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Line 13: − − == Examples == − − *[http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/stokes/stokes.html Calculation of both sides of Stokes' Theorem for a paraboloid, using a circle as the contour] Line 40:
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explained more clearly
'''Stokes' Theorem''' expresses the surface integral of the [[curl]] of [[vector field]] in terms of its easier-to-compute [[circulation]]:
'''Stokes' Theorem''' states that the line integral of a closed path is equal to the surface integral of ''any'' capping surface for that path, provided that the surface normal vectors point in the same general direction as the right-hand direction for the contour:
:<math>\iint_S (\nabla \times \vec F) \cdot \vec{\mathrm{d}S} = \oint_C \vec F \cdot \vec{\mathrm{d}r}\,</math>
:<math>\oint_C \vec F \cdot \vec{\mathrm{d}r}\ = \iint_S (\nabla \times \vec F) \cdot \vec{\mathrm{d}S},</math>
Intuitively, imagine a "capping surface" that is nearly flat with the contour. The curl is the microscopic circulation of the function on tiny loops within that surface, and their sum or integral results in canceling out all the internal circulation paths, leaving only the integration over the outer-most path.<ref>[http://www.robots.ox.ac.uk/~dwm/Courses/2VA/notes78.pdf Notes on Gauss' and Stokes' Theorem, see Section 7.6, p. 87]</ref> This remains true no matter how the capping surface is expanded, provided that the contour remains as its boundary.
Sometimes the [[circulation]] (the left side above) is easier to compute; other times the expresses the surface integral of the [[curl]] of [[vector field]] is easier to computer (particularly when it is zero).<ref>
*[http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/stokes/stokes.html Calculation of both sides of Stokes' Theorem for a paraboloid, using a circle as the contour]</ref>
Stated another way, Stokes' Theorem equates the line integral of a vector fields to a surface integral of the same vector field. For this identity to be true, the ''direction'' of the vector normal ''n'' must obey the right-hand rule for the direction of the contour, ''i.e.'', when walking along the contour the surface must be on your left.
Stated another way, Stokes' Theorem equates the line integral of a vector fields to a surface integral of the same vector field. For this identity to be true, the ''direction'' of the vector normal ''n'' must obey the right-hand rule for the direction of the contour, ''i.e.'', when walking along the contour the surface must be on your left.
Stokes' Theorem is useful in calculating [[circulation]] in mechanical engineering. A [[conservative field]] has a circulation (line integral on a simple, closed curve) of zero, and application of the Stokes' Theorem to such a field proves that the curl of a conservative field over the enclosed surface must also be zero.
Stokes' Theorem is useful in calculating [[circulation]] in mechanical engineering. A [[conservative field]] has a circulation (line integral on a simple, closed curve) of zero, and application of the Stokes' Theorem to such a field proves that the curl of a conservative field over the enclosed surface must also be zero.
== General Form ==
== General Form ==
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 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.
== References ==
<references/>
[[Category:vector analysis]]
[[Category:vector analysis]]
[[Category:calculus]]
[[Category:calculus]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Physics]]
[[Category:Physics]]