Stokes' theorem

Say we have a surface $S$ in $\R^3$ with a closed boundary curve $C$ . Recall that we can orient a curve with a tangent vector $\mathbf T$ and a surface with a normal vector $\mathbf n$ . If we orient the two such that:

If you walk along $C$ in the direction of $\mathbf T$ with the surface to your left, then $\mathbf n$ points up; or
If you point your right thumb in the direction of $\mathbf n$ , your fingers curl in the direction of $\mathbf T$ (right-hand rule).

Then Stokes’ theorem says:

\int_C \mathbf F \cdot d\mathbf r = \iint_S \mathrm{curl}(\mathbf F) \cdot \mathbf n dS.

Note

The boundary of a surface is the set of points where it stops. For example, the boundary of a unit hemisphere centered at the origin is the unit circle. A closed surface doesn’t have a boundary.