# Explicit proof of the Jacobian inverse

## Homework Statement

Given the transformations ##x^2+y^2=2*r*cos(theta)## and ##x*y=r*sin(theta)## prove the Jacobian explicitly

The question then goes on to ask how r and theta are related to the cylindrical coordinates rho and phi. I think ##r=1/2*(x^2+y^2)## and hence ##r=1/2 rho## but Im not really sure about this part I'm afraid, so haven't gotten very far

## Homework Equations

##(partial(x,y)/partial(r,theta))*partial(r,theta)/partial(x,y))=1##

## The Attempt at a Solution

So by multiplying the second expression by two and then squaring both and adding we get ##r=1/2*(x^2+y^2)## from which we can find the first two elements in the second Jacobian to be x and y respectively. By dividing the two transformations we can also get an expression for theta, which is relativity simple to differentiate, however when isolating x and y I seem to be going into pages of algebra, which leads me to think I have got the wrong approach, and there must be a simpler way since this is meant to be a relatively quick question.

Many thanks