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In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a function : of ''n'' variables. If the partial derivative with respect to is denoted with a subscript , then the symmetry is the assertion that the second-order partial derivatives satisfy the identity : so that they form an ''n'' × ''n'' symmetric matrix. This is sometimes known as Young's theorem. In the context of partial differential equations it is called the Schwarz integrability condition. ==Hessian matrix== This matrix of second-order partial derivatives of ''f'' is called the Hessian matrix of ''f''. The entries in it off the main diagonal are the mixed derivatives; that is, successive partial derivatives with respect to different variables. In most "real-life" circumstances the Hessian matrix is symmetric, although there are a great number of functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on ''f'' that goes further than simply stating the existence of the second derivatives at a particular point. Schwarz' theorem gives a sufficient condition on ''f'' for this to occur. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「symmetry of second derivatives」の詳細全文を読む スポンサード リンク
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