Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function f ( x 1 , x 2 , … , x n ) {\displaystyle f\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)} does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities ∂ ∂ x i ( ∂ f ∂ x j ) = ∂ ∂ x j ( ∂ f ∂ x i ) . {\displaystyle {\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{j}}}\right)\ =\ {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right).} In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix.

Source: Wikipedia — Symmetry of second derivatives (CC BY-SA 4.0)

Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function f ( x 1 , x 2 , … , x n ) {\displaystyle f\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)} does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities ∂ ∂ x i ( ∂ f ∂ x j ) = ∂ ∂ x j ( ∂ f ∂ x i ) . {\displaystyle {\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{j}}}\right)\ =\ {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right).} In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix.

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Source: Wikipedia "Symmetry of second derivatives" · CC BY-SA 4.0

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