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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.

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 are the subject of vector calculus.  They play important roles in [[electrodynamics]].  The divergence and curl operations are cornerstones of [[Maxwell's Equations]].

These last two examples (and Stokes' theorem in general) 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 surface integral of the curl of a function to its line integral.  Stokes' Theorem originated in 1850.

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 surface integral of the curl of a function to its line integral.  Stokes' Theorem originated in 1850.

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[[Category:vector analysis]]

[[Category:Vector analysis]]

[[Category:calculus]]

[[Category:Calculus]]

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Physics]]

[[Category:Physics]]