Hessian matrix
The matrix of a function's second partial derivatives; it captures how the gradient itself is changing.
For f:\mathbb R^n\to\mathbb R twice differentiable, the Hessian H(x) is the n\times n matrix of second partial derivatives, H_{ij}=\dfrac{\partial^2 f}{\partial x_i\partial x_j}, symmetric whenever the mixed partials are continuous. Its definiteness at a critical point classifies that point: positive definite means a local minimum, negative definite a local maximum, and indefinite (mixed-sign eigenvalues) means a saddle point.
Point table
Graph settings
Calculator