Stokes' Theorem

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

$$\int_M dw = \int_{\partial M} w,$$ (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 $\Bbb{R}^3$,

$$\mathop{\rm grad}(f) = c^{-1}\,df,$$ (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 $\Bbb{R}^3$,

$$\mathop{\rm div}(f) = {}^*d{}^*c(f),$$ (3)

where $*$ is the Hodge Star operator. If $f$ is a Vector Field on $\Bbb{R}^3$,

$$\mathop{\rm curl}(f) = c^{-1}{}^*dc(f).$$ (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

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

$$\int_{\partial M} f\cdot dA = \int_{\partial M} {}^*cf = \int_M d*cf = \int_M \mathop{\rm div}(f)\,dV,$$ (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

$$\int_{\partial M} f \, dl = \int_{\partial M} cf = \int_M dc(f) = \int_M \mathop{\rm curl}(f) \cdot dA,$$ (7)

which is the Curl Theorem.

Physicists generally refer to the Curl Theorem

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

as Stokes' theorem.

See also Curl Theorem, Divergence Theorem, Gradient Theorem