Vector Calculus

Which of the following conditions must hold for Stokes' theorem $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$ to apply?

Options:

• (A) The vector field $\vec{F}$ has continuous partial derivatives on an open region containing $S$
• (B) The surface $S$ is piecewise smooth and orientable
• (C) The curve $C$ is a simple closed piecewise smooth boundary curve with compatible orientation
• (D) The vector field $\vec{F}$ must be solenoidal