structural:summarization, quality:correction
The Hessian

The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f: R → R, these include the three second-order partials ∂ 2 f ∂ x 2, ∂ 2 f ∂ y 2, and ∂ 2 f ∂ z 2 {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},{\text{ and }}{\frac {\partial ^{2}f}{\partial z^{2}}}} and the mixed partials ∂ 2 f ∂ x ∂ y, ∂ 2 f ∂ x ∂ z, and ∂ 2 f ∂ y ∂ z. {\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},{\text{ and }}{\frac {\partial ^{2}f}{\partial y\,\partial z}}.}

If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.)