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Clairaut's theorem
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(Theorem)
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Clairaut's Theorem If $\mathbf{f}\colon\mathbb{R}^n \to \mathbb{R}^m$ is a function whose second partial derivatives exist and are continuous on a set $S \subseteq \mathbb{R}^n$ , then $$ \frac{\partial^2 f}{\partial x_i \partial x_j} =\frac{\partial^2 f}{\partial x_j \partial x_i} $$ on $S$ , where $1 \leq i,j \leq n$ .
This theorem is commonly referred to as the equality of mixed partials. It is usually first presented in a vector calculus course, and is useful in this context for proving basic properties of the interrelations of gradient, divergence, and curl. For example, if $\mathbf{F}\colon
\mathbb{R}^3 \to \mathbb{R}^3$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times \mathbf{F}) =0$ . Or, if $f\colon\mathbb{R}^3 \to \mathbb{R}$ is a function satisfying the hypothesis, then $\nabla \times \nabla f= \mathbf{0}$ .
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| Other names: |
equality of mixed partials |
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Cross-references: hypothesis, curl, divergence, gradient, properties, Calculus, vector, theorem, continuous, partial derivatives, function
There are 2 references to this entry.
This is version 15 of Clairaut's theorem, born on 2003-08-22, modified 2006-11-13.
Object id is 4642, canonical name is ClairautsTheorem.
Accessed 9003 times total.
Classification:
| AMS MSC: | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) |
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Pending Errata and Addenda
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