Stokes' Theorem

From ProofWiki
Jump to navigation Jump to search

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.