Schwarz’s theorem

If f is twice continuously differentiable, the order of the mixed partial derivatives does not matter, so the Hessian is symmetric. Without continuity it can fail — Sheet 3, Exercise 18.

Chapters

Formulas at a glance · Second derivatives

Hessian & second directional derivatives

D_{u} (D_{v} f) (x) = u^{⊤} H_{f} v = \sum_{i = 1}^{d} \sum_{j = 1}^{d} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} (x) u_{i} v_{j}

Schwarz’s theorem

\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} (x) = \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}} (x)

Schwarz’s theorem

Working

Hessian & second directional derivatives

Schwarz’s theorem