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How to create an interactive Math App in Maple

by: TechnicalSupport 790 Maple 2020
maplecloud embedded-components documenttools
May 13 2020
9  2

Since we are getting many questions on how to create Math apps to add to the Maple Cloud. I wanted to go over the different GUI aspects of how you go about creating a Math App in Maple. The following Document also includes some code examples that are used in the the Math App but doesn't go into them in detail. For more details on the type of coding you do in a Math App see the DocumentTools package help page.

Some of the graphical features of the Math app don't display on Maple Primes so I'd recommend downloading this worksheet from here: HowToMathApp.mw to follow along.


 

NULL

How to make a Math App (An example of using the Document Tools).

 

This Document will provide a beginners guide on one way to make a Math app in Maple.

It will contain some coding examples as well as where to find different options in the user interface.

Step 1 Insert a Table

 

 

• 

When making a Math App in Maple I often start with a table. You can enter a table by going to Insert > Table...

  

 

• 

I often make the table 1 x 2 to start with as this gives an area for input and an area for the output (such as plots).

NULL

 

Add a plot component to one of the cells of the table

 

 

• 

From the Components  Palette you can add a Plot Component . Add it to the table by clicking and dragging it over.

 

 

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Add another table inside the other cell

 

 

• 

In the other cell of the table I'll add another table to organize my use of buttons, sliders, and other components.
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Add some components to the new table

 

 

• 

From the Components Palette I'll add a slider, or dial, or something else for interaction.

 

• 

You may also want a Math region for an area to enter functions and a button to tell Maple to do something with it.

 

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Arrange the Components to look nice

 

 

• 

You can change how the components are placed either by resizing the tables or changing the text orientation of the contents of the cells.

 

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Write some code for the interaction of the buttons.

 

 

• 

Using the DocumentTools  package there are lots of ways you can use the components. I often will start writing my code using a code edit region  as it provides better visualization for syntax. On MaplePrimes these display as collapsed so I will also include code blocks for the code.

 

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Let's write something that takes the value of the slider and applies it to the dial

 

 

• 

Note that the names of the components will change in each section as they are copies of the previous section.

 

with(DocumentTools):

14

with(DocumentTools):
sv:=GetProperty('Slider2',value);
SetProperty('Dial2',value,sv);
• 

This code will only execute when run using the  button. Change the value of the slider below then run the code above to see what happens.

 

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Move the code 'inside' the slider

 

 

• 

Instead of putting the code inside the code edit region where it needs to be executed, we'll next add the code to the value changed code of the slider.

 

• 

Right click the Slider then select "Edit Value Changed Code".

 

 

• 

This will open the code editor for the Slider

 

 

• 

Enter your code (ensuring you're using the correct name for the slider and dial).

 

• 

Notice that you don't need to use the with(DocumentTools): command as "use DocumentTools in ... end use;" is already filled in for you.

 

• 

Save the code in the Slider and hit the  button inside it once.

• 

Now move the slider.

 

• 

On future uses of the App you won't need to hit  as the code will be run on startup.

``

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Add some more details to your App

 

 

• 

Let's make this app do something a bit more interesting than change the contents of a dial when a slider moves.

 

• 

The plan in the next few steps is to make this app allow a user to explore parameters changing in a sinusoidal expression.

 

• 

I'm going to add a second Math Component, put the expression A*sin(t*theta+phi)into both then uncheck the box in the context panel that says "Editable".

 

• 

To make the Math containers fit nicely I'll check the Auto-fit container box and set the Minimum Width Pixels to 200.

 

``

Add code to change the value of phi in the second Math Container when the Slider changes

 

 

Note: Maple uses Radians for trigonometric functions so we should convert the value of phi to Radians.

use DocumentTools in

 

use DocumentTools in 
phi_s:=GetProperty(Slider5,value);
expr:= GetProperty(MathContainer6,expression);
new_expr:=algsubs(phi=phi_s*Pi/180,expr);

SetProperty(MathContainer7,expression,new_expr);
end use:

``

``

Make the Dial go from 0 to 360°

 

 

• 

Click the Dial and look at the options in the context panel on the right.

 

• 

Update the values in the Dial so that the highest position is 360 and the spacing makes sense for the app.

  NULL

``

Have the Dial update the theta value of the expression

 

 

• 

Add the following code to the Dial

 

use DocumentTools in 
use DocumentTools in 
theta_d:=GetProperty(Dial7,value);
phi_s:=GetProperty(Slider7,value); #This is added so that phi also has the value updated

expr:= GetProperty(MathContainer10,expression);
new_expr0:=algsubs(theta=theta_d*Pi/180,expr);
new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0);  #This is added so that phi also has the value updated

SetProperty(MathContainer11,expression,new_expr);
end use:

 

• 

Update the value in the slider to include the value from the dial

 

use DocumentTools in

  

use DocumentTools in 

theta_d:=GetProperty(Dial7,value); #This is added so that theta also has the value updated
phi_s:=GetProperty(Slider7,value); 

expr:= GetProperty(MathContainer10,expression);
new_expr0:=algsubs(theta=theta_d*Pi/180,expr); #This is added so that theta also has the value updated
new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0);  

SetProperty(MathContainer11,expression,new_expr);

end use:

 

``

``

Notice that the code in the Dial and Slider are the same

 

 

• 

Since the code in the Dial and Slider are the same it makes sense to put the code into a procedure that can be called from multiple places.

 

Note: The changes in the code such as local and the single quotes are not needed but make the code easier to read and less likely to run into errors if edited in the future (for example if you create a variable called dial8 it won't interfere now that the names are in quotes).

 

 

UpdateMath:=proc()  

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial8','value'); #Get value of theta from Dial
phi_s:=GetProperty('Slider8','value'); #Get value of phi from slider

expr:= GetProperty('MathContainer12','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
SetProperty('MathContainer13','expression',new_expr); # Update expression
end use:
end proc:

 

• 

Now change the code in the components to call the function using UpdateMath().

 

• 

Since the code above is only defined there it will need to be run once (but only once) before moving the components. Instead of leaving it here you can add it to the Startup code by clicking  or going to Edit > Startup code.  This code will run every time you open the Math App ensuring that it works right away.

 

• 

The startup code isn't defined in this document to allow progression of these steps.

 

``

Make the button initialize the app

 

 

• 

Since the startup code isn't defined in this document we are going to move this function into the button.

 

UpdateMath:=proc()

 

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial9','value'); #Get value of theta from Dial
phi_s:=GetProperty('Slider9','value'); #Get value of phi from slider

expr:= GetProperty('MathContainer14','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
SetProperty('MathContainer15','expression',new_expr); # Update expression
end use:
end proc:
• 

First click the button to rename it, you'll see the  option in the context panel on the right. Then add the code above to the button in the same way as the Slider an Dial (Right click and select Edit Click Code).

 

``

``

Now it is easy to add new components

 

 

• 

Now if we want to add new components we just have to change the one procedure.  Let's add a Volume Gauge to change the value of A.

 

• 

Click in the cell containing the Dial, the context panel will show the option to Insert a row below the Dial.

• 

Now drag a Volume Gauge into the new cell.

 

• 

Click in the cell and choose the alignment (from the context panel) that looks best to you. In this case I chose center:

 

``

 

NULL

``

Update the procedure code for the Gauge

 

 

• 

Add two lines for the volume gauge to get the value and sub it into the expression

UpdateMath:=proc() 

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial11','value'); #Get value of theta from the Dial
phi_s:=GetProperty('Slider11','value'); #Get value of phi from the Slider
A_g:=GetProperty('VolumeGauge1','value'); #Get value of A from the Guage

expr:= GetProperty('MathContainer18','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr1:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
new_expr:=algsubs('A'=A_g,new_expr1);  # Put value of A in expression

SetProperty('MathContainer19','expression',new_expr); # Update expression
end use:
end proc:
• 

Now add

UpdateMath();

  to the Gauge.

  ``

``

Plot the changing expression

 

 

• 

Make a procedure to get the value in the second Math Container and plot it

 

PlotMath:=proc() 

PlotMath:=proc()
	local expr, p;
	use DocumentTools in 

	expr:=GetProperty('MathContainer21','expression'); 

	p:=plot(expr,'t'=-Pi/2..Pi/2,'view'=[-Pi/2..Pi/2,-100..100]):

	SetProperty('Plot14','value',p)
	end use:
end proc:
• 

Put this procedure in the Initialize button and the call to it in the components.

 

NULL

``

Tidy up the app

 

 

• 

Now that we have an interactive app let's tidy it up a bit.

 

• 

The first thing I'd recommend in your own app is moving the code from the initialize button to startup code. In this document we choose to use the button instead to preserve earlier versions.

 

• 

You can also remove the borders around the components by clicking in the table and selecting "Interior Borders" > "None" and "Exterior Borders" > "None" from the context panel.

NULL

``

``

Now you have a Math App

 

 

• 

You can upload your Math App to the Maple Cloud to share with others by going to "File" > "Save to Cloud".

 

• 

I'd recommend also including a description of what your app does. You can do this nicely using another table and Text mode.

 

 

 

``

``

NULL

HowToMathApp.mw

Maple Companion Update

by: Karishma 805 Maple Calculator
education homework phone
May 06 2020
11  0

I’m extremely pleased to introduce the newest update to the Maple Companion. In this time of wide-spread remote learning, tools like the Maple Companion are more important than ever, and I’m happy that our efforts are helping students (and some of their parents!) with at least one small aspect of their life.  Since we’ve added a lot of useful features since I last posted about this free mobile app, I wanted to share the ones I’m most excited about. 

(If you haven’t heard about the Maple Companion app, you can read more about it here.) 

If you use the app primarily to move math into Maple, you’ll be happy to hear that the automatic camera focus has gotten much better over the last couple of updates, and with this latest update, you can now turn on the flash if you need it. For me, these changes have virtually eliminated the need to fiddle with the camera to bring the math in focus, which sometimes happened in earlier versions.

If you use the app to get answers on your phone, that’s gotten much better, too. You can now see plots instantly as you enter your expression in the editor, and watch how the plot changes as you change the expression. You can also get results to many numerical problems results immediately, without having to switch to the results screen. This “calculator mode” is available even if you aren’t connected to the internet.  Okay, so there aren’t a lot of students doing their homework on the bus right now, but someday!

Speaking of plots, you can also now view plots full-screen, so you can see more of plot at once without zooming and panning, squinting, or buying a bigger phone.

Finally, if English is not you or your students’ first language, note that the app was recently made available in Spanish, French, German, Russian, Danish, Japanese, and Simplified Chinese. 

As always, we’d love you hear your feedback and suggestions. Please leave a comment, or use the feedback forms in the app or our web site.

Visit Maple Companion to learn more, find links to the app stores so you can download the app, and access the feedback form. If you already have it installed, you can get the new release simply by updating the app on your phone.

A triangle of maximum area inscribed in an ellipse...

by: one man 1355 Maple 17
optimization area triangle
May 02 2020
4  2

One of the forums asked a question: what is the maximum area of a triangle inscribed in a given ellipse x^2/16 + y^2/3 - 1 = 0? It turned out to be 9, but there are infinitely many such triangles. There was a desire to show them in one of the possible ways. This is a complete (as far as possible) set of such triangles.
(This is not an example of Maple programming; it is just an implementation of a Maple-based algorithm and the work of the Optimization package).
MAX_S_TRIAN_ANINATION.mw

Simulation of the DNA with Maple

by: Lenin Araujo Castillo 1790 Maple 2020
modeling curvature helix
May 01 2020
6  0

This app shows the modeling and simulation of DNA carried out entirely in Maple. The mathematical model is inserted through the combination of trigonometric functions. It shows the graphs of the curvature vs time for its interpretation. Made for engineering and health science students.

MV_AC_R3_UNI_2020.mw

Lenin Araujo Castillo

Ambassador of Maple

 

 

A Brief intro to Student Linear Algebra in Maple...

by: Valerie 132 Maple 2020
April 29 2020
5  0

With the new features added to the Student[LinearAlgebra] package I wanted to go over some of the basics on how someone can do Linear Algebra in Maple without require them to do any programming.  I was recently asked about this and thought that the information may be useful to others.
 

