MaplePrimes - Newest Questions and Posts

How can I get the correct inverse metric with Physics?

Hullzie16 — Tue, 23 Jun 2026 21:44:57 Z

I am looking to do some gravitational perturbations around a generic background spacetime. But before doing that, I wanted to look at just linearized gravity, and make sure all the standard calculations work with Physics before throwing something a little more complicated at it. I went to ?Physics,Library and found the Linearize command, and I thought this was great! When I was reading through it however, I found that the sign infront of the perturbation h in the inverse metric is incorrect. Now, this does not give any invalid results for the Ricci tensor as displayed in the worksheet, since we are only going to linear order, but if we want to go beyond linear order, this will start to cause issues.

Is there a way that Maple can handle this? Or do I have to do some sort of double Define for the metric: one with all downstairs indices, and one with all upstairs indices? If so, how do I do that?

Any help would be greatly appreciated!

>	restart: with(Physics): with(Library):

>	Setup(coordinates = cartesian,signature=`-+++`):

(1)

g_[];

(2)

>	Define(h[mu, nu],symmetric)

(3)

>	Define(eta[mu,nu]=rhs((2)))

(4)

>	g_[mu,nu]=eta[mu,nu]+epsilon*h[mu,nu]

(5)

Lets "define" the inverse metric as it appears from the Library:-Linearize worksheet.

>	g_[~mu,~alpha]=eta[~mu,~alpha]+epsilon*h[~mu,~alpha]

(6)

If we multiply the metric and its inverse together, we should expact that we return the KroneckerDelta by definition -- if we consider only to linear order.

(5)*(6)

(7)

>	expand((7))

(8)

>	Substitute(eta=g_,(8))

(9)

>	subs(epsilon^2=0,(9))

(10)

>	Simplify(%)

(11)

As we can see, we do not get delta alone on the right-hand-side, but instead we still have the perturbation still.

If we instead, use the proper way the inverse should look, which of course comes from the definition of the inverse, it should have minus sign.

>	g_[~mu,~alpha]=eta[~mu,~alpha]-epsilon*h[~mu,~alpha]

(12)

>	subs(epsilon^2=0,expand((5)*(12)))

(13)

>	Simplify(Substitute(eta=g_,(13)))

(14)

Which is the desired result we want. So, my question: is there a way that Maple can produce the correct inverse metric not only to linear order, but to say quadratic, without explicitly deriving it ourselves?

Here is the Physics:-Library(Linearize) Worksheet/Example with some comments

>	restart: with(Physics): with(Library):

>	Setup(coordinates = cartesian);

(1)

The default metric when Physics is loaded is the Minkowski metric, representing a flat (no curvature) spacetime

g_[];

(2)

Suppose you want to define a small perturbation around this metric. For that purpose, define a perturbation tensor , that in the general case depends on the coordinates and is not diagonal, the only requirement is that it is symmetric (to have it diagonal, change symmetric by diagonal; to have it constant, change by )

>	h[mu, nu] = Matrix(4, (i, j) -> delta[i, j](X), shape = symmetric);

(3)

In the above it is understood that are small quantities, so that quadratic or higher powers of it can be approximated to 0 (i.e., discarded). Define the components of accordingly

>	Define((3));

(4)

Define also a tensor representing the unperturbed Minkowski metric

>	eta[mu, nu] = rhs((2));

(5)

>	Define((5));

(6)

The weakly perturbed metric is given by

>	g_[mu, nu] = eta[mu, nu] + h[mu, nu];

(7)

Make this be the definition of the metric

>	Define((7));

(8)

The linearized form of the Ricci tensor is computed by introducing this weakly perturbed metric in the expression of the Ricci tensor as a function of the metric. This can be accomplished in different ways, the simpler being to use the conversion network between tensors, but for illustration purposes, showing steps one at time, a substitution of definitions one into the other one is used

>	Ricci[definition];

(9)

>	Christoffel[~alpha, mu, nu, definition];

(10)

>	Substitute((10), (9));

(11)

Introducing the perturbed metric, and the inert form of Ricci for simplification purposes

>	Substitute((7), Ricci = %Ricci, (11));

(12)

The sign infront of the perturbation in the inverse metric is wrong, it should be minus.

