Stokes' Theorem

Stokes' Theorem, in its most general form, is the fundamental theorem of Exterior Calculus, and is a generalization of the Fundamental Theorem of Calculus. It states that if M is an oriented piecewise smooth manifold of dimension k and <math>\omega</math> is a smooth (k−1)-form with compact support on M, and ∂M denotes the boundary of M with its induced orientation, then

<math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>,

where d is the exterior derivative.

There are a number of well-known special cases of Stokes' theorem:

• When k=1, and the terms appearing in the theorem are translated into their simpler form, this is just the Fundamental Theorem of Calculus.

• When k=3, this is often called Green's Theorem and is useful in vector calculus:

<math>\iiint_R (\nabla \cdot \vec w)\ \mathrm{d}V = \iint_S \vec w \cdot \vec{\mathrm{d}A}\,</math>

Where R is some region of 3-space, S is the boundary surface of R, the triple integral denotes volume integration over R with dV as the volume element, and the double integral denotes surface integration over S with <math>\vec{\mathrm{d}A}</math> as the oriented normal of the surface element. The <math>\nabla \cdot</math> on the left side is the divergence operator, and the <math>\cdot</math> on the right side is the vector dot product.

• When k=2, this is often also called Stokes' Theorem (the less general form):

<math>\iint_S (\nabla \times \vec w) \cdot \vec{\mathrm{d}A} = \oint_E \vec w \cdot \vec{\mathrm{d}l}\,</math>

Here S is a surface, E is the boundary path of S, and the single integral denotes path integration around E with <math>\vec{\mathrm{d}l}</math> as the length element. The <math>\nabla \times</math> on the left side is the curl operator.

These last two examples (and Stokes' theorem in general) are somewhat esoteric, and are the subject of vector calculus. They play important roles in electrodynamics.