if x=f(u,v) and y=g(u,v), find ∂u/∂x and ∂v/∂x

Question

I encountered this problem and tried to partially differentiate both equations against x and see if i can create simultaneous equations to solve for the two, but I don't seem to be getting anywhere and now I'm not sure how I should start.

Answer 1

You are tacitly assuming that $f(u_0,v_0)=x_0$, $\>g(u_0,v_0)=y_0$, and that there are functions $$\phi:\ (x,y)\to u=\phi(x,y),\qquad\psi:\ (x,y)\mapsto v=\psi(x,y)$$ inverting $$(f,g):\quad (u,v)\mapsto(x,y)$$ in the neighborhood of $(u_0,v_0)$. In particular $\phi(x_0,y_0)=u_0$, $\>\psi(x_0,y_0)=v_0$. In such a case the Jacobian of $(\phi,\psi)$ at $(x_0,y_0)$ is the inverse of the Jacobian of $(f,g)$ at $(u_0,v_0)$: $$\left[{\partial (\phi,\psi)\over\partial(x,y)}\right]_{(x_0,y_0)}= \left[{\partial(f,g)\over\partial(u,v)}\right]_{(u_0,v_0)}^{-1}\ .$$ This is a content of the inverse function theorem for maps ${\bf f}:\>{\mathbb R}^n\to{\mathbb R}^n$.