# Stokes' Theorem

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## Theorem

### Classical Stokes' Theorem

Concerning surfaces in 3-space and their boundaries:

Let $S$ be some orientable smooth surface with boundary in $\R^3$.

Let $\mathbf F:\R^3 \to \R^3$ be a vector-valued function with Euclidean coordinate expression:

- $F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$

where $f_i: \R^3 \to \R$.

Then:

- $\displaystyle \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$

where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.

### General Stokes' Theorem

Concerning arbitrary manifolds and their boundaries:

Let $\omega$ be a smooth $\paren {n - 1}$-form with compact support on a smooth $n$-dimensional oriented manifold $X$.

Let the boundary of $X$ be $\partial X$.

Then:

- $\displaystyle \int_{\partial X} \omega = \int_X \rd \omega$

where $\d \omega$ is the exterior derivative of $\omega$.

## Source of Name

This entry was named for George Gabriel Stokes.