Teorema de Clairaut-Schwarz

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Na análise matemática, o teorema de Clairaut-Schwarz é uma condição suficiente para a igualdade das derivadas parciais cruzadas de uma função de várias variáveis. O teorema estabelece que, se as derivadas parciais cruzadas existem e são contínuas, então são iguais. O nome do teorema é uma referência aos não-contemporâneos Alexis Claude de Clairaut e Hermann Amandus Schwarz.

Enunciado

Enunciado geral

Seja $D\subseteq \mathbb {R} ^{n}$ um conjunto aberto e $f:D\to \mathbb {R}$ um campo escalar de classe $C^{2}$ . Então, para qualquer ponto $(d_{1},d_{2},\dots ,d_{n})\in D$ :

${\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(d_{1},\dots ,d_{n})={\frac {\partial ^{2}f}{\partial x_{j}\,\partial x_{i}}}(d_{1},\dots ,d_{n})$

Caso particular a duas variáveis

Seja $D\subseteq \mathbb {R} ^{2}$ um conjunto aberto e $f:D\to \mathbb {R}$ um campo escalar de classe $C^{2}$ . Então, para qualquer ponto $(x,y)\in D$ :

${\frac {\partial ^{2}f}{\partial x\,\partial y}}(x,y)={\frac {\partial ^{2}f}{\partial y\,\partial x}}(x,y)$

Exemplos de aplicações

Aplicando o teorema no operador del de alta ordem se obtêm que:

\nabla \!_{n}\left(\nabla \!_{m}f\right)=\nabla \!_{m}\left(\nabla \!_{n}f\right)

Segundo Stewart, 2007, o teorema de Clairaut-Schwarz é válido se ambas derivadas parciais mistas forem contínuas em seus domínios.^[1]

Seu análogo em integrais duplas/iteradas é o Teorema de Fubini.

O teorema também é fundamental para a física, como exemplo nas relações de Maxwell.

Referências

↑ Stewart, James, Cálculo Vol. 2,5ª ed, 2007, pp.914 .

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