This post will be focussed towards new Maple users. I hope that this will be helpful to students using Maple for the first time and professors who want their students to use Maple without needing to spend time learning the language.
 

In addition to the following post you can find a detailed video on using Maple to do Linear Algebra without programming here. You can also find some of the tools that are new to Maple 2020 for Linear Algebra here.

The biggest tools you'll be using are the Matrix palette on the left of Maple, and the Context Panel on the right of Maple.

First you should load the Student[LinearAlgebra] package by entering:

with(Student[LinearAlgebra]);

at the beginning of your document. If you end it with a colon rather than a semi colon it won't display the commands in the package.

Use the Matrix Palette on the left to input Matrices:
​

 


Once you have a Matrix you can use the context panel on the right to apply a variety of operations to it:

 


The Student Linear Algebra Menu will give you many linear algebra commands.

 


You can also access Maple's Tutors from the Tools Menu

Tools > Tutors > Linear Algebra



If you're interested in using the commands for Student[LinearAlegbra] in Maple you can view the help pages here or by entering:

?Student[LinearAlegbra]

into Maple.

I hope that this helps you get started using Maple for Linear Algebra.

Maple for Beginners

by: ecterrab 14540 Maple
beginners
April 15 2020
11  0



Maple_for_Beginners.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Maple Student Edition is free to use until the...

by: laurent 1020 Maple
April 08 2020
23  2

Over the past weeks, we have spoken with many of our academic customers throughout the world, many of whom have decided to continue their academic years online. As you can imagine, this is a considerable challenge for instructors and students alike. Academia has quickly had to pivot to virtual classrooms, online testing and other collaborative technologies, while at the same time dealing with the stress and uncertainty that has resulted from this crisis.

We have been working with our customers to help them through this time in a variety of ways, but we know that there are still classes and students out there who are having trouble getting all the resources they need to complete their school year. So starting today, Maple Student Edition is being made free for every student, anywhere in the world, until the end of June. It is our hope that this action will remove a barrier for instructors to complete their Maple-led math instruction, and will help make things a bit more simple for everyone.

If you are a student, you can get your free copy of Maple here.

In addition, many of you have asked us about the best way to work on your engineering projects from home and/or teaching and learning remotely during this global crisis. We have put together resources for both that you can use as a starting point, and I invite you to contact us if you have any questions, or are dealing with challenges of your own. We are here to support you, and will be very flexible as we work together through these uncertain times.

I wish you all the best,

Laurent
President & CEO

Vectors in Spherical Coordinates using Tensor...

by: ecterrab 14540 Maple
coordinates orthogonal transformation
April 07 2020
6  3

Vectors in Spherical Coordinates using Tensor Notation

Edgardo S. Cheb-Terrab1 and Pascal Szriftgiser2

(2) Laboratoire PhLAM, UMR CNRS 8523, Université de Lille, F-59655, France

(1) Maplesoft

 

The following is a topic that appears frequently in formulations: given a 3D vector in spherical (or any curvilinear) coordinates, how do you represent and relate, in simple terms, the vector and the corresponding vectorial operations Gradient, Divergence, Curl and Laplacian using tensor notation?

 

The core of the answer is in the relation between the - say physical - vector components and the more abstract tensor covariant and contravariant components. Focusing the case of a transformation from Cartesian to spherical coordinates, the presentation below starts establishing that relationship between 3D vector and tensor components in Sec.I. In Sec.II, we verify the transformation formulas for covariant and contravariant components on the computer using TransformCoordinates. In Sec.III, those tensor transformation formulas are used to derive the vectorial form of the Gradient in spherical coordinates. In Sec.IV, we switch to using full tensor notation, a curvilinear metric and covariant derivatives to derive the 3D vector analysis traditional formulas in spherical coordinates for the Divergence, Curl, Gradient and Laplacian. On the way, some useful technics, like changing variables in 3D vectorial expressions, differential operators, using Jacobians, and shortcut notations are shown.

 

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.640 or newer.

 

Start setting the spacetime to be 3-dimensional, Euclidean, and use Cartesian coordinates

> 

with(Physics); with(Vectors)
> 

Setup(dimension = 3, coordinates = cartesian, g_ = `+`, spacetimeindices = lowercaselatin)

`The dimension and signature of the tensor space are set to `[3, `+ + +`]

  

`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (x, y, z)}

  

`Systems of spacetime coordinates are:`*{X = (x, y, z)}

  

_______________________________________________________

  

`The Euclidean metric in coordinates `*[x, y, z]

  

_______________________________________________________

  

Physics:-g_[mu, nu] = Matrix(%id = 18446744078312229334)

  

(`Defined Pauli sigma matrices (Psigma): `*sigma[1]*`, `*sigma[2]*`, `)*sigma[3]

  

__________________________________________________

  

_______________________________________________________

 (1)

I. The line element in spherical coordinates and the scale-factors

 

 

In vector calculus, at the root of everything there is the line element `#mover(mi("dr"),mo("→"))`, which in Cartesian coordinates has the simple form

> 

dr_ = _i*dx+_j*dy+_k*dz

dr_ = _i*dx+_j*dy+_k*dz

(1.1)

To compute the line element  `#mover(mi("dr"),mo("→"))` in spherical coordinates, the starting point is the transformation

> 

tr := `~`[`=`]([X], ChangeCoordinates([X], spherical))

[x = r*sin(theta)*cos(phi), y = r*sin(theta)*sin(phi), z = r*cos(theta)]

(1.2)
> 

Coordinates(S = [r, theta, phi])

`Systems of spacetime coordinates are:`*{S = (r, theta, phi), X = (x, y, z)}

(1.3)

Since in (dr_ = _i*dx+_j*dy+_k*dz)*[dx, dy, dz] are just symbols with no relationship to "[x,y,z],"start transforming these differentials using the chain rule, computing the Jacobian of the transformation (1.2). In this Jacobian J, the first line is "[(∂x)/(∂r)dr", "(∂x)/(∂theta)"`dθ`, "(∂x)/(∂phi)dphi]"

> 

J := VectorCalculus:-Jacobian(map(rhs, [x = r*sin(theta)*cos(phi), y = r*sin(theta)*sin(phi), z = r*cos(theta)]), [S])

 

So in matrix notation,

> 

Vector([dx, dy, dz]) = J.Vector([dr, dtheta, dphi])

Vector[column](%id = 18446744078518652550) = Vector[column](%id = 18446744078518652790)

(1.4)

To complete the computation of  `#mover(mi("dr"),mo("→"))` in spherical coordinates we can now use ChangeBasis , provided that next we substitute (1.4) in the result, expressing the abstract objects [dx, dy, dz] in terms of [dr, `dθ`, `dφ`].

 

In two steps:

> 

lhs(dr_ = _i*dx+_j*dy+_k*dz) = ChangeBasis(rhs(dr_ = _i*dx+_j*dy+_k*dz), spherical)

dr_ = (dx*sin(theta)*cos(phi)+dy*sin(theta)*sin(phi)+dz*cos(theta))*_r+(dx*cos(phi)*cos(theta)+dy*sin(phi)*cos(theta)-dz*sin(theta))*_theta+(cos(phi)*dy-sin(phi)*dx)*_phi

 (1.5)

The line element

> 

"simplify(subs(convert(lhs(?) =~ rhs(?),set),dr_ = (dx*sin(theta)*cos(phi)+dy*sin(theta)*sin(phi)+dz*cos(theta))*_r+(dx*cos(phi)*cos(theta)+dy*sin(phi)*cos(theta)-dz*sin(theta))*_theta+(cos(phi)*dy-sin(phi)*dx)*_phi))"

dr_ = _phi*dphi*r*sin(theta)+_theta*dtheta*r+_r*dr

(1.6)

This result is important: it gives us the so-called scale factors, the key that connect 3D vectors with the related covariant and contravariant tensors in curvilinear coordinates. The scale factors are computed from (1.6) by taking the scalar product with each of the unit vectors [`#mover(mi("r"),mo("∧"))`, `#mover(mi("θ",fontstyle = "normal"),mo("∧"))`, `#mover(mi("φ",fontstyle = "normal"),mo("∧"))`], then taking the coefficients of the differentials [dr, `dθ`, `dφ`] (just substitute them by the number 1)

> 

h := subs(`~`[`=`]([dr, `dθ`, `dφ`], 1), [seq(rhs(dr_ = _phi*dphi*r*sin(theta)+_theta*dtheta*r+_r*dr).q, q = [`#mover(mi("r"),mo("∧"))`, `#mover(mi("θ",fontstyle = "normal"),mo("∧"))`, `#mover(mi("φ",fontstyle = "normal"),mo("∧"))`])])

[1, r, r*sin(theta)]

(1.7)

The scale factors are relevant because the components of the 3D vector and the corresponding tensor are not the same in curvilinear coordinates. For instance, representing the differential of the coordinates as the tensor dS^j = [dr, `dθ`, `dφ`], we see that corresponding vector, the line element in spherical coordinates `#mover(mi("dS"),mo("→"))`, is not  constructed by directly equating its components to the components of dS^j = [dr, `dθ`, `dφ`], so  

 

 `#mover(mi("dS"),mo("&rarr;"))` <> `d&phi;`*`#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`+dr*`#mover(mi("r"),mo("&and;"))`+`d&theta;`*`#mover(mi("&theta;",fontstyle = "normal"),mo("&and;"))` 

 

The vector `#mover(mi("dS"),mo("&rarr;"))` is constructed multiplying these contravariant components [dr, `d&theta;`, `d&phi;`] by the scaling factors, as

 

 `#mover(mi("dS"),mo("&rarr;"))` = `d&phi;`*`h__&phi;`*`#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`+dr*h__r*`#mover(mi("r"),mo("&and;"))`+`d&theta;`*`h__&theta;`*`#mover(mi("&theta;",fontstyle = "normal"),mo("&and;"))` 

 

This rule applies in general. The vectorial components of a 3D vector in an orthogonal system (curvilinear or not) are always expressed in terms of the contravariant components A^j the same way we did in the line above with the line element, using the scale-factors h__j, so that

 

 `#mover(mi("A"),mo("&rarr;"))` = Sum(h[j]*A^j*`#mover(mi("\`e__j\`"),mo("&circ;"))`, j = 1 .. 3)

 

where on the right-hand side we see the contravariant components "A[]^(j)" and the scale-factors h[j]. Because the system is orthogonal, each vector component `#msub(mi("A",fontstyle = "normal"),mfenced(mi("j")))`satisfies

A__j = h[j]*A[`~j`]

 

The scale-factors h[j] do not constitute a tensor, so on the right-hand side we do not sum over j.  Also, from

 

LinearAlgebra[Norm](`#mover(mi("A"),mo("&rarr;"))`) = A[j]*A[`~j`]

it follows that,

A__j = A__j/h__j

where on the right-hand side we now have the covariant tensor components A__j.

 

• 

This relationship between the components of a 3D vector and the contravariant and covariant components of a tensor representing the vector is key to translate vector-component to corresponding tensor-component formulas.

 

II. Transformation of contravariant and covariant tensors

 

 

Define here two representations for one and the same tensor: A__c will represent A in Cartesian coordinates, while A__s will represent A in spherical coordinates.

