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Stokes' Theorem

For $w$ a Differential k-Form with compact support on an oriented $n$-dimensional Manifold $M$,

\begin{displaymath}
\int_M dw = \int_{\partial M} w,
\end{displaymath} (1)

where $dw$ is the Exterior Derivative of the differential form $w$. This connects to the ``standard'' Gradient, Curl, and Divergence Theorems by the following relations. If $f$ is a function on $\Bbb{R}^3$,
\begin{displaymath}
\mathop{\rm grad}(f) = c^{-1}\,df,
\end{displaymath} (2)

where $c: \Bbb{R}^3 \to \Bbb{R}^3{}^*$ (the dual space) is the duality isomorphism between a Vector Space and its dual, given by the Euclidean Inner Product on $\Bbb{R}^3$. If $f$ is a Vector Field on a $\Bbb{R}^3$,
\begin{displaymath}
\mathop{\rm div}(f) = {}^*d{}^*c(f),
\end{displaymath} (3)

where $*$ is the Hodge Star operator. If $f$ is a Vector Field on $\Bbb{R}^3$,
\begin{displaymath}
\mathop{\rm curl}(f) = c^{-1}{}^*dc(f).
\end{displaymath} (4)


With these three identities in mind, the above Stokes' theorem in the three instances is transformed into the Gradient, Curl, and Divergence Theorems respectively as follows. If $f$ is a function on $\Bbb{R}^3$ and $\gamma$ is a curve in $\Bbb{R}^3$, then

\begin{displaymath}
\int_\gamma \mathop{\rm grad}(f) \cdot d{\bf l} = \int_\gamma\, df = f(\gamma(1)) - f(\gamma(0)),
\end{displaymath} (5)

which is the Gradient Theorem. If $f: \Bbb{R}^3 \to \Bbb{R}^3$ is a Vector Field and $M$ an embedded compact 3-manifold with boundary in $\Bbb{R}^3$, then
\begin{displaymath}
\int_{\partial M} f\cdot dA = \int_{\partial M} {}^*cf = \int_M d*cf = \int_M \mathop{\rm div}(f)\,dV,
\end{displaymath} (6)

which is the Divergence Theorem. If $f$ is a Vector Field and $M$ is an oriented, embedded, compact 2-Manifold with boundary in $\Bbb{R}^3$, then
\begin{displaymath}
\int_{\partial M} f \, dl = \int_{\partial M} cf = \int_M dc(f) = \int_M \mathop{\rm curl}(f) \cdot dA,
\end{displaymath} (7)

which is the Curl Theorem.


Physicists generally refer to the Curl Theorem

\begin{displaymath}
\int_S(\nabla \times {\bf F})\cdot d{\bf a} = \int_{\partial S}{\bf F}\cdot d{\bf s}
\end{displaymath} (8)

as Stokes' theorem.

See also Curl Theorem, Divergence Theorem, Gradient Theorem



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© 1996-9 Eric W. Weisstein
1999-05-26