Stokes's theorem
The Stokes theoremAlso called Kelvin-Stokes theorem, is a theorem in vector calculus R3{displaystyle mathbb {R} ^{3}. Given a vector field, the theorem relates the integral of the rotational field of a vector field on a surface, with the line integral of the vector field on the border of the surface.
Stokes' theorem is a special case of the generalized Stokes' theorem.
Theorem
Sea ・ ・ {displaystyle sigma } a smooth surface facing R3{displaystyle mathbb {R} ^{3} border ▪ ▪ ・ ・ {displaystyle partial sigma }. If a vector field F=(P(x,and,z),Q(x,and,z),R(x,and,z)){displaystyle mathbf {F} =(P(x,y,z),Q(x,y,z),R(x,y,z)} is defined and has continuous partial derivatives in an open region containing a ・ ・ {displaystyle sigma } then.
- ♫ ♫ ▪ ▪ ・ ・ F⋅ ⋅ dr=∫ ∫ ・ ・ (► ► × × F)⋅ ⋅ dS{displaystyle oint _{partial sigma }mathbf {F} cdot dmathbf {r} =iint _{sigma }left(nabla times mathbf {F} right)cdot dmathbf {S}} }
more explicitly, the equality above says that
- ♫ ♫ ▪ ▪ ・ ・ (Pdx+Qdand+Rdz)=∫ ∫ ・ ・ [chuckles](▪ ▪ R▪ ▪ and− − ▪ ▪ Q▪ ▪ z)danddz+(▪ ▪ P▪ ▪ z− − ▪ ▪ R▪ ▪ x)dzdx+(▪ ▪ Q▪ ▪ x− − ▪ ▪ P▪ ▪ and)dxdand]{cHFFFFFF}{cHFFFFFF}{cHFFFFFF00}{cHFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFFFFFFFF00}{cHFFFFFFFFFF00}{cHFFFFFFFFFFFFFF00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{c}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{c}{c}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
![{displaystyle {begin{aligned}&oint _{partial Sigma }left(Pdx+Qdy+Rdzright)\&=iint _{Sigma }left[left({frac {partial R}{partial y}}-{frac {partial Q}{partial z}}right)dydz+left({frac {partial P}{partial z}}-{frac {partial R}{partial x}}right)dzdx+left({frac {partial Q}{partial x}}-{frac {partial P}{partial y}}right)dxdyright]end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e7d3b31b8de0edb1bc1f84d47e3a6b9b2a1320)
Applications
Maxwell's Equations
In electromagnetism, Stokes' theorem justifies the equivalence between the differential form of the Maxwell-Faraday equation and the Maxwell-Ampère equation and the integral form of these equations.
For Faraday law, Stokes theorem applies to the electric field E{displaystyle mathbf {E} }
- ♫ ♫ ▪ ▪ ・ ・ E⋅ ⋅ dr=∫ ∫ ・ ・ (► ► × × E)⋅ ⋅ dS{displaystyle oint _{partial sigma }mathbf {E} cdot dmathbf {r} =iint _{sigma }left(nabla times mathbf {E} right)cdot dmathbf {S}} }
For the law of Ampère, the Stokes theorem applies to the magnetic field B{displaystyle mathbf {B} }
- ♫ ♫ ▪ ▪ ・ ・ B⋅ ⋅ dr=∫ ∫ ・ ・ (► ► × × B)⋅ ⋅ dS{displaystyle oint _{partial sigma }mathbf {B} cdot dmathbf {r} =iint _{sigma }left(nabla times mathbf {B} right)cdot dmathbf {S}} }