> 

Define(A__c[j], A__s[j])

`Defined objects with tensor properties`

 

{A__c[j], A__s[j], Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](S), Physics:-SpaceTimeVector[a](X)}

(2.1)

Transformation rule for a contravariant tensor

 

We know, by definition, that the transformation rule for the components of a contravariant tensor is `#mrow(msup(mi("A"),mi("&mu;",fontstyle = "normal")),mo("&ApplyFunction;"),mfenced(mi("y")),mo("&equals;"),mfrac(mrow(mo("&PartialD;"),msup(mi("y"),mi("&mu;",fontstyle = "normal"))),mrow(mo("&PartialD;"),msup(mi("x"),mi("&nu;",fontstyle = "normal"))),linethickness = "1"),mo("&InvisibleTimes;"),mo("&InvisibleTimes;"),msup(mi("A"),mi("&nu;",fontstyle = "normal")),mfenced(mi("x")))`, that is the same as the rule for the differential of the coordinates. Then, the transformation rule from "`A__c`[]^(j)" to "`A__s`[]^(j)"computed using TransformCoordinates should give the same relation (1.4). The application of the command, however, requires attention, because, as in (1.4), we want the Cartesian (not the spherical) components isolated. That is like performing a reversed transformation. So we will use

 

"TensorArray(`A__c`[]^(j))=TransformCoordinates(tr,`A__s`[]^(j),[X],[S])"

where on the left-hand side we get, isolated, the three components of A in Cartesian coordinates, and on the right-hand side we transform the spherical components "`A__c`[]^(j)", from spherical S = (r, theta, phi) (4th argument) to Cartesian X = (x, y, z) (3rd argument), which according to the 5th bullet of TransformCoordinates  will result in a transformation expressed in terms of the old coordinates (here the spherical [S]). Expand things to make the comparison with (1.4) possible by eye

 

> 

Vector[column](TensorArray(A__c[`~j`])) = TransformCoordinates(tr, A__s[`~j`], [X], [S], simplifier = expand)

Vector[column](%id = 18446744078459463070) = Vector[column](%id = 18446744078459463550)

(2.2)

We see that the transformation rule for a contravariant vector "`A__c`[]^(j)"is, indeed, as the transformation (1.4) for the differential of the coordinates.

Transformation rule for a covariant tensor

 

For the transformation rule for the components of a covariant tensor A__c[j], we know, by definition, that it is `#mrow(msub(mi("A"),mi("&mu;",fontstyle = "normal")),mo("&ApplyFunction;"),mfenced(mi("y")),mo("&equals;"),mfrac(mrow(mo("&PartialD;"),msup(mi("x"),mi("&nu;",fontstyle = "normal"))),mrow(mo("&PartialD;"),msup(mi("y"),mi("&mu;",fontstyle = "normal"))),linethickness = "1"),mo("&InvisibleTimes;"),mo("&InvisibleTimes;"),msub(mi("A"),mi("&nu;",fontstyle = "normal")),mfenced(mi("x")))`, so the same transformation rule for the gradient [`&PartialD;`[x], `&PartialD;`[y], `&PartialD;`[z]], where `&PartialD;`[x] = (proc (u) options operator, arrow; diff(u, x) end proc) and so on. We can experiment this by directly changing variables in the differential operators [`&PartialD;`[x], `&PartialD;`[y], `&PartialD;`[z]], for example

> 

d_[x] = PDEtools:-dchange(tr, proc (u) options operator, arrow; diff(u, x) end proc, simplify)

Physics:-d_[x] = (proc (u) options operator, arrow; ((-r*cos(theta)^2+r)*cos(phi)*(diff(u, r))+sin(theta)*cos(phi)*cos(theta)*(diff(u, theta))-(diff(u, phi))*sin(phi))/(r*sin(theta)) end proc)

(2.3)

This result, and the equivalent ones replacing x by y or z in the input above can be computed in one go, in matricial and simplified form, using the Jacobian of the transformation computed in . We need to take the transpose of the inverse of J (because now we are transforming the components of the gradient   [`&PartialD;`[x], `&PartialD;`[y], `&PartialD;`[z]])

> 

H := simplify(LinearAlgebra:-Transpose(1/J))

> 

Vector([d_[x], d_[y], d_[z]]) = H.Vector([d_[r], d_[theta], d_[phi]])

Vector[column](%id = 18446744078518933014) = Vector[column](%id = 18446744078518933254)

(2.4)

The corresponding transformation equations relating the tensors A__c and A__s in Cartesian and spherical coordinates is computed with TransformCoordinates  as in (2.2), just lowering the indices on the left and right hand sides (i.e., remove the tilde ~)

> 

Vector[column](TensorArray(A__c[j])) = TransformCoordinates(tr, A__s[j], [X], [r, theta, phi], simplifier = expand)

Vector[column](%id = 18446744078557373854) = Vector[column](%id = 18446744078557374334)

(2.5)

We see that the transformation rule for a covariant vector A__c[j] is, indeed, as the transformation rule (2.4) for the gradient.

 

To the side: once it is understood how to compute these transformation rules, we can have the inverse of (2.5) as follows

> 

Vector[column](TensorArray(A__s[j])) = TransformCoordinates(tr, A__c[j], [S], [X], simplifier = expand)

Vector[column](%id = 18446744078557355894) = Vector[column](%id = 18446744078557348198)

(2.6)

III. Deriving the transformation rule for the Gradient using TransformCoordinates

 

 

Turn ON the CompactDisplay  notation for derivatives, so that the differentiation variable is displayed as an index:

> 

ON


The gradient of a function f in Cartesian coordinates and spherical coordinates is respectively given by

> 

(%Nabla = Nabla)(f(X))

%Nabla(f(X)) = (diff(f(X), x))*_i+(diff(f(X), y))*_j+(diff(f(X), z))*_k

(3.1)
> 

(%Nabla = Nabla)(f(S))

%Nabla(f(S)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

(3.2)

What we want now is to depart from (3.1) in Cartesian coordinates and obtain (3.2) in spherical coordinates using the transformation rule for a covariant tensor computed with TransformCoordinates in (2.5). (An equivalent derivation, simpler and with less steps is done in Sec. IV.)

 

Start changing the vector basis in the gradient (3.1)

> 

lhs(%Nabla(f(X)) = (diff(f(X), x))*_i+(diff(f(X), y))*_j+(diff(f(X), z))*_k) = ChangeBasis(rhs(%Nabla(f(X)) = (diff(f(X), x))*_i+(diff(f(X), y))*_j+(diff(f(X), z))*_k), spherical)

%Nabla(f(X)) = ((diff(f(X), x))*sin(theta)*cos(phi)+(diff(f(X), y))*sin(theta)*sin(phi)+(diff(f(X), z))*cos(theta))*_r+((diff(f(X), x))*cos(phi)*cos(theta)+(diff(f(X), y))*sin(phi)*cos(theta)-(diff(f(X), z))*sin(theta))*_theta+(-(diff(f(X), x))*sin(phi)+cos(phi)*(diff(f(X), y)))*_phi

 (3.3)

By eye, we see that in this result the coefficients of [`#mover(mi("r"),mo("&and;"))`, `#mover(mi("&theta;",fontstyle = "normal"),mo("&and;"))`, `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`] are the three lines in the right-hand side of (2.6) after replacing the covariant components A__j by the derivatives of f with respect to the jth coordinate, here displayed using indexed notation due to using CompactDisplay

> 

`~`[`=`]([A__s[1], A__s[2], A__s[3]], [diff(f(S), r), diff(f(S), theta), diff(f(S), phi)])

[A__s[1] = Physics:-Vectors:-diff(f(S), r), A__s[2] = Physics:-Vectors:-diff(f(S), theta), A__s[3] = Physics:-Vectors:-diff(f(S), phi)]

(3.4)
> 

`~`[`=`]([A__c[1], A__c[2], A__c[3]], [diff(f(X), x), diff(f(X), y), diff(f(X), z)])

[A__c[1] = Physics:-Vectors:-diff(f(X), x), A__c[2] = Physics:-Vectors:-diff(f(X), y), A__c[3] = Physics:-Vectors:-diff(f(X), z)]

(3.5)

So since (2.5) is the inverse of (2.6), replace A by ∂ f in (2.5), the formula computed using TransformCoordinates, then insert the result in (3.3) to relate the gradient in Cartesian and spherical coordinates. We expect to arrive at the formula for the gradient in spherical coordinates (3.2) .

> 

"subs([A__s[1] = Physics:-Vectors:-diff(f(S),r), A__s[2] = Physics:-Vectors:-diff(f(S),theta), A__s[3] = Physics:-Vectors:-diff(f(S),phi)],[A__c[1] = Physics:-Vectors:-diff(f(X),x), A__c[2] = Physics:-Vectors:-diff(f(X),y), A__c[3] = Physics:-Vectors:-diff(f(X),z)],?)"

Vector[column](%id = 18446744078344866862) = Vector[column](%id = 18446744078344866742)

(3.6)
> 

"subs(convert(lhs(?) =~ rhs(?),set),%Nabla(f(X)) = (diff(f(X),x)*sin(theta)*cos(phi)+diff(f(X),y)*sin(theta)*sin(phi)+diff(f(X),z)*cos(theta))*_r+(diff(f(X),x)*cos(phi)*cos(theta)+diff(f(X),y)*sin(phi)*cos(theta)-diff(f(X),z)*sin(theta))*_theta+(-diff(f(X),x)*sin(phi)+cos(phi)*diff(f(X),y))*_phi)"

%Nabla(f(X)) = ((sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(theta)*cos(phi)+(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(theta)*sin(phi)+(cos(theta)*(diff(f(S), r))-sin(theta)*(diff(f(S), theta))/r)*cos(theta))*_r+((sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*cos(phi)*cos(theta)+(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(phi)*cos(theta)-(cos(theta)*(diff(f(S), r))-sin(theta)*(diff(f(S), theta))/r)*sin(theta))*_theta+(-(sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(phi)+cos(phi)*(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta))))*_phi

 (3.7)

Simplifying, we arrive at (3.2)

> 

(lhs = `@`(`@`(expand, simplify), rhs))(%Nabla(f(X)) = ((sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(theta)*cos(phi)+(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(theta)*sin(phi)+(cos(theta)*(diff(f(S), r))-sin(theta)*(diff(f(S), theta))/r)*cos(theta))*_r+((sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*cos(phi)*cos(theta)+(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(phi)*cos(theta)-(cos(theta)*(diff(f(S), r))-sin(theta)*(diff(f(S), theta))/r)*sin(theta))*_theta+(-(sin(theta)*cos(phi)*(diff(f(S), r))+cos(theta)*cos(phi)*(diff(f(S), theta))/r-sin(phi)*(diff(f(S), phi))/(r*sin(theta)))*sin(phi)+cos(phi)*(sin(theta)*sin(phi)*(diff(f(S), r))+cos(theta)*sin(phi)*(diff(f(S), theta))/r+cos(phi)*(diff(f(S), phi))/(r*sin(theta))))*_phi)

%Nabla(f(X)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

(3.8)
> 

%Nabla(f(S)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

%Nabla(f(S)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

(3.9)

IV. Deriving the transformation rule for the Divergence, Curl, Gradient and Laplacian, using TransformCoordinates and Covariant derivatives

 

 

• 

The Divergence

 

Introducing the vector A in spherical coordinates, its Divergence is given by

> 

A__s_ := A__r(S)*_r+`A__&theta;`(S)*_theta+`A__&phi;`(S)*_phi

A__r(S)*_r+`A__&theta;`(S)*_theta+`A__&phi;`(S)*_phi

(4.1)
> 

CompactDisplay(%)

` A__r`(S)*`will now be displayed as`*A__r

 

` A__&phi;`(S)*`will now be displayed as`*`A__&phi;`

 

` A__&theta;`(S)*`will now be displayed as`*`A__&theta;`

(4.2)
> 

%Divergence(%A__s_) = Divergence(A__s_)

%Divergence(%A__s_) = ((diff(A__r(S), r))*r+2*A__r(S))/r+((diff(`A__&theta;`(S), theta))*sin(theta)+`A__&theta;`(S)*cos(theta))/(r*sin(theta))+(diff(`A__&phi;`(S), phi))/(r*sin(theta))

(4.3)

We want to see how this result, (4.3), can be obtained using TransformCoordinates and departing from a tensorial representation of the object, this time the covariant derivative "`&dtri;`[j](`A__s`[]^(j))". For that purpose, we first transform the coordinates and the metric introducing nonzero Christoffel symbols

> 

TransformCoordinates(tr, g_[j, k], [S], setmetric)

`Systems of spacetime coordinates are:`*{S = (r, theta, phi), X = (x, y, z)}

 

`Changing the differentiation variables used to compute the Christoffel symbols from `[x, y, z]*` to `[r, theta, phi]*` while the spacetime metric depends on `[r, theta]

  

`Default differentiation variables for d_, D_ and dAlembertian are:`*{S = (r, theta, phi)}

 

_______________________________________________________

 

`Coordinates: `[r, theta, phi]*`. Signature: `(`+ + -`)

