291 Stokes’ Theorem for Flux Integral
\[\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \dfrac{d\mathbf{r}}{ds}(s) \, ds\].
Or,
\[\iint_S (\text{curl} \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \dfrac{d\mathbf{r}}{ds}(s) \, ds\]
where
- \(S\) a piecewise smooth oriented surface in space
- \(C\) a piecewise smooth simple closed curve boundary of \(S\)
- \(\mathbf{n}\) is the unit normal vector of \(S\).
- \(\dfrac{d\mathbf{r}}{ds}(s)\) is the unit tangent vector of \(C\).
- \(s\) is the arc length of \(C\).
- \(\mathbf{F}(x, y, z)\) is a continuous vector function