Hessian at a critical point under a change of coordinates ($H_{f\circ \varphi }(u_{\ast })=J_{\varphi }(u_{\ast }{)}^T{H}_f(\varphi (u_{\ast }))J_{\varphi }(u_{\ast })$)

Let $\varphi$ be a twice continuously differentiable local diffeomorphism, and suppose $\nabla f(\varphi (u_{\ast }))=0$. The chain rule gives the displayed transformation of the Hessian matrix: terms containing second partial derivatives of $\varphi$ are multiplied by first partial derivatives of $f$ and vanish at the critical point. By Sylvester's law of inertia, the numbers of positive, negative and zero eigenvalues are unchanged. Independently of nondegeneracy, the local diffeomorphism preserves the critical point's status as a local minimum, local maximum or saddle point of a scalar function, because it maps neighbourhoods to neighbourhoods.