Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 2 / b

The partial derivatives areThus the Jacobian determinant isThis vanishing reflects the functional dependence found next.

The tangent addition formula givesHence $\boxed{f(v)=\tan v}$. The restriction $xy\ne 1$ excludes the poles of tangent that can arise in the possible range $-\pi <v<\pi$. One should not replace this identity by $v=\arctan u$ globally: the principal inverse tangent can differ from $v$ by an integer multiple of $\pi$.