Exercise 3.04 (with Spherical coordinates)

This is a model for doing the real exercise 3.4 which was on paraboloidal coordinates ##\left(u,v,\phi\right)##. I did it because I know, or can find, the answers and so I can check my methods. There is very little information on  paraboloidal coordinates that I can find😖. I was particularly hazy on how to calculate the basis vectors. Now I know! It appears that basis one-forms point in the same direction as basis vectors and the ##r## basis one-form is the same as the vector. I probably should think about that harder. I also had quite a lot of problems with signs on terms such as ##\sqrt{x^2+y^2}##. Inattentive use of https://www.derivative-calculator.net/ gave wrong answers.

Hopefully I will very soon complete the nasty real question.

Question

In Euclidean three-space we can define spherical coordinates ##\left(r,\theta,\phi\right)## via
\begin{align}
x=r\sin{\theta}\cos{\phi}&\phantom {10000}(1)\nonumber\\
y=r\sin{\theta}\sin{\phi}&\phantom {10000}(2)\nonumber\\
z=r\cos{\theta}&\phantom {10000}(3)\nonumber
\end{align}Note: we are using the 'physicist' convention as shown on the video. ## \theta## is the colatitude (polar) and ## \phi## the longitude (azimuthal).

(a) Find the coordinate transformation matrix between paraboloidal and Cartesian coordinates ##\frac{\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##.

Full details at Ex 3.04 Spherical coordinates.pdf. 7 pages including link to Excel file for generating the video.