This expression contains several terms quadratic in the small perturbation . The routine to filter out those terms is Linearize, that takes as second argument the symbol representing the small quantities (perturbation)

Lets look at the metric times inverse in this setup

>	g_[mu,nu,definition]*g_[~mu,~alpha,definition]

(13)

>	Linearize((13),h)

(14)

>	Simplify(subs(eta=g_,(14)))

(15)

The result is not correct, left-hand-side does not match right-hand-side, this is because the inverse metric has the wrong. If it were a minus, we would get:

>	g_[mu, nu]g_[~alpha, ~mu] = eta[mu, nu]eta[~alpha, ~mu] - eta[mu, nu]h[~alpha, ~mu] + eta[~alpha, ~mu]h[mu, nu]

(16)

>	Simplify(subs(eta=g_,(16)))

(17)

Which is correct. The continued calculation from the Help page is below.

>	Linearize((12), h);

(18)

In this result, is the flat Minkowski metric. To further simplify this expression using the internal algorithms for a flat metric it is practical to reintroduce representing that Minkowski metric

g_[min];

(19)

Replace in the expression for the Ricci tensor the intermediate Minkowski by

>	subs(eta = g_, (18));

(20)

Simplifying, results in the linearized form of the Ricci tensor

>	Simplify((20));

(21)

This is correct result, because we are going to linear order only the +/- does not have an effect on the end result.

Download LinearizedWorksheet-Comments.mw

Download LinearQuestion.mw

What if my permutation has a fixed point?

wkehowski — Tue, 23 Jun 2026 06:13:03 Z

I am trying to use the Perm command in the GroupTheory package to create permutations. The problem is when the permutation has fixed points. For example, neither of the forms

[[1,4,7],[2,8,5],[3],[6]]

[[1,4,7],[2,8,5],[3,3],[6,6]]

will work. Any suggestions?

repeated equation labels

C_R — Mon, 22 Jun 2026 16:38:53 Z

Has anybody seen something like that? I do not use Maple 2026 very often.

Does this vanish when the document is executed on another machine?

repeated_equation_labels.mw

Update:

expanding the document block by "show command" makes the equation labels disappear.
copying the input to another document block seems to fix the problem

How to get consistent output from dsolve for system of ode's?

nm — Mon, 22 Jun 2026 06:41:21 Z

Consider these two output, both for solving system of 2 first order different equations.

Why is the first result is put in a list, then each solution is in a set inside the list, while the second one is just a set of the two solutions?

My guess is that because the first system is non-linear. Is this why?

This makes it little harder to parse the result later on, as it can change each time.

Is there a way to get same output for the first example as in the second example?

Mapkle 2026.1

ode:=diff(x(t),t) = x(t)^2, diff(y(t),t) = exp(t);
sol:=dsolve([ode],[x(t),y(t)])

ode:=diff(x(t),t) = x(t), diff(y(t),t) = t;
sol:=dsolve([ode],[x(t),y(t)])

ps. the ode's are not even coupled in these example. So each can be solved on its own if needed.

And if there is one ode with multiple solutions, now dsolve returns expression sequence. No set, no list.

ode:=2*x*diff(y(x),x)*diff(diff(y(x),x),x) = -1+diff(y(x),x)^2; 
dsolve(ode,y(x));

This whole thing is a mess.

There should be one consistent way to return solutions for all cases.

Regadless if it is one ode with one solution, or one ode with mutliple solutions, or coupled systems of odes, linear or not and so on.

The output should be the same form in all cases. A list of lists or list of sets or whatever it is decided on.

But it should not change.

what is mistake in this transformation ?

salim-barzani — Sun, 21 Jun 2026 17:51:22 Z

i do same trasnformation but i don't know what is issue the result is near to same but parameter (t) appear in my which that make my ode not be correct so i can't see the problem in here and i am intrested in finding this, regarding to this i will put here my result and the papers result rregarding to resolve the issue

pde1

pde2

T1.mw

Series Solutions of ODEs in Maple Followup

senthooran — Sun, 21 Jun 2026 11:39:05 Z

Little bit of a followup on the "Series Solutions of ODEs in Maple" online seminar.

According to Mathematical Methods for Physicists, 7th Edition by Arfken, Weber and Harris,

Pages 343-345,

Singular points are classified as regular or irregular

Irregular points are called essential singularies.

They show how to apply these to famous differential equations in Quantum Mechanics and other physical applications (examples given in Farlow's Partial Differential Equations for Scientists and Engineers).

In Section 12.1 of Mathematical Methods for Physicists, the complex series Laurent expansion (chapter 11 of the book) is applied to generalized to the complex plane (see Saff and Snider Fundamentals of Complex Analysis for Mathematics, Science and Engineering, 2nd Edition). Not too sure how Maple handles contour integrals though.

It seems that a regular point is the same as a ordinary point, as per Elementary Differential Equations and Boundary Value Problems, 8th Edition by Boyce and DiPrima, Chapter 5.