Transformation to elliptical coordinates

Question

I'm currently struggling to make any progress with this question. I'm a little bit thrown by the inclusion of cosh and sinh. I am aware of all of the definitions, just need guidance with approach.

Answer 1

We have $$\begin{eqnarray}x =& a \cosh \mu \cos \nu,\\ y =& a \sinh \mu \sin \nu.\end{eqnarray}$$

For part (a), if we choose a value of $\mu$ and hold $\mu$ constant at that value, then $a\cosh \mu$ and $a\sinh \mu$ are simply two constants. If we label those constants $m$ and $n$, the curve that we trace as we vary $\nu$ is just the curve described by the parametric equations $x = m \cos\nu, \; y = n \sin\nu,$ which should be familiar. If you'd rather have a non-parametric equation of the curve, let $$h = \left(\frac xm\right)^2 + \left(\frac yn\right)^2$$ and show that $h$ is a constant.

For the case where we hold $\nu$ constant, it may help to rewrite the equations of the coordinates as $$\begin{eqnarray}x =& (a \cos \nu) \cosh \mu,\\ y =& (a \sin \nu) \sinh \mu.\end{eqnarray}$$ Now realize that if we hold $\nu$ constant, then $a \cos \nu$ and $a \sin \nu$ are constant, so we have $x = p \cosh \mu, \; y = q \sinh \mu$ for some constants $p$ and $q.$ If you're not familiar with this parameterization of a hyperbola, let $$k = \left(\frac xp\right)^2 - \left(\frac yq\right)^2.$$ Evaluating $k$ (it helps to use the facts $\cosh\mu = \frac{e^\mu+e^{-\mu}}2$ and $\sinh\mu = \frac{e^\mu-e^{-\mu}}2$ while doing this), we can show that $k$ is a positive constant, and so we have obtained a non-parametric equation of a hyperbola.

Part (b) seems to be fairly straightforward (take partial derivatives to compute basis vectors, then find the magnitudes of the vectors and their inner product), though you will want to know the derivatives of $\cosh\mu$ and $\sinh\mu$ in order to compute the basis vector $\hat\mu.$