| Version current |
Version 14 |
| \begin{cthm*} |
\begin{cthm*} |
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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
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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
|
| \[ |
\[ |
| \frac{\partial^2 f}{\partial x_i \partial x_j} |
\frac{\partial^2 f}{\partial x_i \partial x_j} |
| =\frac{\partial^2 f}{\partial x_j \partial x_i} |
=\frac{\partial^2 f}{\partial x_j \partial x_i} |
| \] |
\] |
| on $S$, where $1 \leq i,j \leq n$. |
on $S$, where $1 \leq i,j \leq n$. |
| \end{cthm*} |
\end{cthm*} |
|
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| This theorem is commonly referred to as \emph{the equality of mixed partials}. |
This theorem is commonly referred to as \emph{the equality of mixed partials}. |
| It is usually first presented in a vector calculus course, |
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. |
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 $\nabla \cdot (\nabla \times \mathbf{F}) =0$. |
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 $\nabla \times \nabla f= \mathbf{0}$. |
Or, if $f\colon\mathbb{R}^3 \to \mathbb{R}$ is a function satisfying the hypothesis, then $\nabla \times \nabla f= \mathbf{0}$. |
