Past exam of the options I course of the University of Oxford / 2008 / F (Multivariable Calculus)

For $f=(f_1,\dots ,f_k)$, the Jacobian matrix is the matrix of its derivative in standard bases, with entries $(Df{)}_{ij}=\partial f_i/\partial x_j$. Here

The inverse function theorem states: if $f$ is continuously differentiable on an open subset of ${\mathbb{R}}^n$ and $Df(a)$ is invertible, then some open neighbourhood of $a$ is mapped bijectively onto an open neighbourhood of $f(a)$, with continuously differentiable inverse. Its derivative at $f(a)$ is $(Df(a){)}^{-1}$.

The present map is smooth, and at $a=(\pi /2,\pi )$,Also $f(a)=(0,0)$, so the theorem applies locally and givesThe inverse is defined near the image point $(0,0)$ and takes values near $a$; no assertion of a global inverse is required.