Bill Goodwine's Courses

Unless otherwise indicated, the problems are from the course text, pages 46 and 47.

1. 1 (a) and (b)
2. 8
3. 10
4. 13
5. 16
6. 17
7. 18
8. Consider the transformation between rectangular Cartesian coordinates and spherical coordinates given by
   • 
   with inverse given by
   • 
   Compute the components of the Euclidean metric tensor.

I do not believe that you need the inverse transformation to do this problem. To keep things straight, think of the \xi's as the Cartesian coordinates and the x's as the elliptic coordinates.

So, all you need are the Christoffel symbols as a function of the x's (since you want the expression in elliptic coordinates). So, the d \xi/dx terms are naturally in the correct coordinates. The dx/d \xi terms aren't. However, remember that the "Jacobain of the inverse transformation" is the "inverse of the Jacobain" and that the Jacobain will naturally be in the coordinates you want, i.e., in terms of the x's. So, compute the Jacobain and invert it, and the elements of the matrix will be the terms you need and will be a function of the correct variables.

Note: the Jacobain is perhaps messy enough that you may want to use a computer algebra program (like Mathematica) to compute it.