 

_______________________________________________________

 

Physics:-g_[a, b] = Matrix(%id = 18446744078312216446)

 

_______________________________________________________

 

`Setting `*greek*` letters to represent `*space*` indices`

(4.4)

To the side: despite having nonzero Christoffel symbols, the space still has no curvature, all the components of the Riemann tensor are equal to zero

> 

Riemann[nonzero]

Physics:-Riemann[a, b, c, d] = {}

(4.5)

Consider now the divergence of the contravariant "`A__s`[]^(j)"tensor, computed in tensor notation

> 

CompactDisplay(A__s(S))

` A__s`(S)*`will now be displayed as`*A__s

(4.6)
> 

D_[j](A__s[`~j`](S))

Physics:-D_[j](A__s[`~j`](S), [S])

(4.7)

To the side: the covariant derivative  expressed using the D_  operator can be rewritten in terms of the non-covariant d_  and Christoffel  symbols as follows

> 

D_[j](A__s[`~j`](S), [S]) = convert(D_[j](A__s[`~j`](S), [S]), d_)

Physics:-D_[j](A__s[`~j`](S), [S]) = Physics:-d_[j](A__s[`~j`](S), [S])+Physics:-Christoffel[`~j`, a, j]*A__s[`~a`](S)

(4.8)

Summing over the repeated indices in (4.7), we have

> 

%D_[j](%A__s[`~j`]) = SumOverRepeatedIndices(D_[j](A__s[`~j`](S), [S]))

%D_[j](%A__s[`~j`]) = diff(A__s[`~1`](S), r)+diff(A__s[`~2`](S), theta)+diff(A__s[`~3`](S), phi)+2*A__s[`~1`](S)/r+cos(theta)*A__s[`~2`](S)/sin(theta)

(4.9)

How is this related to the expression of the VectorCalculus[Nabla].`#mover(mi("\`A__s\`"),mo("&rarr;"))` in (4.3) ? The answer is in the relationship established at the end of Sec I between the components of the tensor "`A__s`[]^(j)"and the components of the vector `#mover(mi("\`A__s\`"),mo("&rarr;"))`, namely that the vector components are obtained multiplying the contravariant tensor components by the scale-factors h__j. So, in the above we need to substitute the contravariant "`A__s`[]^(j)" by the vector components A__j divided by the scale-factors

> 

[seq(A__s[Library:-Contravariant(j)](S) = Component(A__s_, j)/h[j], j = 1 .. 3)]

[A__s[`~1`](S) = A__r(S), A__s[`~2`](S) = `A__&theta;`(S)/r, A__s[`~3`](S) = `A__&phi;`(S)/(r*sin(theta))]

(4.10)
> 

subs[eval]([A__s[`~1`](S) = A__r(S), A__s[`~2`](S) = `A__&theta;`(S)/r, A__s[`~3`](S) = `A__&phi;`(S)/(r*sin(theta))], %D_[j](%A__s[`~j`]) = diff(A__s[`~1`](S), r)+diff(A__s[`~2`](S), theta)+diff(A__s[`~3`](S), phi)+2*A__s[`~1`](S)/r+cos(theta)*A__s[`~2`](S)/sin(theta))

%D_[j](%A__s[`~j`]) = diff(A__r(S), r)+(diff(`A__&theta;`(S), theta))/r+(diff(`A__&phi;`(S), phi))/(r*sin(theta))+2*A__r(S)/r+cos(theta)*`A__&theta;`(S)/(sin(theta)*r)

(4.11)

Comparing with (4.3), we see these two expressions are the same:

> 

expand(%Divergence(%A__s_) = ((diff(A__r(S), r))*r+2*A__r(S))/r+((diff(`A__&theta;`(S), theta))*sin(theta)+`A__&theta;`(S)*cos(theta))/(r*sin(theta))+(diff(`A__&phi;`(S), phi))/(r*sin(theta)))

%Divergence(%A__s_) = diff(A__r(S), r)+(diff(`A__&theta;`(S), theta))/r+(diff(`A__&phi;`(S), phi))/(r*sin(theta))+2*A__r(S)/r+cos(theta)*`A__&theta;`(S)/(sin(theta)*r)

(4.12)
• 

The Curl

 

The Curl of the the vector `#mover(mi("\`A__s\`"),mo("&rarr;"))` in spherical coordinates is given by

> 

%Curl(%A__s_) = Curl(A__s_)

%Curl(%A__s_) = ((diff(`A__&phi;`(S), theta))*sin(theta)+`A__&phi;`(S)*cos(theta)-(diff(`A__&theta;`(S), phi)))*_r/(r*sin(theta))+(diff(A__r(S), phi)-(diff(`A__&phi;`(S), r))*r*sin(theta)-`A__&phi;`(S)*sin(theta))*_theta/(r*sin(theta))+((diff(`A__&theta;`(S), r))*r+`A__&theta;`(S)-(diff(A__r(S), theta)))*_phi/r

 (4.13)

 

One could think that the expression for the Curl in tensor notation is as in a non-curvilinear system

 

"`&epsilon;`[i,j,k] `&dtri;`[]^(j)(`A__s`[]^(k))"

 

But in a curvilinear system `&epsilon;`[i, j, k] is not a tensor, we need to use the non-Galilean form Epsilon[i, j, k] = sqrt(%g_[determinant])*`&epsilon;`[i, j, k], where %g_[determinant] is the determinant of the metric. Moreover, since the expression "Epsilon[i,j,k] `&dtri;`[]^(j)(`A__s`[]^(k))"has one free covariant index (the first one), to compare with the vectorial formula (4.12) this index also needs to be rewritten as a vector component as discussed at the end of Sec. I, using

A__j = A__j/h__j

The formula (4.13) for the vectorial Curl is thus expressed using tensor notation as

> 

Setup(levicivita = nongalilean)

[levicivita = nongalilean]

(4.14)
> 

%Curl(%A__s_) = LeviCivita[i, j, k]*D_[`~j`](A__s[`~k`](S))/%h[i]

%Curl(%A__s_) = Physics:-LeviCivita[i, j, k]*Physics:-D_[`~j`](A__s[`~k`](S), [S])/%h[i]

(4.15)

followed by replacing the contravariant tensor components "`A__s`[]^(k)" by the vector components A__k/h__k using (4.10). Proceeding the same way we did with the Divergence, expand this expression. We could use TensorArray , but Library:-TensorComponents places a comma between components making things more readable in this case

> 

lhs(%Curl(%A__s_) = Physics[LeviCivita][i, j, k]*D_[`~j`](A__s[`~k`](S), [S])/%h[i]) = Library:-TensorComponents(rhs(%Curl(%A__s_) = Physics[LeviCivita][i, j, k]*D_[`~j`](A__s[`~k`](S), [S])/%h[i]))

%Curl(%A__s_) = [(sin(theta)^3*(diff(A__s[`~3`](S), theta))*r^2+2*sin(theta)^2*cos(theta)*A__s[`~3`](S)*r^2-(diff(A__s[`~2`](S), phi))*sin(theta)*r^2)/(%h[1]*sin(theta)^2*r^2), (-sin(theta)^3*(diff(A__s[`~3`](S), r))*r^4-2*sin(theta)^3*A__s[`~3`](S)*r^3+(diff(A__s[`~1`](S), phi))*sin(theta)*r^2)/(%h[2]*sin(theta)^2*r^2), (sin(theta)^3*(diff(A__s[`~2`](S), r))*r^4+2*sin(theta)^3*A__s[`~2`](S)*r^3-sin(theta)^3*(diff(A__s[`~1`](S), theta))*r^2)/(%h[3]*sin(theta)^2*r^2)]

 (4.16)

Replace now the components of the tensor "`A__s`[]^(j)" by the components of the 3D vector `#mover(mi("\`A__s\`"),mo("&rarr;"))` using (4.10)

> 

lhs(%Curl(%A__s_) = [(sin(theta)^3*(diff(A__s[`~3`](S), theta))*r^2+2*sin(theta)^2*cos(theta)*A__s[`~3`](S)*r^2-(diff(A__s[`~2`](S), phi))*sin(theta)*r^2)/(%h[1]*sin(theta)^2*r^2), (-sin(theta)^3*(diff(A__s[`~3`](S), r))*r^4-2*sin(theta)^3*A__s[`~3`](S)*r^3+(diff(A__s[`~1`](S), phi))*sin(theta)*r^2)/(%h[2]*sin(theta)^2*r^2), (sin(theta)^3*(diff(A__s[`~2`](S), r))*r^4+2*sin(theta)^3*A__s[`~2`](S)*r^3-sin(theta)^3*(diff(A__s[`~1`](S), theta))*r^2)/(%h[3]*sin(theta)^2*r^2)]) = value(subs[eval]([A__s[`~1`](S) = A__r(S), A__s[`~2`](S) = `A__&theta;`(S)/r, A__s[`~3`](S) = `A__&phi;`(S)/(r*sin(theta))], rhs(%Curl(%A__s_) = [(sin(theta)^3*(diff(A__s[`~3`](S), theta))*r^2+2*sin(theta)^2*cos(theta)*A__s[`~3`](S)*r^2-(diff(A__s[`~2`](S), phi))*sin(theta)*r^2)/(%h[1]*sin(theta)^2*r^2), (-sin(theta)^3*(diff(A__s[`~3`](S), r))*r^4-2*sin(theta)^3*A__s[`~3`](S)*r^3+(diff(A__s[`~1`](S), phi))*sin(theta)*r^2)/(%h[2]*sin(theta)^2*r^2), (sin(theta)^3*(diff(A__s[`~2`](S), r))*r^4+2*sin(theta)^3*A__s[`~2`](S)*r^3-sin(theta)^3*(diff(A__s[`~1`](S), theta))*r^2)/(%h[3]*sin(theta)^2*r^2)])))

%Curl(%A__s_) = [(sin(theta)^3*((diff(`A__&phi;`(S), theta))/(r*sin(theta))-`A__&phi;`(S)*cos(theta)/(r*sin(theta)^2))*r^2+2*sin(theta)*cos(theta)*`A__&phi;`(S)*r-(diff(`A__&theta;`(S), phi))*r*sin(theta))/(h[1]*sin(theta)^2*r^2), (-sin(theta)^3*((diff(`A__&phi;`(S), r))/(r*sin(theta))-`A__&phi;`(S)/(r^2*sin(theta)))*r^4-2*sin(theta)^2*`A__&phi;`(S)*r^2+(diff(A__r(S), phi))*sin(theta)*r^2)/(h[2]*sin(theta)^2*r^2), (sin(theta)^3*((diff(`A__&theta;`(S), r))/r-`A__&theta;`(S)/r^2)*r^4+2*sin(theta)^3*`A__&theta;`(S)*r^2-sin(theta)^3*(diff(A__r(S), theta))*r^2)/(h[3]*sin(theta)^2*r^2)]

 (4.17)
> 

(lhs = `@`(simplify, rhs))(%Curl(%A__s_) = [(sin(theta)^3*((diff(`A__&phi;`(S), theta))/(r*sin(theta))-`A__&phi;`(S)*cos(theta)/(r*sin(theta)^2))*r^2+2*sin(theta)*cos(theta)*`A__&phi;`(S)*r-(diff(`A__&theta;`(S), phi))*r*sin(theta))/(h[1]*sin(theta)^2*r^2), (-sin(theta)^3*((diff(`A__&phi;`(S), r))/(r*sin(theta))-`A__&phi;`(S)/(r^2*sin(theta)))*r^4-2*sin(theta)^2*`A__&phi;`(S)*r^2+(diff(A__r(S), phi))*sin(theta)*r^2)/(h[2]*sin(theta)^2*r^2), (sin(theta)^3*((diff(`A__&theta;`(S), r))/r-`A__&theta;`(S)/r^2)*r^4+2*sin(theta)^3*`A__&theta;`(S)*r^2-sin(theta)^3*(diff(A__r(S), theta))*r^2)/(h[3]*sin(theta)^2*r^2)])

