical-align:-6px" width="418"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="[coordinatesystems = {X}]" height="23" src="/view.aspx?sf=243656_question/7283a8af3423657b797fa502f1c22e1c.gif" style="vertical-align:-6px" width="172"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(1)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The default metric when Physics is loaded is the Minkowski metric, representing a flat (no curvature) spacetime&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="g_[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/5ef096b89530302530c2ae5904589b7d.gif" style="vertical-align:-46px" width="162"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(2)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Suppose you want to define a small perturbation around this metric. For that purpose, define a perturbation tensor &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/88d43a56eaede21235c2e4725f734215.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;, that in the general case depends on the coordinates and is not diagonal, the only requirement is that it is symmetric (to have it diagonal, change symmetric by diagonal; to have it constant, change &lt;/span&gt;&lt;img alt="delta[i, j](X)" height="31" src="/view.aspx?sf=243656_question/faf9236d814da6be25c86a31f86455a4.gif" style="vertical-align:-12px" width="50"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;by &lt;/span&gt;&lt;img alt="delta[i, j]" height="31" src="/view.aspx?sf=243656_question/db07e4d487f0635ea03bc39af520fdc4.gif" style="vertical-align:-12px" width="25"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;h[mu, nu] = Matrix(4, (i, j) -&amp;gt; delta[i, j](X), shape = symmetric);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="h[mu, nu] = (Matrix(4, 4, {(1, 1) = delta[1, 1](x, y, z, t), (1, 2) = delta[1, 2](x, y, z, t), (1, 3) = delta[1, 3](x, y, z, t), (1, 4) = delta[1, 4](x, y, z, t), (2, 1) = delta[1, 2](x, y, z, t), (2, 2) = delta[2, 2](x, y, z, t), (2, 3) = delta[2, 3](x, y, z, t), (2, 4) = delta[2, 4](x, y, z, t), (3, 1) = delta[1, 3](x, y, z, t), (3, 2) = delta[2, 3](x, y, z, t), (3, 3) = delta[3, 3](x, y, z, t), (3, 4) = delta[3, 4](x, y, z, t), (4, 1) = delta[1, 4](x, y, z, t), (4, 2) = delta[2, 4](x, y, z, t), (4, 3) = delta[3, 4](x, y, z, t), (4, 4) = delta[4, 4](x, y, z, t)}))" height="135" src="/view.aspx?sf=243656_question/d3a384982e56dc179292fa75137073bf.gif" style="vertical-align:-62px" width="292"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(3)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;In the above it is understood that &lt;/span&gt;&lt;img alt="delta[i, j]" height="31" src="/view.aspx?sf=243656_question/8f1f919532ec4e69bbc21cb059453f55.gif" style="vertical-align:-12px" width="25"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;are small quantities, so that quadratic or higher powers of it can be approximated to 0 (i.e., discarded). Define the components of &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/e5830ba45cafcf7c8f060fc572475409.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;accordingly&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(3)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/109619a20335d25c9dcd231bca961162.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/713ccba30a16b2408b45fff4b0a4cd7e.gif" style="vertical-align:-16px" width="217"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(4)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Define also a tensor &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/411d04182a0db120bece5de6552912f4.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;representing the unperturbed Minkowski metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;eta[mu, nu] = rhs(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(2)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="eta[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))" height="103" src="/view.aspx?sf=243656_question/72b8f3f6b0cbb13a2fe4058310aa4f4d.gif" style="vertical-align:-46px" width="164"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(5)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(5)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/48ba24bdc6e6805159b7bcbe9970eee9.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/f14dcad620490a352a44d4d5c0d93866.gif" style="vertical-align:-16px" width="251"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(6)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The weakly perturbed metric is given by&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu, nu] = eta[mu, nu] + h[mu, nu];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = eta[mu, nu]+h[mu, nu]" height="34" src="/view.aspx?sf=243656_question/9c69391c3b31468de474dd218babec60.gif" style="vertical-align:-15px" width="119"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(7)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Make this be the definition of the metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Define(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/4b0a7119fbf07258703361d0a1818841.