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IEVref: | 102-05-31 | ID: | |

Language: | en | Status: Standard | |

Term: | Stokes theorem | ||

Synonym1: | circulation theorem | ||

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Definition: | for a vector field that is given at each point of a surface S limited by an oriented closed curve C, theorem stating that the surface integral over S of the rotation of the field U is equal to the circulation of this field along the curve CU$\underset{\text{S}}{\iint}\mathrm{rot}}\text{\hspace{0.17em}}U\cdot {e}_{\text{n}}dA={\displaystyle \underset{\text{C}}{\oint}U\cdot dr$ where Note 1 to entry: The orientation of the surface S with respect to the curve C is chosen such that, at any point of C, the vector line element, the unit vector normal to S and defining its orientation, and the unit vector normal to these two vectors and oriented towards the exterior of the curve, form a right-handed or a left-handed trihedron according to space orientation. Note 2 to entry: The Stokes theorem can be generalized to the Note 3 to entry: In magnetostatics, the Stokes theorem is applied to express that the magnetic flux through the surface S is equal to the circulation over C of the magnetic vector potential. This circulation defines the linked flux. See IEV 121-11-24. | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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$\underset{\text{S}}{\iint}\mathrm{rot}}\text{\hspace{0.17em}}U\cdot {e}_{\text{n}}dA={\displaystyle \underset{\text{C}}{\oint}U\cdot dr$

where *e*_{n}d*A* is the vector surface element and d** r** is the vector line element

Note 1 to entry: The orientation of the surface S with respect to the curve C is chosen such that, at any point of C, the vector line element, the unit vector normal to S and defining its orientation, and the unit vector normal to these two vectors and oriented towards the exterior of the curve, form a right-handed or a left-handed trihedron according to space orientation.

Note 2 to entry: The Stokes theorem can be generalized to the *n*-dimensional Euclidean space.

Note 3 to entry: In magnetostatics, the Stokes theorem is applied to express that the magnetic flux through the surface S is equal to the circulation over C of the magnetic vector potential. This circulation defines the linked flux. See IEV 121-11-24.