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Stokes' theorem

It states that the flux of the curl of a vector field through a surface equals the line integral of that field around the surface's boundary curve. It is used to convert between surface and line integrals in vector calculus and electromagnetism.

\iint_S (\nabla \times \boldsymbol{F}) \cdot \mathrm{d}\boldsymbol{\sigma} = \oint_{\partial S} \boldsymbol{F} \cdot \mathrm{d}\boldsymbol{r}

What the symbols mean

vector field
curl of the vector field
oriented surface bounded by the curve
boundary curve of S, oriented consistently with S
oriented surface element (normal vector times area)
line element along the boundary curve

About this formula

  • FieldMathematics
  • TopicVector calculus
  • LevelUniversity
  • Also calledcirculation theorem, curl theorem, Kelvin-Stokes theorem, fundamental theorem for curls, surface integral of curl equals line integral, flux of curl
  • SourceWikidata Q467756

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