Difference between revisions of "Stokes' Theorem"

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(I was mistaken about "Green's Theorem", but the k=2 case really is called Stokes' Theorem in many textbooks.)
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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>
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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>
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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 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]]

Revision as of 20:39, 24 May 2007

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 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.