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This 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 $\mathbf{F}$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times \mathbf{F}) =0$ and $\nabla \times \nabla \mathbf{F} = \mathbf{0}$.
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This 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 inter-relations of gradient, divergence, and curl. I.e., if $\mathbf{F}$ is a function satisfying the hypothesis, then $$\nabla \cdot (\nabla \times \mathbf{F}) =0$$ and $$\nabla \times \nabla \mathbf{F} = \mathbf{0}$$.
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