Hessian & second directional derivatives

Differentiate the partial derivatives again: the $d \times d$ matrix of all second-order partials. It gives the second-order directional derivative $D_{u} (D_{v} f) = u^{⊤} H_{f} v$ .

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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)

Hessian & second directional derivatives

Working

Hessian & second directional derivatives

Schwarz’s theorem