## 45 Reputation

0 years, 261 days

## Expanding a derivative onto an expressio...

Maple 2020

Hello I am trying to compute the following expression;

(x^2r+y^2+z^2) gamma^{0} partial_{x} (-xy(1-r)) gamma^{2} partial_{y} \Psi

in which r = sqrt(1-K(x^2+y^2+z^2)), K is a constant, the gamma's are the Dirac gamma matrices and \Psi is the wave function. Essentially I would like to have partial_{x} operate on the expression to the right of it and apply product rule where there will be three terms with the last one being partia_{x} partial_{y} \Psi where it evaluates on everything expect the partial and the wavefunction, combining into the term partial_{x} partial_{y} \Psi, and if the partials are the same they combine into a single second order derivative with respect to the given variable. I have done this calculation by hand but seeing that there is an extensive amount of terms I would like to automate it using Maple. I have tried using the D() calling sequence but it doesn't quite give me what I am looking for. Is there anything documentation which will give me what I am looking for?

## Trouble Simplifying expressions in Maple...

Maple 2019

Hello, I am having a bit of difficulty simplifying some calculations in Maple 2019. In short, in order to verify that the tensors that I am trying to use are indeed inverses of each other, I am simply trying to multiply component wise, for example the tensor component e[2,~2] with the tensor component f[~2,2], since they are essentially inverses of each other, i.e. the matrix defining f is actually the inverse of the matrix e, i.e. f=e^(-1), should give back 1 as an answer. Nonetheless, when I attempt to take this simple multiplication Maple does not reduce it, but rather just gives multiplies the terms with no simplification. Is there anything I can do so that Maple may simplify its calculations? I have already tried the "eval" calling sequence but that didn't do the trick, and I fear that when escalating the calculations I will get a bunch of long expressions rather than concise solutions. Thank you for your help in advance,

Christoffel_symbols_of_de_Sitter_metric_research.mw

## Defining a new tensor using the inverse ...

Maple 2019

I need to take the inverse of a tensor which I have denoted as e[~mu,nu] which is defined by a rather larger Matrix. I had computed this matrix using Mathematica and then simply transferred the resulting matrix by using the calling sequence

with(MmaTranslator):

which worked swell for transferring said matrix in to Maple. Then using the Physics package I was able to define it as a tensor, with a contravariant and covariant index, respectively. Now, when trying to transfer the inverse of said matrix into Maple to define as a new tensor which I intend to call f[mu,~nu], I get an error saying that the number of free indices on the left hand side does not coincide with the number of free indices on the right hand side. Since, this "new" tensor will really just be the inverse of the matrix which I used to define e[~mu,nu], I was wondering if there was any way in which I can simply compute the inverse of the matrix defining e[~mu,nu] in Maple and then let it be equivalent to f[mu,~nu], afterwhich I would then define it as a new tensor itself.

Research_project.mw

Any help would be greatly appreciated.

## Changing the coordinate system in the Ph...

Maple 2019

I am trying to write a metric in Maple 2019, using the Physics package in Cartesian coordinates. I know that there is already a pre-installed way in which to set the coordinates to Cartesian, namely, if we have something like

Setup(coordinates=cartesian);

or similarly, using the Coordiantes calling sequence, such that we may define something like

Coordinates(A=cartesian);Setup(coordinates=A);

yet, if I am not mistaken, this gives the coordinates in the following form: {x,y,z,t}, yet I seek to define the metric in the following, perhaps more arbitrary coordinate system, {x1,x2,x3,t}. Is there any way to setup the coordinate system to read like the set I just described, such that I can then take a norm of these three spatial coordaintes to be |x|? Any help would be greatly appreciated.

## Maple Inner product for Tensors ...

Maple 2019

I'm fairly new to using Maple and am having a bit of a hard time calculating the following inner product. Firstly, I define the tensors (which to this end I'm not certain they are correctly defined),

```with(Physics):

Setup(mathematicalnotation=true)

Setup(coordinatesystems=spherical):

ds2 := - dt^2 + a(t)^2 /( 1-k*r^2)*dr^2 + a(t)^2*r^2*dtheta^2 + a(t)^2*r^2*sin(theta)^2*dphi^2;
Setup(coordinates = spherical, metric = ds2);

e[mu, `~nu`] = Matrix(4, {(1,1)= a(t)/sqrt(1-k*r^2), (2,2)=a(t)*r, (3,3)=a(t)*r*sin(theta), (4,4)=1}, fill=0); (15)
Define((15))
f[`~mu`, nu] = Matrix(4, {(1,1)=sqrt(1-k*r^2)/(a(t)), (2,2)= 1/(a(t)*r), (3,3)=1/(a(t)*r*sin(theta)), (4,4)=1}, fill=0); (28)
```
```Define((28))
```

Thus, I defined two mixed tensors e[mu, `~nu`] (one covariant and one contravariant index ) and f[`~mu`, nu] (one contravariant and one covariant index).

Then, I try to take the following inner product between the two mixed tensors and the Christoffel symbols of the second kind, namely,

`e[nu, `~alpha`].f[`~sigma`, beta].Christoffel [`~nu`, sigma, mu];`

where I used the Physics['.'] command . However, when I try taking this inner product, it returns unevaluated.

Did I define the mixed tensors incorrectly? Does it matter how you define the indices when you're gonna take the inner product? Because taking the inner product of simply e[mu, `~nu`].f[`~mu`, nu] also returns unevaluated. Also, I should mention that  e[mu, `~nu`] and f[`~mu`, nu] are inverses of each other, is there any way to define one and get the other, since, simply changing the way in which the indices are raised and lowered doesn't take the reciprocal of the components.

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