Question: Prove that if (mathbf{F}) is a gradient vector field, then the flux of (operatorname{curl}(mathbf{F})) through a smooth surface (mathcal{S}) (whether closed or not) is equal

Prove that if \(\mathbf{F}\) is a gradient vector field, then the flux of \(\operatorname{curl}(\mathbf{F})\) through a smooth surface \(\mathcal{S}\) (whether closed or not) is equal to zero.

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