= Hessian at a critical point under a change of coordinates
{c}
{title2=$H_{f\circ\phi}(u_*)=J_\phi(u_*)^T H_f(\phi(u_*))J_\phi(u_*)$}
Let $\phi$ be a twice continuously differentiable local <diffeomorphism>, and suppose $\nabla f(\phi(u_*))=0$. The <chain rule> gives the displayed transformation of the <Hessian matrix>: terms containing second <partial derivatives> of $\phi$ 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.
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