Stokes' Theorem

Stokes' theorem relates a surface integral of the curl to a line integral:

$$\oint_C \vec{F}\cdot d\vec{r} = \iint_S (\nabla\times\vec{F})\cdot d\vec{S}$$

where $C = \partial S$ is the boundary, oriented by the right-hand rule.

Physical meaning — Circulation around a loop equals the total curl through any surface it bounds.

Special cases

• $\nabla\times\vec{F} = \vec{0}$ → $\oint \vec{F}\cdot d\vec{r} = 0$ (conservative field)
• Green's theorem is Stokes in $\mathbb{R}^{2}$

Green's theorem

$$\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$$

The three integral theorems:

• Fundamental thm: $f(b)-f(a) = \int f'\,dx$
• Stokes: boundary line integral = interior curl integral
• Divergence: boundary flux = interior divergence integral

Key insight: What happens on the boundary determines what happens inside.

Common pitfall: In Stokes’ theorem, the curve’s direction and the surface’s normal are chained by the right-hand rule. Choosing them independently produces answers that are perfectly computed and off by a sign.