A two$-$dimensional $(u,v)$ elliptical coordinate system can be related to an $(x,y)$

Cartesian system by the coordinate equations

$x=coshucosv$

$y=sinhusinv$

Determine the elements of the transformation matrix a that converts vector

components in the Cartesian system to vector components in this elliptical sys$-$

tem. What is $[a{]}^{-1}?$