Consider a hanging rectangular rubber sheet.
We will consider there to be two ways to get energy into or out of
this sheet: you can either stretch it vertically or horizontally.
The distance of vertical stretch we will call \(y\), and the distance
of horizontal stretch we will call \(x\).
If I pull the bottom down by a small distance \(\Delta y\), with no
horizontal force, what is the resulting change in width \(\Delta x\)?
Express your answer in terms of partial derivatives of the potential
energy \(U(x,y)\).