Friday, 20 September 2019

Exercise 3.04 Paraboloidal coordinates

Question

In Euclidean three-space we can define paraboloidal coordinates ##\left(u,v,\phi\right)## via$$
x=uv\cos{\phi}\ \ y=uv\sin{\phi}\ \ z=\frac{1}{2}\left(u^2-v^2\right)
$$(a) Find the coordinate transformation matrix between paraboloidal and Cartesian coordinates ##{\partial x^\alpha}/{\partial x^{\beta^\prime}}## and the inverse transformation. Are there any singular points in the map?

(b) Find the basis vectors and the basis one-forms in terms of Cartesian basis vectors and forms.

(c) Find the metric and inverse metric in paraboloidal coordinates.

(d) Calculate the Christoffel symbols.

(e) Calculate the divergence ##\nabla_\mu V^\mu## and Laplacian ##\nabla_\mu\nabla^\mu f##.

Overview

I completely screwed up calculation of the metric, getting one that was very non-diagonal and would have been hard work and I had to go back to spherical polar coordinates to see where I had gone wrong, then it all worked except for a sign error in the inverse transformation matrix. It was fairly obvious which component contained the error.

The way that a covariant tensor ends up in the wrong coordinates after using the general tensor transformation law (e.g. at (81)) is very odd and needs further investigation. I think it's just inevitable and would often need transformation back into the correct coordinate system.

I learned a bit more about basis vectors, but this exercise mainly seemed to be teacher-torture with plenty of exercise on differentiation. The Laplacian ##\nabla_\mu\nabla^\mu f## is not in the index of this book. I hope it is discussed somewhere. I spent about half the time developing drawings of the basis vectors and one-forms but sadly the animation broke down. It was a surprise that the coordinate transformation matrix and its inverse not only take tensors back and forth between coordinate systems but are also inverses in the matrix sense. I suppose I should have known.

When calculating the Christoffel symbols, my first shot was 77% correct. I suppose if this was an exam question that would be quite good, except that I have taken too much time!

Answer

See: Ex 3.04 Paraboloidal coordinates.pdf (15 pages, 147 equations)

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