%Curl(%A__s_) = [((diff(`A__&phi;`(S), theta))*sin(theta)+`A__&phi;`(S)*cos(theta)-(diff(`A__&theta;`(S), phi)))/(r*sin(theta)), (diff(A__r(S), phi)-(diff(`A__&phi;`(S), r))*r*sin(theta)-`A__&phi;`(S)*sin(theta))/(r*sin(theta)), ((diff(`A__&theta;`(S), r))*r+`A__&theta;`(S)-(diff(A__r(S), theta)))/r]

(4.18)

We see these are exactly the components of the Curl (4.13)

> 

%Curl(%A__s_) = ((diff(`A__&phi;`(S), theta))*sin(theta)+`A__&phi;`(S)*cos(theta)-(diff(`A__&theta;`(S), phi)))*_r/(r*sin(theta))+(diff(A__r(S), phi)-(diff(`A__&phi;`(S), r))*r*sin(theta)-`A__&phi;`(S)*sin(theta))*_theta/(r*sin(theta))+((diff(`A__&theta;`(S), r))*r+`A__&theta;`(S)-(diff(A__r(S), theta)))*_phi/r

%Curl(%A__s_) = ((diff(`A__&phi;`(S), theta))*sin(theta)+`A__&phi;`(S)*cos(theta)-(diff(`A__&theta;`(S), phi)))*_r/(r*sin(theta))+(diff(A__r(S), phi)-(diff(`A__&phi;`(S), r))*r*sin(theta)-`A__&phi;`(S)*sin(theta))*_theta/(r*sin(theta))+((diff(`A__&theta;`(S), r))*r+`A__&theta;`(S)-(diff(A__r(S), theta)))*_phi/r

 (4.19)
• 

The Gradient

 

Once the problem is fully understood, it is easy to redo the computations of Sec.III for the Gradient, this time using tensor notation and the covariant derivative. In tensor notation, the components of the Gradient are given by the components of the right-hand side

> 

%Nabla(f(S)) = `&dtri;`[j](f(S))/%h[j]

%Nabla(f(S)) = Physics:-d_[j](f(S), [S])/%h[j]

(4.20)

where on the left-hand side we have the vectorial Nabla  differential operator and on the right-hand side, since f(S) is a scalar, the covariant derivative `&dtri;`[j](f) becomes the standard derivative `&PartialD;`[j](f).

> 

lhs(%Nabla(f(S)) = Physics[d_][j](f(S), [S])/%h[j]) = eval(value(Library:-TensorComponents(rhs(%Nabla(f(S)) = Physics[d_][j](f(S), [S])/%h[j]))))

%Nabla(f(S)) = [Physics:-Vectors:-diff(f(S), r), (diff(f(S), theta))/r, (diff(f(S), phi))/(r*sin(theta))]

(4.21)

The above is the expected result (3.2)

> 

%Nabla(f(S)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

%Nabla(f(S)) = (diff(f(S), r))*_r+(diff(f(S), theta))*_theta/r+(diff(f(S), phi))*_phi/(r*sin(theta))

(4.22)
• 

The Laplacian

 

Likewise we can compute the Laplacian directly as

> 

%Laplacian(f(S)) = D_[j](D_[j](f(S)))

%Laplacian(f(S)) = Physics:-D_[j](Physics:-d_[`~j`](f(S), [S]), [S])

(4.23)

In this case there are no free indices nor tensor components to be rewritten as vector components, so there is no need for scale-factors. Summing over the repeated indices,

> 

SumOverRepeatedIndices(%Laplacian(f(S)) = D_[j](Physics[d_][`~j`](f(S), [S]), [S]))

%Laplacian(f(S)) = Physics:-dAlembertian(f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.24)

Evaluating the  Vectors:-Laplacian on the left-hand side,

> 

value(%Laplacian(f(S)) = Physics[dAlembertian](f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2))

((diff(diff(f(S), r), r))*r+2*(diff(f(S), r)))/r+((diff(diff(f(S), theta), theta))*sin(theta)+cos(theta)*(diff(f(S), theta)))/(r^2*sin(theta))+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2) = Physics:-dAlembertian(f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.25)

On the right-hand side we see the dAlembertian , "`&square;`(f(S)),"in curvilinear coordinates; rewrite it using standard diff  derivatives and expand both sides of the equation for comparison

> 

expand(convert(((diff(diff(f(S), r), r))*r+2*(diff(f(S), r)))/r+((diff(diff(f(S), theta), theta))*sin(theta)+cos(theta)*(diff(f(S), theta)))/(r^2*sin(theta))+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2) = Physics[dAlembertian](f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2), diff))

diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2) = diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.26)

This is an identity, the left and right hand sides are equal:

> 

evalb(diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2) = diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2))

true

(4.27)


 

Download Vectors_and_Spherical_coordinates_in_tensor_notation.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Нoles on an implicit plot

by: one man 1355 Maple 17
implicit surface plotting
April 01 2020
2  0

A way of cutting holes on an implicit plot. This is from the field of numerical parameterization of surfaces. On the example of the surface  x3 = 0.01*exp (x1) / (0.01 + x1^4 + x2^4 + x3^4)  consider the approach to producing holes. The surface is locally parameterized in some suitable way and the place for the hole and its size are selected. In the first example, the parametrization is performed on the basis of the section of the initial surface by perpendicular planes. In the second example, "round"  parametrization. It is made on the basis of the cylinder and the planes passing through its axis. Holes can be of any size and any shape. In the figures, the cut out surface sections are colored green and are located above their own holes at an equidistant to the original surface.
HOLE_1.mwHOLE_2.mw

Symbol - Utilization

by: Ramakrishnan 399 Maple
symbol plotting arrow
March 28 2020
5  0

 

In maple plot, very many symbols like, diamond, star, solidcircle are available. Many of them may have been used also for teaching purposes.

Recently, someone encountered the need to draw graphs with arrowheads and many solutions may be available as well. But it requires a thorough understanding of maple's features which are infinitely many. My feeling was that an arrow symbol also could be added in the symbol feature so that the option can be used as a plot point in the graph at the graph end points very easily. It can be just like adding a solidbox symbol at any point on the curve.

Hope my suggestions are in order.

Thanks.

Ramakrishnan V

Puzzle - cut it into 2 equal parts

by: Kitonum 21435
March 23 2020
5  0

The following puzzle prompted me to write this post: "A figure is drawn on checkered paper that needs to be cut into 2 equal parts (the cuts must pass along the sides of the squares.)" (parts are called equal if, after cutting, they can be superimposed on one another, that is, if one of them can be moved, rotated and (if need to) flip so that they completely coincide) (see the first picture below). 
I could not solve it manually and wrote a procedure called  CutTwoParts  that does this automatically (of course, this procedure applies to other similar puzzles). This procedure uses my procedure  AreIsometric  published earlier  https://www.mapleprimes.com/posts/200157-Testing-Of-Two-Plane-Sets-For-Isometry  (for convenience, I have included its text here). In the procedure  CutTwoParts  the figure is specified by the coordinates of the centers of the squares of which it consists).

I advise everyone to first try to solve this puzzle manually in order to feel its non-triviality, and only then load the worksheet with the procedure for automatic solution.


For some reason, the worksheet did not load and I was only able to insert the link.

Cuttings.mw



 

First equilibrium condition with Maple

by: Lenin Araujo Castillo 1790 Maple 2020
engineering physics education
March 20 2020
2  0

With this application our students of science and engineering in the areas of physics will check the first condition of balance using Maple technology. Only with entering mass and angles we obtain graphs and data for a better interpretation.

First_equilibrium_condition.zip

Lenin AC

Ambassador of Maple

The Two Kinds of Loops in Maple

by: stefanv 357 Maple
March 04 2020
9  9

When discussing Maple programming, we often refer to for-loops, while-loops, until-loops, and do-loops (the latter being an infinite loop). But under the hood, Maple has only two kinds of loop, albeit very flexible and powerful ones that can combine the capabilities of any or all of the above, making it possible to write very concise code in a natural way.

Before looking at some actual examples, here is the formal definition of the loops' syntax, expressed in Wirth Syntax Notation, where "|" denotes alternatives, "[...]" denotes an optional part, "(...)" denotes grouping, and Maple keywords are in boldface:

[ for  ] [ from  ] [ by  ] [ to  ]
    [ while  ]
do
    
( end do | until  )

[ for  [ , variable ] ] in 
    [ while  ]
do
    
( end do | until  )

In the first form, every part of the loop syntax is optional, except the do keyword before the body of the loop, and either end do or an until clause after the body. (For those who prefer it, end do can also be written as od.) In the second form, only the in clause is required.

The simplest loop is just:

do
    
end do

This will repeat the forever, unless a break or return statement is executed, or an error occurs.

One or two loop termination conditions can be added:

  • A while clause can be written before the do, specifying a condition that is tested before each iteration begins. If the condition evaluates to false, the loop ends.
  • An until clause can be written instead of the end do, specifying a condition that is tested after each iteration finishes. If the condition evaluates to true, the loop ends.

A so-called for-loop is just a loop to which iteration clauses have been added. These can take one of two forms:

  • Any combination of for (with a single variable), from, by, and to clauses. The last three can appear in any order.
  • A for clause with one or two variables, followed by an in clause.

The following for-loop executes 10 times:

for  from 1 to 10 do
    
end do

However, if the doesn't depend on the value of , both the for and from clauses can be omitted:

to 10 do
    
end do

In this case, Maple supplies an implicit for clause (with an inaccessible internal variable), as well as an implicit "from 1" clause. In fact, all of the clauses are optional, and the infinite loop shown earlier is understood by Maple in exactly the same way as:

for  from 1 by 1 to infinity while true do
    
until false

When looping over the contents of a container, such as a one-dimensional array A, there are several possible approaches. The one closest to how it would be done in most other programming languages is (this example and those that follow can be copied and pasted into a Maple session):

 := Array([,"foo",42]);
for  from lowerbound() to upperbound() do
    print([],[])
end do;

If only the entries in the container are of interest, it is not necessary to loop over the indices. Instead, one can write:

 := Array([,"foo",42]);
for  in  do
    print()
end do;

If both the indices and values are needed, one can write:

 := Array([,"foo",42]);
for ,  in  do
    print([],)
end do;

For a numerically indexed container such as an Array, this is equivalent to the for-from-to example. However, this method also works with arbitrarily indexed containers such as a Matrix or table:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    print([],)
end do;

 := table({1="one","hello"="world",=42});
for ,  in eval() do
    print([],)
end do;

(The second example requires the call to eval due to last-name evaluation of tables in Maple, a topic for another post.)

As with a simple do-loop, a while and/or until clause can be added. For example, the following finds the first negative entry, if any, in a Matrix (traversing the Matrix in storage order):

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    # nothing to do here
until  < 0;
if  < 0 then
    print([],)
end if;

Notice that the test, < 0, is written twice, since it is possible that the Matrix has no negative entry. Another way to write the same loop but only perform the test once is as follows:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    if  < 0 then
	print([],);
	break
    end if;
end do;

Here, we perform the test within the loop, perform the desired processing on the found value (just printing in this case), and use a break statement to terminate the loop.

Sometimes, it is useful to abort the current iteration of the loop and move on to the next one. The next statement does exactly that. The following loop prints all the indices but only the positive values in a Matrix:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    print(=[]);
    if  < 0 then
	next
    end if;
    print(=);
end do;

(Note that a simple example like this would be better written by enclosing the printing of the value in an if-statement instead of using next. The latter is generally only used if the former is not possible.)

Maple's loop statements are very flexible and powerful, making it possible to write loops with complex combinations of termination conditions in a concise yet readable way. The ability to use while and/or until in conjunction with for means that break statements are often unnecessary, further improving clarity.

Binary Search Algorithm

by: radaar 202 Maple
array index
February 29 2020
2  4

The binary search algorithm is used to obtain the index of a given number by dividing the search bound in half over iteration. If the value entered in the array a message pop up telling that ''value is not present in the array". Please see the code. 
 