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Coordinates: `[x, y, z, t]*`. Signature: `(`- - - +`)" height="23" src="/view.aspx?sf=243656_question/81c7d230652109120279afef36712980.gif" style="vertical-align:-6px" width="271"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/9c8682d4cb3c1a19e6f3bb24c78f52c1.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = Matrix(%id = 36893488152142178892)" height="135" src="/view.aspx?sf=243656_question/a752436901d582200090a689f254b260.gif" style="vertical-align:-62px" width="430"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/a37a7ef8626bc9dc6340bbbba1db6beb.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Setting `*lowercaselatin_is*` letters to represent `*space*` indices`" height="23" src="/view.aspx?sf=243656_question/282f14cc4c6cff23eb601c7b98d9ba62.gif" style="vertical-align:-6px" width="356"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Defined objects with tensor properties`" height="23" src="/view.aspx?sf=243656_question/d917bc089756c2ecbac6d5fa5b6547e9.gif" style="vertical-align:-6px" width="235"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="{Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], h[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}" height="35" src="/view.aspx?sf=243656_question/8d5812b423fc420b5d7d418da5aa0158.gif" style="vertical-align:-16px" width="538"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(8)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;The linearized form of the Ricci tensor is computed by introducing this weakly perturbed metric in the expression of the &lt;/span&gt;&lt;!-- HelpHyperlink topic=Ricci --&gt; &lt;span style="color:#008080;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&lt;u&gt;Ricci&lt;/u&gt;&lt;/span&gt; &lt;!-- /HelpHyperlink --&gt; &lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;tensor as a function of the metric. This can be accomplished in different ways, the simpler being to use the conversion network between tensors, but for illustration purposes, showing steps one at time, a substitution of definitions one into the other one is used&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Ricci[definition];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-Ricci[mu, nu] = Physics:-d_[alpha](Physics:-Christoffel[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~alpha`, mu, alpha], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, alpha]-Physics:-Christoffel[`~beta`, mu, alpha]*Physics:-Christoffel[`~alpha`, nu, beta]" height="42" src="/view.aspx?sf=243656_question/eb3df9d8664a0ba8828f4a5c95ba113c.gif" style="vertical-align:-16px" width="403"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(9)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Christoffel[~alpha, mu, nu, definition];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-Christoffel[`~alpha`, mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~beta`]*(Physics:-d_[nu](Physics:-g_[beta, mu], [X])+Physics:-d_[mu](Physics:-g_[beta, nu], [X])-Physics:-d_[beta](Physics:-g_[mu, nu], [X]))" height="59" src="/view.aspx?sf=243656_question/39d599bc82ae28128c982c2c47a0bc4a.gif" style="vertical-align:-16px" width="319"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(10)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(10)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(9)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="Physics:-Ricci[mu, nu] = Physics:-d_[alpha]((1/2)*Physics:-g_[`~alpha`, `~kappa`]*(Physics:-d_[nu](Physics:-g_[kappa, mu], [X])+Physics:-d_[mu](Physics:-g_[kappa, nu], [X])-Physics:-d_[kappa](Physics:-g_[mu, nu], [X])), [X])-Physics:-d_[nu]((1/2)*Physics:-g_[`~alpha`, `~tau`]*(Physics:-d_[mu](Physics:-g_[tau, alpha], [X])+Physics:-d_[alpha](Physics:-g_[tau, mu], [X])-Physics:-d_[tau](Physics:-g_[alpha, mu], [X])), [X])+(1/4)*Physics:-g_[`~beta`, `~iota`]*(Physics:-d_[nu](Physics:-g_[iota, mu], [X])+Physics:-d_[mu](Physics:-g_[iota, nu], [X])-Physics:-d_[iota](Physics:-g_[mu, nu], [X]))*Physics:-g_[`~alpha`, `~lambda`]*(Physics:-d_[beta](Physics:-g_[lambda, alpha], [X])+Physics:-d_[alpha](Physics:-g_[lambda, beta], [X])-Physics:-d_[lambda](Physics:-g_[alpha, beta], [X]))-(1/4)*Physics:-g_[`~beta`, `~omega`]*(Physics:-d_[mu](Physics:-g_[omega, alpha], [X])+Physics:-d_[alpha](Physics:-g_[omega, mu], [X])-Physics:-d_[omega](Physics:-g_[alpha, mu], [X]))*Physics:-g_[`~alpha`, `~chi`]*(Physics:-d_[nu](Physics:-g_[chi, beta], [X])+Physics:-d_[beta](Physics:-g_[chi, nu], [X])-Physics:-d_[chi](Physics:-g_[beta, nu], [X]))" height="165" src="/view.aspx?sf=243656_question/f65889bbff45c4cccce7308270597a12.gif" style="vertical-align:-122px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(11)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Introducing