Expain stokes theeorem
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generally differential geometry, Stokes' theorem(sometimes spelled Stokes's theorem, and also called the generalized Stokes theorem or the Stokes–Cartan theorem[) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivativedω over the whole of Ω, i.e.,
{\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.}
Stokes' theorem was formulated in its modern form by Élie Cartan in 1945,[] following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.
This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861.This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface in Euclidean three-space to the line integral of the vector field over its boundary: Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theoremimplies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))and F is a smooth vector field on R3, then:
{\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }
This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.
{\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.}
Stokes' theorem was formulated in its modern form by Élie Cartan in 1945,[] following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.
This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861.This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface in Euclidean three-space to the line integral of the vector field over its boundary: Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theoremimplies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))and F is a smooth vector field on R3, then:
{\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }
This classical statement, along with the classical divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.
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