> 

restart; with(ArrayTools); AA := Array(1 .. 10, [20, 2, 30, 4, 50, 7, 60, 8, 90, 100]); AA := sort(AA); KEYVALUE := 200; DUP_KEYVALUE := infinity; low := 1; high := NumElems(AA); while DUP_KEYVALUE <> KEYVALUE do mid := floor((low+high)*(1/2)); if AA[mid] = KEYVALUE then DUP_KEYVALUE := KEYVALUE; printf("%s\n %a\n", "the index is ", mid) elif AA[mid] < KEYVALUE then low := mid+1 elif AA[mid] > KEYVALUE then high := mid-1 end if; if low > high then printf("%s\n", "value not present in the array"); break end if end do


 

Download BINARY_SEARCH1.mw

 

Feynman Diagrams - the scattering matrix in coordi...

by: ecterrab 14540 Maple
physics feynman-diagrams momentum-representation
January 25 2020
7  0
 

Feynman Diagrams
The scattering matrix in coordinates and momentum representation

  

Mathematical methods for particle physics was one of the weak spots in the Physics package. There existed a FeynmanDiagrams command, but its capabilities were too minimal. People working in the area asked for more functionality. These diagrams are the cornerstone of calculations in particle physics (collisions involving from the electron to the Higgs boson), for example at the CERN. As an introduction for people curious, not working in the area, see "Why Feynman Diagrams are so important".
  

This post is thus about a new development in Physics: a full rewriting of the FeynmanDiagrams command, now including a myriad of new capabilities (mainly a. b. and c. in the Introduction), reversing the previous status of things entirely. This is work in collaboration with Davide Polvara from Durham University, Centre for Particle Theory.

  


The complexity of this material is high, so the introduction to the presentation below is as brief as it can get, emphasizing the examples instead. This material is reproducible in Maple 2019.2 after installing the Physics Updates, v.598 or higher.
  

 

  

At the end they are attached the worksheet corresponding to this presentation and a PDF version of it, as well as the new FeynmanDiagrams help page with all the explanatory details.

Introduction

  

A scattering matrix S relates the initial and final states, `#mfenced(mrow(mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&verbar;",close = "&rang;")` and `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;")),open = "&verbar;",close = "&rang;")`, of an interacting system. In an 4-dimensional spacetime with coordinates X, S can be written as:

S = T(exp(i*`#mrow(mo("&int;"),mi("L"),mo("&ApplyFunction;"),mfenced(mi("X")),mo("&DifferentialD;"),msup(mi("X"),mn("4")))`))

  

where i is the imaginary unit  and L is the interaction Lagrangian, written in terms of quantum fields  depending on the spacetime coordinates  X. The T symbol means time-ordered. For the terminology used in this page, see for instance chapter IV, "The Scattering Matrix", of ref.[1] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields.

  

This exponential can be expanded as

S = 1+S[1]+S[2]+S[3]+`...`

  

where

S[n] = `#mrow(mo("&ApplyFunction;"),mfrac(msup(mi("i"),mi("n")),mrow(mi("n"),mo("&excl;")),linethickness = "1"),mo("&InvisibleTimes;"),mo("&int;"),mi("&hellip;"),mo("&InvisibleTimes;"),mo("&int;"),mi("T"),mo("&ApplyFunction;"),mfenced(mrow(mi("L"),mo("&ApplyFunction;"),mfenced(mi("\`X__1\`")),mo("&comma;"),mi("&hellip;"),mo("&comma;"),mi("L"),mo("&ApplyFunction;"),mfenced(mi("\`X__n\`")))),mo("&InvisibleTimes;"),mo("&DifferentialD;"),msup(mi("\`X__1\`"),mn("4")),mo("&InvisibleTimes;"),mi("&hellip;"),mo("&InvisibleTimes;"),mo("&DifferentialD;"),msup(mi("\`X__n\`"),mn("4")))`

  

and T(L(X[1]), `...`, L(X[n])) is the time-ordered product of n interaction Lagrangians evaluated at different points. The S matrix formulation is at the core of perturbative approaches in relativistic Quantum Field Theory.
  

In connection, the FeynmanDiagrams  command has been rewritten entirely for Maple 2020. In brief, the new functionality includes computing:
a. 

The expansion S = 1+S[1]+S[2]+S[3]+`...` in coordinates representation up to arbitrary order (the limitation is now only your hardware)
b. 

The S-matrix element `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` in momentum representation up to arbitrary order for given number of loops and initial and final particles (the contents of the `#mfenced(mrow(mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&verbar;",close = "&rang;")` and `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;")),open = "&verbar;",close = "&rang;")` states); optionally, also the transition probability density, constructed using the square of the scattering matrix element abs(`#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")`)^2, as shown in formula (13) of sec. 21.1 of ref.[1].
c. 

The Feynman diagrams (drawings) related to the different terms of the expansion of S or of its matrix elements `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")`.
  

Interaction Lagrangians involving derivatives of fields, typically appearing in non-Abelian gauge theories, are also handled, and several options are provided enabling restricting the outcome in different ways, regarding the incoming and outgoing particles, the number of loops, vertices or external legs, the propagators and normal products, or whether to compute tadpoles and 1-particle reducible terms.

 

Examples

 

For illustration purposes set three coordinate systems , and set phi to represent a quantum operator

> 

with(Physics)

> 

Setup(mathematicalnotation = true, coordinates = [X, Y, Z], quantumoperators = phi)

`Systems of spacetime coordinates are:`*{X = (x1, x2, x3, x4), Y = (y1, y2, y3, y4), Z = (z1, z2, z3, z4)}

 

_______________________________________________________

 

[coordinatesystems = {X, Y, Z}, mathematicalnotation = true, quantumoperators = {phi}]

(1.1)

Let L be the interaction Lagrangian

> 

L := lambda*phi(X)^4

lambda*Physics:-`^`(phi(X), 4)

(1.2)

The expansion of S in coordinates representation, computed by default up to order = 3 (you can change that using the option order = n), by definition containing all possible configurations of external legs, displaying the related Feynman Diagrams, is given by

> 

%eval(S, `=`(order, 3)) = FeynmanDiagrams(L, diagrams)

 

 

 

%eval(S, order = 3) = 1+%FeynmanIntegral(lambda*_GF(_NP(phi(X), phi(X), phi(X), phi(X))), [[X]])+%FeynmanIntegral(16*lambda^2*_GF(_NP(phi(X), phi(X), phi(X), phi(Y), phi(Y), phi(Y)), [[phi(X), phi(Y)]])+96*lambda^2*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Y)]])+72*lambda^2*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(1728*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Z)], [phi(X), phi(Y)], [phi(Z), phi(Y)]])+2592*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Z), phi(Y)], [phi(Z), phi(Y)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+576*lambda^3*_GF(_NP(phi(X), phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z), phi(Z)), [[phi(X), phi(Y)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])

 (1.3)


The expansion of S  in coordinates representation to a specific order shows in a compact way the topology of the underlying Feynman diagrams. Each integral is represented with a new command, FeynmanIntegral , that works both in coordinates and momentum representation. To each term of the integrands corresponds a diagram, and the correspondence is always clear from the symmetry factors.

In a typical situation, one wants to compute a specific term, or scattering process, instead of the S matrix up to some order with all possible configurations of external legs. For example, to compute only the terms of this result that correspond to diagrams with 1 loop use numberofloops = 1 (for tree-level, use numerofloops = 0)

> 

%eval(S, [`=`(order, 3), `=`(loops, 1)]) = FeynmanDiagrams(L, numberofloops = 1, diagrams)

%eval(S, [order = 3, loops = 1]) = %FeynmanIntegral(72*lambda^2*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(1728*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Z)], [phi(X), phi(Y)], [phi(Z), phi(Y)]]), [[X], [Y], [Z]])

 (1.4)


In the result above there are two terms, with 4 and 6 external legs respectively.

A scattering process with matrix element `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` in momentum representation, corresponding to the term with 4 external legs (symmetry factor = 72), could be any process where the total number of incoming + outgoing parties is equal to 4. For example, one with 2 incoming and 2 outgoing particles. The transition probability for that process is given by

> 

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&comma;",mathcolor = "olive"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&comma;",mathcolor = "olive"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&InvisibleTimes;",mathcolor = "olive")),open = "&lang;",close = "&rang;")` = FeynmanDiagrams(L, incomingparticles = [phi, phi], outgoingparticles = [phi, phi], numberofloops = 1, diagrams)

 

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&comma;",mathcolor = "olive"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&comma;",mathcolor = "olive"),mi("&phi;",fontstyle = "normal",mathcolor = "olive"),mo("&InvisibleTimes;",mathcolor = "olive")),open = "&lang;",close = "&rang;")` = %FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-P__2-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__3-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__4-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])

 (1.5)

When computing in momentum representation, only the topology of the corresponding Feynman diagrams is shown (i.e. the diagrams associated to the corresponding Feynman integral in coordinates representation).

The transition matrix element `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` is related to the transition probability density dw (formula (13) of sec. 21.1 of ref.[1]) by

dw = (2*Pi)^(3*s-4)*n__1*`...`*n__s*abs(F(p[i], p[f]))^2*delta(sum(p[i], i = 1 .. s)-(sum(p[f], f = 1 .. r)))*` d `^3*p[1]*` ...`*`d `^3*p[r]

where n__1*`...`*n__s represent the particle densities of each of the s particles in the initial state `#mfenced(mrow(mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&verbar;",close = "&rang;")`, the delta (Dirac) is the expected singular factor due to the conservation of the energy-momentum and the amplitude F(p[i], p[f])is related to `#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` via

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` = F(p[i], p[f])*delta(sum(p[i], i = 1 .. s)-(sum(p[f], f = 1 .. r)))

To directly get the probability density dw instead of`#mfenced(mrow(mo("&InvisibleTimes;"),mi("f"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("i"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")`use the option output = probabilitydensity

> 

FeynmanDiagrams(L, incomingparticles = [phi, phi], outgoingparticles = [phi, phi], numberofloops = 1, output = probabilitydensity)

Physics:-FeynmanDiagrams:-ProbabilityDensity(4*Pi^2*%mul(n[i], i = 1 .. 2)*abs(F)^2*Dirac(-P__3-P__4+P__1+P__2)*%mul(dP_[f]^3, f = 1 .. 2), F = %FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-P__2-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__3-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__4-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]))

 (1.6)

In practice, the most common computations involve processes with 2 or 4 external legs. To restrict the expansion of the scattering matrix in coordinates representation to that kind of processes use the numberofexternallegs option. For example, the following computes the expansion of S up to order = 3, restricting the outcome to the terms corresponding to diagrams with only 2 external legs

> 

%eval(S, [`=`(order, 3), `=`(legs, 2)]) = FeynmanDiagrams(L, numberofexternallegs = 2, diagrams)

%eval(S, [order = 3, legs = 2]) = %FeynmanIntegral(96*lambda^2*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])

 (1.7)


This result shows two Feynman integrals, with respectively 2 and 3 loops, the second integral with two terms. The transition probability density in momentum representation for a process related to the first integral (1 term with symmetry factor = 96) is then

> 

FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 2, diagrams, output = probabilitydensity)

Physics:-FeynmanDiagrams:-ProbabilityDensity((1/2)*%mul(n[i], i = 1 .. 1)*abs(F)^2*Dirac(-P__2+P__1)*%mul(dP_[f]^3, f = 1 .. 1)/Pi, F = %FeynmanIntegral(%FeynmanIntegral(((3/8)*I)*lambda^2/(Pi^7*(E__1*E__2)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-p__2-p__3)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]), [[p__3]]))

 (1.8)

In the above, for readability, the contracted spacetime indices in the square of momenta entering the amplitude F (as denominators of propagators) are implicit. To make those indices explicit, use the option putindicesinsquareofmomentum

> 

F = FeynmanDiagrams(L, incoming = [phi], outgoing = [phi], numberofloops = 2, indices)

`* Partial match of '`*indices*`' against keyword '`*putindicesinsquareofmomentum*`' `

 

F = %FeynmanIntegral(%FeynmanIntegral(((3/8)*I)*lambda^2*Dirac(-P__2[`~kappa`]+P__1[`~kappa`])/(Pi^7*(E__1*E__2)^(1/2)*(p__2[mu]*p__2[`~mu`]-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__3[nu]*p__3[`~nu`]-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1[beta]-p__2[beta]-p__3[beta])*(-P__1[`~beta`]-p__2[`~beta`]-p__3[`~beta`])-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]), [[p__3]])

