Stokes' Theorem

If S is an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C with positive orientation. and F is a vector field then, Stokes’ theorem says that

$$ \int_C \vec{F} \cdot d\vec{r} = \iint_S \nabla \times \vec{F} \cdot d\vec{S} $$

Definitely read Paul’s Online Notes, they’re great.

Stokes’ theorem can be thought of as a 3D generalization of Green’s theorem.

Sources #

• Wikipedia
• Paul’s Online Notes

See also #

• This Khan Academy video provides great intuition: https://www.youtube.com/watch?v=9iaYNaENVH4
• Divergence Theorem
• Green’s Theorem