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For my homework, I'm given a question that wants me to find the flux of F × ∇φ through a surface S enclosed by a curve C, where F is irrotaional and φ is a scalar function.

Flux is $\iint_S u \cdot n dS$, which in this case, u = F × ∇φ so the flux is$\iint_S F × ∇φ \cdot n dS$. This looks very similar to Stoke's thm,

$\iint_S ∇ × u \cdot n dS = \int_C F \cdot dr$

but is missing ∇ × in front of u, so I was wondering if there is some property or theorem of line integrals that I'm missing to move ∇ × inside the integral. I also know that since F is conservative, F = ∇f for a scalar function f, and that ∇f × ∇φ = 0, which would make the integral zero, but I feel like I'm not supposed to do that since the question wants us to express the flux as a line integral. Any help would be much appreciated, thanks!

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Hint: F is irrotational so F can be written as gradient of some scalar field f. What is the cross product of the gradient of two scalar fields?

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