Stokes' Theorem

Positive Orientation of a Boundary Curve

Let $\sigma$ be an oriented surface with unit normal field $\hat{n}$, and let $C$ be the boundary curve of $\sigma$. The positive orientation of $C$ is the direction determined by the chosen orientation of $\sigma$.

Equivalently, walking along $C$ with the head pointing in the direction of $\hat{n}$ keeps the surface on the left.

Stokes’ Theorem

Let $\sigma$ be a piecewise smooth oriented surface bounded by a positively oriented, piecewise smooth, simple closed curve $C$. Suppose

is a vector field whose component functions have continuous first-order partial derivatives on an open set containing $\sigma$. If $\hat{n}$ is the chosen unit normal field for $\sigma$,