the perturbed metric, and the inert form of Ricci for simplification purposes&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Substitute(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(7)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, Ricci = %Ricci, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(11)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*Physics:-d_[alpha](eta[`~alpha`, `~kappa`]+h[`~alpha`, `~kappa`], [X])*(Physics:-d_[nu](eta[kappa, mu]+h[kappa, mu], [X])+Physics:-d_[mu](eta[kappa, nu]+h[kappa, nu], [X])-Physics:-d_[kappa](eta[mu, nu]+h[mu, nu], [X]))+(1/2)*(eta[`~alpha`, `~kappa`]+h[`~alpha`, `~kappa`])*(Physics:-d_[alpha](Physics:-d_[nu](eta[kappa, mu]+h[kappa, mu], [X]), [X])+Physics:-d_[alpha](Physics:-d_[mu](eta[kappa, nu]+h[kappa, nu], [X]), [X])-Physics:-d_[alpha](Physics:-d_[kappa](eta[mu, nu]+h[mu, nu], [X]), [X]))-(1/2)*Physics:-d_[nu](eta[`~alpha`, `~tau`]+h[`~alpha`, `~tau`], [X])*(Physics:-d_[mu](eta[alpha, tau]+h[alpha, tau], [X])+Physics:-d_[alpha](eta[mu, tau]+h[mu, tau], [X])-Physics:-d_[tau](eta[alpha, mu]+h[alpha, mu], [X]))-(1/2)*(eta[`~alpha`, `~tau`]+h[`~alpha`, `~tau`])*(Physics:-d_[mu](Physics:-d_[nu](eta[alpha, tau]+h[alpha, tau], [X]), [X])+Physics:-d_[alpha](Physics:-d_[nu](eta[mu, tau]+h[mu, tau], [X]), [X])-Physics:-d_[nu](Physics:-d_[tau](eta[alpha, mu]+h[alpha, mu], [X]), [X]))+(1/4)*(eta[`~beta`, `~iota`]+h[`~beta`, `~iota`])*(Physics:-d_[nu](eta[iota, mu]+h[iota, mu], [X])+Physics:-d_[mu](eta[iota, nu]+h[iota, nu], [X])-Physics:-d_[iota](eta[mu, nu]+h[mu, nu], [X]))*(eta[`~alpha`, `~lambda`]+h[`~alpha`, `~lambda`])*(Physics:-d_[beta](eta[alpha, lambda]+h[alpha, lambda], [X])+Physics:-d_[alpha](eta[beta, lambda]+h[beta, lambda], [X])-Physics:-d_[lambda](eta[alpha, beta]+h[alpha, beta], [X]))-(1/4)*(eta[`~beta`, `~omega`]+h[`~beta`, `~omega`])*(Physics:-d_[mu](eta[alpha, omega]+h[alpha, omega], [X])+Physics:-d_[alpha](eta[mu, omega]+h[mu, omega], [X])-Physics:-d_[omega](eta[alpha, mu]+h[alpha, mu], [X]))*(eta[`~alpha`, `~chi`]+h[`~alpha`, `~chi`])*(Physics:-d_[nu](eta[beta, chi]+h[beta, chi], [X])+Physics:-d_[beta](eta[chi, nu]+h[chi, nu], [X])-Physics:-d_[chi](eta[beta, nu]+h[beta, nu], [X]))" height="334" src="/view.aspx?sf=243656_question/ee96858b01f691d0375584bdf698017e.gif" style="vertical-align:-289px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(12)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;The sign infront of the perturbation in the inverse metric is wrong, it should be minus. &lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;This expression contains several terms quadratic in the small perturbation &lt;/span&gt;&lt;img alt="h[mu, nu]" height="31" src="/view.aspx?sf=243656_question/a983bc035c44215bd1a5c39435625910.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;. The routine to filter out those terms is &lt;/span&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:italic;"&gt;Linearize&lt;/span&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;, that takes as second argument the symbol representing the small quantities (perturbation)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;Lets look at the metric times inverse in this setup&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu,nu,definition]*g_[~mu,~alpha,definition]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~mu`, `~alpha`] = (eta[mu, nu]+h[mu, nu])*(eta[`~mu`, `~alpha`]+h[`~mu`, `~alpha`])" height="41" src="/view.aspx?sf=243656_question/fddc2df8d0afa5e197ee55322fbb4afb.gif" style="vertical-align:-15px" width="272"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(13)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Linearize(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(13)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;,h)&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = eta[mu, nu]*eta[`~alpha`, `~mu`]+eta[mu, nu]*h[`~alpha`, `~mu`]+eta[`~alpha`, `~mu`]*h[mu, nu]" height="41" src="/view.aspx?sf=243656_question/39f753644af5069020e2d3989f4dc70b.gif" style="vertical-align:-15px" width="298"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(14)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(subs(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(14)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]+2*h[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/c9dff1e3442c26c3e7e53bb7ddef6d0d.gif" style="vertical-align:-15px" width="112"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(15)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;The