Stokes' Theorem

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Stokes' Theorem holds that the double integral of the curl of a vector field with respect to a surface is equal to its line integral with respect to a simple curve enclosing the surface. This is the analog in two dimensions of the Divergence Theorem. Stokes' Theorem is useful in calculating circulation in mechanical engineering. Stokes' Theorem has no application to conservative fields.

In its most general form, this theorem 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 is a smooth (k−1)-form with compact support on M, and ∂M denotes the boundary of M with its induced orientation, then


where d is the exterior derivative.

There are a number of well-known special cases of Stokes' theorem, including one that is referred to simply as "Stokes' theorem" in less advanced treatments of mathematics, physics, and engineering:

  • 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 Gauss' Theorem or the Divergence Theorem and is useful in vector calculus:

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 as the oriented normal of the surface element. The on the left side is the divergence operator, and the on the right side is the vector dot product.

  • When k=2, this is often just called Stokes' Theorem:

Here S is a surface, E is the boundary path of S, and the single integral denotes path integration around E with as the length element. The 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. The divergence and curl operations are cornerstones of Maxwell's Equations.

Stokes' Theorem is a lower-dimension version of the Divergence Theorem, and a higher-dimension version of Green's Theorem. Green’s Theorem relates a line integral to a double integral over a region, while Stokes' Theorem relates a line integral to a surface integral.