# Expressing Flux of irrotational vector field crossed with gradient of scalar field

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!

## 1 Answer

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?