result is not correct, left-hand-side does not match right-hand-side, this is because the inverse metric has the wrong. If it were a minus, we would get:&lt;/span&gt;&lt;/p&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[mu, nu]*g_[~alpha, ~mu] = eta[mu, nu]*eta[~alpha, ~mu] - eta[mu, nu]*h[~alpha, ~mu] + eta[~alpha, ~mu]*h[mu, nu]&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu]*Physics:-g_[`~alpha`, `~mu`] = eta[mu, nu]*eta[`~alpha`, `~mu`]-eta[mu, nu]*h[`~alpha`, `~mu`]+eta[`~alpha`, `~mu`]*h[mu, nu]" height="41" src="/view.aspx?sf=243656_question/3780cec4644972a2ab9aaf5dfdd6cffc.gif" style="vertical-align:-15px" width="298"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(16)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(subs(eta=g_,&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(16)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;))&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[nu, `~alpha`] = Physics:-g_[nu, `~alpha`]" height="41" src="/view.aspx?sf=243656_question/36728c4f2f9fab825be9f4d947a209a2.gif" style="vertical-align:-15px" width="62"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(17)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;Which is correct. The continued calculation from the Help page is below. &lt;/span&gt;&lt;/p&gt;
			&amp;nbsp;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Linearize(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(12)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;, h);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[nu](Physics:-d_[tau](h[alpha, mu], [X]), [X])-(1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[mu](Physics:-d_[nu](h[alpha, tau], [X]), [X])-(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[kappa](h[mu, nu], [X]), [X])+(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[nu](h[kappa, mu], [X]), [X])+(1/2)*eta[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[mu](h[kappa, nu], [X]), [X])-(1/2)*eta[`~alpha`, `~tau`]*Physics:-d_[alpha](Physics:-d_[nu](h[mu, tau], [X]), [X])" height="116" src="/view.aspx?sf=243656_question/161b36d55a53a47f0d0114712b250360.gif" style="vertical-align:-71px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(18)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;
			&amp;nbsp;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;In this result, &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/d343e2294efb91788cb58aa443ffe126.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;is the flat Minkowski metric. To further simplify this expression using the internal algorithms for a flat metric it is practical to reintroduce &lt;/span&gt;&lt;img alt="g[mu, nu]" height="31" src="/view.aspx?sf=243656_question/4620c519c121dd9a6f413f1991adb15a.gif" style="vertical-align:-14px" width="31"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;representing that Minkowski metric&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;g_[min];&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/c7f34f870c0debd7898318d0a9871c18.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`The Minkowski metric in coordinates `*[x, y, z, t]" height="23" src="/view.aspx?sf=243656_question/15bee490de5359843d002ba72da94e9e.gif" style="vertical-align:-6px" width="294"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="`Signature: `(`- - - +`)" height="23" src="/view.aspx?sf=243656_question/faacf14c02220772a09fb186eb7f3dfe.gif" style="vertical-align:-6px" width="127"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="_______________________________________________________" height="23" src="/view.aspx?sf=243656_question/d630f95755e537c67cfd5567963fcc23.gif" style="vertical-align:-6px" width="421"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;&amp;nbsp;&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="Physics:-g_[mu, nu] = Matrix(%id = 36893488152069364060)" height="103" src="/view.aspx?sf=243656_question/d808bef8d776fe521927d2a268bcb52b.gif" style="vertical-align:-46px" width="162"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(19)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Replace in the expression for the Ricci tensor the intermediate Minkowski &lt;/span&gt;&lt;img alt="eta[mu, nu]" height="34" src="/view.aspx?sf=243656_question/bc5114140920505409534b9e719710b6.gif" style="vertical-align:-15px" width="33"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;by &lt;/span&gt;&lt;img alt="g[mu, nu]" height="31" src="/view.aspx?sf=243656_question/2dafbc0ed5f2199727add3c84366ccd2.gif" style="vertical-align:-14px" width="31"&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;subs(eta = g_, &lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(18)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img align="middle" alt="%Ricci[mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[nu](Physics:-d_[tau](h[alpha, mu], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[mu](Physics:-d_[nu](h[alpha, tau], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[kappa](h[mu, nu], [X]), [X])+(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[nu](h[kappa, mu], [X]), [X])+(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-d_[alpha](Physics:-d_[mu](h[kappa, nu], [X]), [X])-(1/2)*Physics:-g_[`~alpha`, `~tau`]*Physics:-d_[alpha](Physics:-d_[nu](h[mu, tau], [X]), [X])" height="112" src="/view.aspx?sf=243656_question/247ace3da0af4e444b7180f89774c547.gif" style="vertical-align:-69px" width="728"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(20)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:3px; padding-bottom:3px"&gt;&lt;span style="color:#000000;font-size: 100%;font-family: Times New Roman,serif;font-weight:normal;font-style:normal;"&gt;Simplifying, results in the linearized form of the Ricci tensor&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table style="margin-left:0px;margin-right:0px"&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;&amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
						&lt;td&gt;
						&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;Simplify(&lt;/span&gt;&lt;span style="color:#000000; font-weight:bold; font-style:normal;"&gt;(20)&lt;/span&gt;&lt;span style="color:#78000e;font-size: 100%;font-family: Courier New,monospace;font-weight:bold;font-style:normal;"&gt;);&lt;/span&gt;&lt;/p&gt;
						&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;table&gt;
				&lt;tbody&gt;
					&lt;tr valign="baseline"&gt;
						&lt;td&gt;
						&lt;p align="center" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;img alt="%Ricci[mu, nu] = -(1/2)*Physics:-d_[mu](Physics:-d_[nu](h[tau, `~tau`], [X]), [X])-(1/2)*Physics:-dAlembertian(h[mu, nu], [X])+(1/2)*Physics:-d_[nu](Physics:-d_[tau](h[mu, `~tau`], [X]), [X])+(1/2)*Physics:-d_[mu](Physics:-d_[tau](h[nu, `~tau`], [X]), [X])" height="59" src="/view.aspx?sf=243656_question/682cfa427f7b91c481419818d27e7bff.gif" style="vertical-align:-16px" width="457"&gt;&lt;/p&gt;
						&lt;/td&gt;
						&lt;td align="right" style="color:#000000; font-family:Times, serif; font-weight:bold; font-style:normal;"&gt;(21)&lt;/td&gt;
					&lt;/tr&gt;
				&lt;/tbody&gt;
			&lt;/table&gt;

			&lt;p align="left" style="margin:0 0 0 0; padding-top:0px; padding-bottom:0px"&gt;&lt;span style="color:#000000;font-size: 133%;font-family: Times New Roman,serif;font-weight:bold;font-style:normal;"&gt;This is correct result, because we are going to linear order only the +/- does not have an effect on the end result. &lt;/span&gt;&lt;/p&gt;
			&lt;/td&gt;
		&lt;/tr&gt;
	&lt;/tbody&gt;
&lt;/table&gt;
&lt;input name="sequence" type="hidden" value="1"&gt; &lt;input name="cmd" type="hidden" value="none"&gt;&lt;/form&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243656_question/LinearizedWorksheet-Comments.mw"&gt;Download LinearizedWorksheet-Comments.mw&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href="/view.aspx?sf=243656_question/LinearQuestion.mw"&gt;Download LinearQuestion.mw&lt;/a&gt;&lt;/p&gt;
</description>
      <guid>243656</guid>
      <pubDate>Tue, 23 Jun 2026 21:44:57 Z</pubDate>
      <itunes:author>Hullzie16</itunes:author>
      <author>Hullzie16</author>
    </item>
    <item>
      <title>What if my permutation has a fixed point?</title>
      <link>http://www.mapleprimes.com/questions/243655-What-If-My-Permutation-Has-A-Fixed-Point?ref=Feed:MaplePrimes:New%20Questions</link>
      <itunes:summary>&lt;p&gt;I am trying to use the Perm command in the GroupTheory package to create permutations. The problem is when the permutation has fixed points. For example, neither of the forms&lt;/p&gt;

&lt;p&gt;[[1,4,7],[2,8,5],[3],[6]]&lt;/p&gt;

&lt;p&gt;[[1,4,7],[2,8,5],[3,3],[6,6]]&lt;/p&gt;

&lt;p&gt;will work. Any suggestions?&lt;/p&gt;
</itunes:summary>
      <description>&lt;p&gt;I am trying to use the Perm command in the GroupTheory package to create permutations. The problem is when the permutation has fixed points. For example, neither of the forms&lt;/p&gt;

&lt;p&gt;[[1,4,7],[2,8,5],[3],[6]]&lt;/p&gt;

&lt;p&gt;[[1,4,7],[2,8,5],[3,3],[6,6]]&lt;/p&gt;

&lt;p&gt;will work. Any suggestions?&lt;/p&gt;
</description>
      <guid>243655</guid>
      <pubDate>Tue, 23 Jun 2026 06:13:03 Z</pubDate>
      <itunes:author>wkehowski</itunes:author>
      <author>wkehowski</author>
    </item>
  </channel>
</rss>