 (1.9)

This computation can also be performed to higher orders. For example, with 3 loops, in coordinates and momentum representations, corresponding to the other two terms and diagrams in (1.7)

> 

%eval(S[3], [`=`(legs, 2), `=`(loops, 3)]) = FeynmanDiagrams(L, legs = 2, loops = 3)

`* Partial match of '`*legs*`' against keyword '`*numberoflegs*`' `

 

`* Partial match of '`*loops*`' against keyword '`*numberofloops*`' `

 

%eval(S[3], [legs = 2, loops = 3]) = %FeynmanIntegral(3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])

 (1.10)

A corresponding S-matrix element in momentum representation:

> 

%eval(%Bracket(phi, S[3], phi), `=`(loops, 3)) = FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 3)

%eval(%Bracket(phi, S[3], phi), loops = 3) = %FeynmanIntegral(%FeynmanIntegral(%FeynmanIntegral((9/32)*lambda^3*Dirac(-P__2+P__1)/(Pi^11*(E__1*E__2)^(1/2)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__3-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__2+p__3+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__3]]), [[p__4]]), [[p__5]])+2*%FeynmanIntegral(%FeynmanIntegral(%FeynmanIntegral((9/32)*lambda^3*Dirac(-P__2+P__1)/(Pi^11*(E__1*E__2)^(1/2)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__3-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__3]]), [[p__4]]), [[p__5]])+%FeynmanIntegral(%FeynmanIntegral((1/2048)*lambda*Dirac(-P__2+P__1)*%FeynmanIntegral(576*lambda^2/((p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__2-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])/(Pi^11*(E__1*E__2)^(1/2)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__4]]), [[p__5]])

 (1.11)

Consider the interaction Lagrangian of Quantum Electrodynamics (QED). To formulate this problem on the worksheet, start defining the vector field A[mu].

> 

Define(A[mu])

`Defined objects with tensor properties`

 

{A[mu], Physics:-Dgamma[mu], P__1[mu], P__2[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], p__1[mu], p__2[mu], p__3[mu], p__4[mu], p__5[mu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X), Physics:-SpaceTimeVector[mu](Y), Physics:-SpaceTimeVector[mu](Z)}

(1.12)

Set lowercase Latin letters from i to s to represent spinor indices (you can change this setting according to your preference, see Setup ), also the (anticommutative) spinor field will be represented by psi, so set psi as an anticommutativeprefix, and set A and psi as quantum operators

> 

Setup(spinorindices = lowercaselatin_is, anticommutativeprefix = psi, op = {A, psi})

`* Partial match of '`*op*`' against keyword '`*quantumoperators*`' `

 

_______________________________________________________

 

[anticommutativeprefix = {psi}, quantumoperators = {A, phi, psi}, spinorindices = lowercaselatin_is]

(1.13)

The matrix indices of the Dirac matrices  are written explicitly and use conjugate  to represent the Dirac conjugate conjugate(psi[j])

> 

L__QED := alpha*conjugate(psi[j](X))*Dgamma[mu][j, k]*psi[k](X)*A[mu](X)

alpha*Physics:-`*`(conjugate(psi[j](X)), psi[k](X), A[mu](X))*Physics:-Dgamma[`~mu`][j, k]

(1.14)

Compute S[2], only the terms with 4 external legs, and display the diagrams: all the corresponding graphs have no loops

> 

%eval(S[2], `=`(legs, 4)) = FeynmanDiagrams(L__QED, numberofvertices = 2, numberoflegs = 4, diagrams)

%eval(S[2], legs = 4) = %FeynmanIntegral(-2*alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(psi[k](X), A[mu](X), conjugate(psi[i](Y)), A[alpha](Y)), [[psi[l](Y), conjugate(psi[j](X))]])+alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(conjugate(psi[j](X)), psi[k](X), conjugate(psi[i](Y)), psi[l](Y)), [[A[mu](X), A[alpha](Y)]]), [[X], [Y]])

 (1.15)

The same computation but with only 2 external legs results in the diagrams with 1 loop that correspond to the self-energy of the electron and the photon (page 218 of ref.[1])

> 

%eval(S[2], `=`(legs, 2)) = FeynmanDiagrams(L__QED, numberofvertices = 2, numberoflegs = 2, diagrams)

 

 

%eval(S[2], legs = 2) = %FeynmanIntegral(-2*alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(psi[k](X), conjugate(psi[i](Y))), [[A[mu](X), A[alpha](Y)], [psi[l](Y), conjugate(psi[j](X))]])-alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(A[mu](X), A[alpha](Y)), [[psi[l](Y), conjugate(psi[j](X))], [psi[k](X), conjugate(psi[i](Y))]]), [[X], [Y]])

 (1.16)

where the diagram with two spinor legs is the electron self-energy. To restrict the output furthermore, for example getting only the self-energy of the photon, you can specify the normal products you want:

> 

%eval(S[2], [`=`(legs, 2), `=`(products, _NP(A, A))]) = FeynmanDiagrams(L__QED, numberofvertices = 2, numberoflegs = 2, normalproduct = _NP(A, A))

`* Partial match of '`*normalproduct*`' against keyword '`*normalproducts*`' `

 

%eval(S[2], [legs = 2, products = _NP(A, A)]) = %FeynmanIntegral(alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(A[mu](X), A[alpha](Y)), [[conjugate(psi[j](X)), psi[l](Y)], [psi[k](X), conjugate(psi[i](Y))]]), [[X], [Y]])

 (1.17)

The corresponding S-matrix elements in momentum representation

> 

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("&psi;",fontstyle = "normal"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("&psi;",fontstyle = "normal"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` = FeynmanDiagrams(L__QED, incomingparticles = [psi], outgoing = [psi], numberofloops = 1, diagrams)

 

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("&psi;",fontstyle = "normal"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("&psi;",fontstyle = "normal"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` = -%FeynmanIntegral((1/8)*Physics:-FeynmanDiagrams:-Uspinor[psi][i](P__1_)*conjugate(Physics:-FeynmanDiagrams:-Uspinor[psi][l](P__2_))*(-Physics:-g_[alpha, nu]+p__2[nu]*p__2[alpha]/m__A^2)*alpha^2*Physics:-Dgamma[`~alpha`][l, m]*Physics:-Dgamma[`~nu`][n, i]*((P__1[beta]+p__2[beta])*Physics:-Dgamma[`~beta`][m, n]+m__psi*Physics:-KroneckerDelta[m, n])*Dirac(-P__2+P__1)/(Pi^3*(p__2^2-m__A^2+I*Physics:-FeynmanDiagrams:-epsilon)*((P__1+p__2)^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])

 (1.18)


In this result we see u[psi] spinor (see ref.[2]), and the propagator of the field A[mu] with a mass m[A]. To indicate that this field is massless use

> 

Setup(massless = A)

`* Partial match of '`*massless*`' against keyword '`*masslessfields*`' `

 

_______________________________________________________

 

[masslessfields = {A}]

(1.19)

Now the propagator for A[mu] is the one of a massless vector field:

> 

FeynmanDiagrams(L__QED, incoming = [psi], outgoing = [psi], numberofloops = 1)

-%FeynmanIntegral(-(1/8)*Physics:-FeynmanDiagrams:-Uspinor[psi][i](P__1_)*conjugate(Physics:-FeynmanDiagrams:-Uspinor[psi][l](P__2_))*Physics:-g_[alpha, nu]*alpha^2*Physics:-Dgamma[`~alpha`][l, m]*Physics:-Dgamma[`~nu`][n, i]*((P__1[beta]+p__2[beta])*Physics:-Dgamma[`~beta`][m, n]+m__psi*Physics:-KroneckerDelta[m, n])*Dirac(-P__2+P__1)/(Pi^3*(p__2^2+I*Physics:-FeynmanDiagrams:-epsilon)*((P__1+p__2)^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])

(1.20)

The self-energy of the photon:

> 

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("A"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("A"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` = FeynmanDiagrams(L__QED, incomingparticles = [A], outgoing = [A], numberofloops = 1)

`#mfenced(mrow(mo("&InvisibleTimes;"),mi("A"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("S"),mo("&InvisibleTimes;"),mo("&verbar;"),mo("&InvisibleTimes;"),mi("A"),mo("&InvisibleTimes;")),open = "&lang;",close = "&rang;")` = -%FeynmanIntegral((1/16)*Physics:-FeynmanDiagrams:-PolarizationVector[A][nu](P__1_)*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[A][alpha](P__2_))*(m__psi*Physics:-KroneckerDelta[l, n]+p__2[beta]*Physics:-Dgamma[`~beta`][l, n])*alpha^2*Physics:-Dgamma[`~alpha`][n, i]*Physics:-Dgamma[`~nu`][m, l]*((P__1[tau]+p__2[tau])*Physics:-Dgamma[`~tau`][i, m]+m__psi*Physics:-KroneckerDelta[i, m])*Dirac(-P__2+P__1)/(Pi^3*(E__1*E__2)^(1/2)*(p__2^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((P__1+p__2)^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])

 (1.21)

where epsilon[A] is the corresponding polarization vector.

When working with non-Abelian gauge fields, the interaction Lagrangian involves derivatives. FeynmanDiagrams  can handle that kind of interaction in momentum representation. Consider for instance a Yang-Mills theory with a massless field B[mu, a] where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as follows

> 

Setup(su2indices = lowercaselatin_ah, massless = B, op = B)

`* Partial match of '`*massless*`' against keyword '`*masslessfields*`' `

 

`* Partial match of '`*op*`' against keyword '`*quantumoperators*`' `

 

_______________________________________________________

 

[masslessfields = {A, B}, quantumoperators = {A, B, phi, psi, psi1}, su2indices = lowercaselatin_ah]

(1.22)
> 

Define(B[mu, a], quiet)

> 

F__B[mu, nu, a] := d_[mu](B[nu, a](X))-d_[nu](B[mu, a](X))

Physics:-d_[mu](B[nu, a](X), [X])-Physics:-d_[nu](B[mu, a](X), [X])

(1.23)
> 

L := (1/2)*g*LeviCivita[a, b, c]*F__B[mu, nu, a]*B[mu, b](X)*B[nu, c](X)+(1/4)*g^2*LeviCivita[a, b, c]*LeviCivita[a, e, f]*B[mu, b](X)*B[nu, c](X)*B[mu, e](X)*B[nu, f](X)

(1/2)*g*Physics:-LeviCivita[a, b, c]*Physics:-`*`(Physics:-d_[mu](B[nu, a](X), [X])-Physics:-d_[nu](B[mu, a](X), [X]), B[`~mu`, b](X), B[`~nu`, c](X))+(1/4)*g^2*Physics:-LeviCivita[a, b, c]*Physics:-LeviCivita[a, e, f]*Physics:-`*`(B[mu, b](X), B[nu, c](X), B[`~mu`, e](X), B[`~nu`, f](X))

 (1.24)

The transition probability density at tree-level for a process with two incoming and two outgoing B particles is given by

> 

FeynmanDiagrams(L, incomingparticles = [B, B], outgoingparticles = [B, B], numberofloops = 0, output = probabilitydensity, factor, diagrams)

`* Partial match of '`*factor*`' against keyword '`*factortreelevel*`' `

(1.25)

 

 

