Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / F / 1

For a differentiable map $f:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^k$, the Jacobian matrix represents its derivative and has entries $(Df{)}_{ij}=\partial f_i/\partial x_j$. Differentiating the two components gives

The inverse function theorem states that if $f$ is continuously differentiable near $a\in {\mathbb{R}}^n$ and $Df(a)$ is invertible, then neighborhoods of $a$ and $f(a)$ exist on which $f$ is a bijection with a continuously differentiable inverse, satisfying $D(f^{-1})(f(a))=Df(a{)}^{-1}$.

Here $a=(\pi /2,\pi )$, $f(a)=(0,0)$, andThe hypotheses hold because both components are smooth. Inverting the matrix yields