Torre

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These are answers submitted by Torre

Unless I am misreading your expression, the integrand is a constant (assuming of course that all your parameters are real, as would be the case in a quantum mechanical situation).  

The integral of a constant over [0, infinity] is infinite. 

Such an integral shouldn't appear in a quantum mechanical analysis - what is it you are trying to do?

I don't have a spreadsheet, but I made a worksheet. The woksheet, attached below, contains a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. It's a little rough around the edges, but hopefully it gets you started. Of course, any decent textbook will derive such a result. My analysis aligns with that found in the old text "Introduction to General Relativity", by Adler, Bazin and Schiffer.  The worksheet was created with Maple 17 (but should work with earlier versions) and uses the DifferentialGeometry and DifferentialGeometry:-Tensor packages.

Perihelion.mw

If you have Maple 17, one implementation of quaternions can be found in the package LieAlgebras, which is a sub-package of DifferentialGeometry.  Look at the help page for the command AlgebraLibraryData to see an example.  See also the document

How To Create The Quaternion & Octonion Algebras

for more information.

 

CT

 

Depending upon your background and specific needs, another option is the DifferentialGeometry package.  See the help page for details.

 

Charles Torre

If you use the DifferentialGeometry package one method for solving tensor equations is via the command DGsolve.  See the DifferentialGeometry, DGsolve help page for details.

By the way, there are various ways to convert lists (of lists) of expressions into tensors of various types within the DifferentialGeometry package (and its sub-package Tensor - capital T).  For instance, look at the DifferentialGeometry, Convert help page and see the DifferentialGeometry, DGzip help page.

One way to do this uses the functionality provided by the DifferentialGemetry package and its sub-package, Tensor.

In Maple help, search on "CovariantDerivative" for instructions/examples on how to compute the covariant derivative on a manifold with respect to a given connection.  Search on "Christoffel" for instructions/examples on how to compute the connection defined by a given metric. 

charlie

 

 

The problem arises because you are using different functions in the metric and in the tetrad.  In the metric you have u(r) and v(r). In the tetrad you have u(r, t) and v(r,t).  I investigated this using the command TensorInnerProduct(g, NT, NT).

How are you trying to calculate the spin coefficients?  Are you using the DifferentialGeometry package (and its sub-package Tensor)?  If not, that should be the package to use.

After loading DifferentialGeometry and then Tensor (capital T), you need to define your null tetrad as a list of vector fields. Then you use the command NPSpinCoefficients to compute all the usual spin coefficients of the given null tetrad. Look at the NPSpinCoefficients help page for details and examples.   

Let me know if this doesn't help, or if you need further assistance.

charlie

 

[Converted from Reply to Answer.---Carl Love as a Moderator]

If you are using the DifferentialGeometry package and its sub-package Tensor...

Define the scalar phi, the metric, its Christoffel symbol and Ricci tensor, and the second covariant derivative of phi.

Use the command ContractIndices to make the desired scalar. 

Alternatively, use the command TensorInnerProduct applied to the Ricci tensor and the second derivative of phi.

What sort of computations are you trying to do?

Maple 13 has a pretty extensive suite of tensor analysis commands in its built-in DifferentialGeometry package.   Subsequent releases of Maple have even more functionality.

Without seeing the computation it is hard to give advice. Here are some random comments.

 

The case {a=a, b=b} should give the same output as without the "parameters" keyword.

 

Is it possible that the result of the Lie derivative IS zero, but needs simpification?

 

You don't need the "map" command when taking the Lie Derivative. If you take the Lie derivative along a list of vectors, the command knows to compute a list of Lie derivatives.

 

charlie

s1 := solve(x^2+ 3*x + 2 >0, {x});

s2 := solve(x^2 - 9>=0, {x});

s3 := s1 union s2;

Try simplify( * , symbolic)

I just tried simplify(Z, symbolic) and got zero.

This worked with or without the assume you had in your worksheet.

Maple 15.01. 

charlie

BTW,  I think the commands you should have used were:

 

with(DifferentialGeometry):


DGsetup([x,y],E2):

w := ExteriorDerivative(x^2);

 

Hook(D_x, w);

 

charlie

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