User:Egm6321.f11.team2.rho/HW7

=R*7.4 - Laplacian in elliptic coordinates=

Given
The most common definition of elliptic coordinates $$\displaystyle (\mu, \nu)$$ is

$$ x = a \ \cosh \mu \ \cos \nu $$

$$ y = a \ \sinh \mu \ \sin \nu $$

where $$\displaystyle \mu$$ is a nonnegative real number and $$\displaystyle \nu \in [0, 2\pi].$$

Find
Verify the Laplacian in elliptic coordinates given by

Solution
Elliptical coordinates are

First, we assume

And then, we differentiate (7.4.2) and (7.4.3) with respect to each direction (i.e. x1 and x2).

And then,

Since $$\displaystyle ds^2 = \sum_{i} (dx_i)^2= \sum_{k} {(h_k)^2 (d \xi _k)^2} $$,

And the Laplacian equals

Thus,