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- Jan 26, 2012

- 995

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**Problem**: Let $f,g : \mathbb{R}^3\rightarrow\mathbb{R}$ be differentiable functions. Given that $\mathbf{F}=\langle F_1,F_2,F_3\rangle$ is a differentiable vector field and $\text{div}\,(\mathbf{F})=\nabla\cdot\mathbf{F} = \dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}$, show that $\text{div}\,(\nabla f\times\nabla g) = 0$.

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