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Symmetry of second derivatives
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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
does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix.
Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem.
In the context of partial differential equations, it is called the Schwarz integrability condition.
In symbols, the symmetry may be expressed as:
Another notation is:
In terms of composition of the differential operator Di which takes the partial derivative with respect to xi:
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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
does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix.
Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem.
In the context of partial differential equations, it is called the Schwarz integrability condition.
In symbols, the symmetry may be expressed as:
Another notation is:
In terms of composition of the differential operator Di which takes the partial derivative with respect to xi: