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where ''d'' is the [[exterior derivative]].

where ''d'' is the [[exterior derivative]].

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

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=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]]:

*When k=3, this is often called '''Gauss' Theorem''' or the '''Divergence 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>

:<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 [[inner product|dot product]].

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 [[inner product|dot product]].

*When k=2, this is often also called '''Gauss' Theorem''':

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

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

:<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.

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]].

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]].

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Physics]]

[[Category:Physics]]