Physics:-FeynmanDiagrams:-ProbabilityDensity(4*Pi^2*%mul(n[i], i = 1 .. 2)*abs(F)^2*Dirac(-P__3[`~sigma`]-P__4[`~sigma`]+P__1[`~sigma`]+P__2[`~sigma`])*%mul(dP_[f]^3, f = 1 .. 2), F = (((1/8)*I)*Physics:-LeviCivita[a1, a3, h]*((-P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics:-g_[`~lambda`, `~tau`]+(P__1[`~lambda`]+P__2[`~lambda`]+P__3[`~lambda`])*Physics:-g_[`~kappa`, `~tau`]-Physics:-g_[`~kappa`, `~lambda`]*(P__3[`~tau`]-P__4[`~tau`]))*Physics:-LeviCivita[a2, d, g]*((P__1[`~beta`]+(1/2)*P__2[`~beta`])*Physics:-g_[`~alpha`, `~sigma`]+(-(1/2)*P__1[`~sigma`]+(1/2)*P__2[`~sigma`])*Physics:-g_[`~alpha`, `~beta`]-(1/2)*Physics:-g_[`~beta`, `~sigma`]*(P__1[`~alpha`]+2*P__2[`~alpha`]))*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]-P__2[chi])*(-P__1[`~chi`]-P__2[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*((-P__1[`~beta`]+P__3[`~beta`]-P__4[`~beta`])*Physics:-g_[`~lambda`, `~tau`]+(P__1[`~lambda`]-P__2[`~lambda`]-P__3[`~lambda`])*Physics:-g_[`~beta`, `~tau`]+Physics:-g_[`~beta`, `~lambda`]*(P__2[`~tau`]+P__4[`~tau`]))*Physics:-LeviCivita[a1, a3, g]*((P__1[`~sigma`]+P__3[`~sigma`])*Physics:-g_[`~alpha`, `~kappa`]+(-2*P__1[`~kappa`]+P__3[`~kappa`])*Physics:-g_[`~alpha`, `~sigma`]+Physics:-g_[`~kappa`, `~sigma`]*(P__1[`~alpha`]-2*P__3[`~alpha`]))*Physics:-LeviCivita[a2, d, h]*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]+P__3[chi])*(-P__1[`~chi`]+P__3[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*((-P__1[`~beta`]-P__3[`~beta`]+P__4[`~beta`])*Physics:-g_[`~kappa`, `~tau`]+(P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics:-g_[`~beta`, `~tau`]+Physics:-g_[`~beta`, `~kappa`]*(P__2[`~tau`]+P__3[`~tau`]))*Physics:-LeviCivita[a3, g, h]*((P__1[`~sigma`]+P__4[`~sigma`])*Physics:-g_[`~alpha`, `~lambda`]+(P__1[`~alpha`]-2*P__4[`~alpha`])*Physics:-g_[`~lambda`, `~sigma`]-2*Physics:-g_[`~alpha`, `~sigma`]*(P__1[`~lambda`]-(1/2)*P__4[`~lambda`]))*Physics:-LeviCivita[a1, a2, d]*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]+P__4[chi])*(-P__1[`~chi`]+P__4[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*(Physics:-KroneckerDelta[g, h]*Physics:-KroneckerDelta[a1, d]*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]+Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`]-2*Physics:-g_[`~alpha`, `~lambda`]*Physics:-g_[`~beta`, `~kappa`])+Physics:-KroneckerDelta[d, h]*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]-2*Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`]+Physics:-g_[`~alpha`, `~lambda`]*Physics:-g_[`~beta`, `~kappa`])*Physics:-KroneckerDelta[a1, g]-2*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]-(1/2)*Physics:-g_[`~beta`, `~kappa`]*Physics:-g_[`~alpha`, `~lambda`]-(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`])*Physics:-KroneckerDelta[d, g]*Physics:-KroneckerDelta[a1, h]))*g^2*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[B][kappa, h](P__3_))*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[B][lambda, a1](P__4_))*Physics:-FeynmanDiagrams:-PolarizationVector[B][alpha, d](P__1_)*Physics:-FeynmanDiagrams:-PolarizationVector[B][beta, g](P__2_)/(Pi^2*(E__1*E__2*E__3*E__4)^(1/2)))

 (1.26)

To simplify the repeated indices, us the option simplifytensorindices. To check the indices entering a result like this one use Check ; there are no free indices, and regarding the repeated indices:

> 

Check(Physics[FeynmanDiagrams]:-ProbabilityDensity(4*Pi^2*%mul(n[i], i = 1 .. 2)*abs(F)^2*Dirac(-P__3[`~sigma`]-P__4[`~sigma`]+P__1[`~sigma`]+P__2[`~sigma`])*%mul(dP_[f]^3, f = 1 .. 2), F = (((1/8)*I)*Physics[LeviCivita][a1, a3, h]*((-P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics[g_][`~lambda`, `~tau`]+(P__1[`~lambda`]+P__2[`~lambda`]+P__3[`~lambda`])*Physics[g_][`~kappa`, `~tau`]-Physics[g_][`~kappa`, `~lambda`]*(P__3[`~tau`]-P__4[`~tau`]))*Physics[LeviCivita][a2, d, g]*((P__1[`~beta`]+(1/2)*P__2[`~beta`])*Physics[g_][`~alpha`, `~sigma`]+(-(1/2)*P__1[`~sigma`]+(1/2)*P__2[`~sigma`])*Physics[g_][`~alpha`, `~beta`]-(1/2)*Physics[g_][`~beta`, `~sigma`]*(P__1[`~alpha`]+2*P__2[`~alpha`]))*Physics[g_][sigma, tau]*Physics[KroneckerDelta][a2, a3]/((-P__1[chi]-P__2[chi])*(-P__1[`~chi`]-P__2[`~chi`])+I*Physics[FeynmanDiagrams]:-epsilon)-((1/16)*I)*((-P__1[`~beta`]+P__3[`~beta`]-P__4[`~beta`])*Physics[g_][`~lambda`, `~tau`]+(P__1[`~lambda`]-P__2[`~lambda`]-P__3[`~lambda`])*Physics[g_][`~beta`, `~tau`]+Physics[g_][`~beta`, `~lambda`]*(P__2[`~tau`]+P__4[`~tau`]))*Physics[LeviCivita][a1, a3, g]*((P__1[`~sigma`]+P__3[`~sigma`])*Physics[g_][`~alpha`, `~kappa`]+(-2*P__1[`~kappa`]+P__3[`~kappa`])*Physics[g_][`~alpha`, `~sigma`]+Physics[g_][`~kappa`, `~sigma`]*(P__1[`~alpha`]-2*P__3[`~alpha`]))*Physics[LeviCivita][a2, d, h]*Physics[g_][sigma, tau]*Physics[KroneckerDelta][a2, a3]/((-P__1[chi]+P__3[chi])*(-P__1[`~chi`]+P__3[`~chi`])+I*Physics[FeynmanDiagrams]:-epsilon)-((1/16)*I)*((-P__1[`~beta`]-P__3[`~beta`]+P__4[`~beta`])*Physics[g_][`~kappa`, `~tau`]+(P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics[g_][`~beta`, `~tau`]+Physics[g_][`~beta`, `~kappa`]*(P__2[`~tau`]+P__3[`~tau`]))*Physics[LeviCivita][a3, g, h]*((P__1[`~sigma`]+P__4[`~sigma`])*Physics[g_][`~alpha`, `~lambda`]+(P__1[`~alpha`]-2*P__4[`~alpha`])*Physics[g_][`~lambda`, `~sigma`]-2*Physics[g_][`~alpha`, `~sigma`]*(P__1[`~lambda`]-(1/2)*P__4[`~lambda`]))*Physics[LeviCivita][a1, a2, d]*Physics[g_][sigma, tau]*Physics[KroneckerDelta][a2, a3]/((-P__1[chi]+P__4[chi])*(-P__1[`~chi`]+P__4[`~chi`])+I*Physics[FeynmanDiagrams]:-epsilon)-((1/16)*I)*(Physics[KroneckerDelta][g, h]*Physics[KroneckerDelta][a1, d]*(Physics[g_][`~alpha`, `~beta`]*Physics[g_][`~kappa`, `~lambda`]+Physics[g_][`~alpha`, `~kappa`]*Physics[g_][`~beta`, `~lambda`]-2*Physics[g_][`~alpha`, `~lambda`]*Physics[g_][`~beta`, `~kappa`])+Physics[KroneckerDelta][d, h]*(Physics[g_][`~alpha`, `~beta`]*Physics[g_][`~kappa`, `~lambda`]-2*Physics[g_][`~alpha`, `~kappa`]*Physics[g_][`~beta`, `~lambda`]+Physics[g_][`~alpha`, `~lambda`]*Physics[g_][`~beta`, `~kappa`])*Physics[KroneckerDelta][a1, g]-2*(Physics[g_][`~alpha`, `~beta`]*Physics[g_][`~kappa`, `~lambda`]-(1/2)*Physics[g_][`~alpha`, `~lambda`]*Physics[g_][`~beta`, `~kappa`]-(1/2)*Physics[g_][`~alpha`, `~kappa`]*Physics[g_][`~beta`, `~lambda`])*Physics[KroneckerDelta][d, g]*Physics[KroneckerDelta][a1, h]))*g^2*conjugate(Physics[FeynmanDiagrams]:-PolarizationVector[B][kappa, h](P__3_))*conjugate(Physics[FeynmanDiagrams]:-PolarizationVector[B][lambda, a1](P__4_))*Physics[FeynmanDiagrams]:-PolarizationVector[B][alpha, d](P__1_)*Physics[FeynmanDiagrams]:-PolarizationVector[B][beta, g](P__2_)/(Pi^2*(E__1*E__2*E__3*E__4)^(1/2))), all)

`The repeated indices per term are: `[{`...`}, {`...`}, `...`]*`, the free indices are: `*{`...`}

 

[{a1, a2, a3, alpha, beta, chi, d, g, h, kappa, lambda, sigma, tau}], {}

(1.27)


This process can be computed with 1 or more loops, in which case the number of terms increases significantly. As another interesting non-Abelian model, consider the interaction Lagrangian of the electro-weak part of the Standard Model

> 

Coordinates(clear, Z)

`Unaliasing `*{Z}*` previously defined as a system of spacetime coordinates`

(1.28)
> 

Setup(quantumoperators = {W, Z})

[quantumoperators = {A, B, W, Z, phi, psi, psi1}]

(1.29)
> 

Define(W[mu], Z[mu])

`Defined objects with tensor properties`

 

{A[mu], B[mu, a], Physics:-Dgamma[mu], P__1[mu], P__2[mu], P__3[alpha], P__4[alpha], Physics:-Psigma[mu], W[mu], Z[mu], Physics:-d_[mu], Physics:-g_[mu, nu], p__1[mu], p__2[mu], p__3[mu], p__4[mu], p__5[mu], psi[j], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X), Physics:-SpaceTimeVector[mu](Y)}

(1.30)
> 

CompactDisplay((W, Z)(X))

` W`(X)*`will now be displayed as`*W

 

` Z`(X)*`will now be displayed as`*Z

(1.31)
> 

F__W[mu, nu] := d_[mu](W[nu](X))-d_[nu](W[mu](X))

Physics:-d_[mu](W[nu](X), [X])-Physics:-d_[nu](W[mu](X), [X])

(1.32)
> 

F__Z[mu, nu] := d_[mu](Z[nu](X))-d_[nu](Z[mu](X))

Physics:-d_[mu](Z[nu](X), [X])-Physics:-d_[nu](Z[mu](X), [X])

(1.33)
> 

L__WZ := I*g*cos(`&theta;__w`)*((Dagger(F__W[mu, nu])*W[mu](X)-Dagger(W[mu](X))*F__W[mu, nu])*Z[nu](X)+W[nu](X)*Dagger(W[mu](X))*F__Z[mu, nu])

I*g*cos(theta__w)*(Physics:-`*`(Physics:-`*`(Physics:-d_[mu](Physics:-Dagger(W[nu](X)), [X])-Physics:-d_[nu](Physics:-Dagger(W[mu](X)), [X]), W[`~mu`](X))-Physics:-`*`(Physics:-Dagger(W[mu](X)), Physics:-d_[`~mu`](W[nu](X), [X])-Physics:-d_[nu](W[`~mu`](X), [X])), Z[`~nu`](X))+Physics:-`*`(W[nu](X), Physics:-Dagger(W[mu](X)), Physics:-d_[`~mu`](Z[`~nu`](X), [X])-Physics:-d_[`~nu`](Z[`~mu`](X), [X])))

(1.34)

This interaction Lagrangian contains six different terms. The S-matrix element for the tree-level process with two incoming and two outgoing W particles is shown in the help page for FeynmanDiagrams .

> 

NULL

References

 

[1] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.

[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.

 

FeynmanDiagrams_and_the_Scattering_Matrix.PDF

FeynmanDiagrams_and_the_Scattering_Matrix.mw

FeynmanDiagrams_-_help_page.mw


Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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