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Revision difference : Clairaut's theorem |
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Version 9 |
| \paragraph{Theorem.}(Clairaut's Theorem) |
\paragraph{Theorem.}(Clairaut's Theorem) |
| If $\mathbf{F}:\mathbb{R}^n \to \mathbb{R}^m$ is a function whose second partial derivatives exist and are continuous on a set $S \subseteq \mathbb{R}^n$, then |
If $\mathbf{F}:\mathbb{R}^n \to \mathbb{R}^m$ 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 $1 \leq i,j \leq n$).\\ |
$$ \frac{\partial^2 f}{\partial x_i \partial x_j}=\frac{\partial^2 f}{\partial x_j \partial x_i} $$ on $S$ (where $1 \leq i,j \leq n$).\\ |
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| This theorem is commonly referred to as simply '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. I.e., if $\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times \mathbf{F}) =0$. Or, if $f:\mathbb{R}^3 \to \mathbb{R}$ is a function satisfying the hypothesis, $\nabla \times \nabla f= \mathbf{0}$. |
This theorem is commonly referred to as simply '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. I.e., if $\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times \mathbf{F}) =0$. Or, if $f:\mathbb{R}^3 \to \mathbb{R}$ is a function satisfying the hypothesis, $\nabla \times \nabla f= \mathbf{0}$. |
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