Stokes theorem Definition and 83 Threads

In vector calculus and differential geometry, the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called 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. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.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 derivative dω over the whole of Ω, i.e.,








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{\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 (that is, the flux of curl F) in Euclidean three-space to the line integral of the vector field over its boundary (also known as the loop integral).
Simple classical vector analysis example
Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies 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:







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{\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }
This classical statement, is a special case of the general formulation stated above after making an identification of vector field with a 1-form and its curl with a two form through






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{\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\\\end{pmatrix}}\cdot d\Gamma \to F_{x}\,dx+F_{y}\,dy+F_{z}\,dz}






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{\displaystyle \nabla \times {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}\cdot d\mathbf {S} ={\begin{pmatrix}\partial _{y}F_{z}-\partial _{z}F_{y}\\\partial _{z}F_{x}-\partial _{x}F_{z}\\\partial _{x}F_{y}-\partial _{y}F_{x}\\\end{pmatrix}}\cdot d\mathbf {S} \to }





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{\displaystyle d(F_{x}\,dx+F_{y}\,dy+F_{z}\,dz)=(\partial _{y}F_{z}-\partial _{z}F_{y})\,dy\wedge dz+(\partial _{z}F_{x}-\partial _{x}F_{z})\,dz\wedge dx+(\partial _{x}F_{y}-\partial _{y}F_{x})\,dx\wedge dy}
.Other classical generalisations of the fundamental theorem of calculus like the divergence theorem, and Green's theorem are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).

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  1. N

    Stokes theorem and downward orientation problem

    Homework Statement From Calculus:Concepts and Contexts 4th Edition by James Stewart. Pg.965 #13 Verify that Stokes' Theorem is true for the given vector field F and surface S F(x,y,z)= -yi+xj-2k S is the cone z^2= x^2+y^2, o<=z<=4, oriented downwards Homework Equations The Attempt at a...
  2. P

    Is This Special Case of Stokes Theorem True?

    Do you agree that the following identity is true: \int_S (\nabla_\mu X^\mu) \Omega = \int_{\partial S} X \invneg \lrcorner \Omega where \Omega is volume form and X\invneg \lrcorner \Omega is contraction of volume form with vector X.
  3. C

    On an alternative Stokes theorem

    I find in a homework an alternative Stokes theorem tha i wasn't knew before. I wold like to know that it is really true. Any can give me a proof please? It is: Int (line) dℓ′× A = Int (surface)dS′×∇′× A
  4. C

    How Does Stokes' Theorem Relate to Vorticity in Fluid Dynamics?

    Homework Statement Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...
  5. H

    Verifying Stokes Theorem on Paraboloid z=0.5(x^2+y^2)

    Homework Statement Verify Stokes Theorem ∬(∇xF).N dA where surface S is the paraboloid z = 0.5(x^2 + y^2) bound by the plane z=2, Cis its boundary, and the vector field F = 3yi - xzj + yzk. The Attempt at a Solution I had found (∇xF) = (z+x)i + (-z-3)k r = [u, v, 0.5(u^2 + v^2)]...
  6. I

    Using stokes theorem to find magnetic field

    Homework Statement magnetic field is azimuthal B(r) = B(p,z) \phi current density J(r) = Jp(p,z) p + Jz(p,z) z = p*exp[-p] p + (p-2)*z*exp[-p] z use stokes theorem to find B-filed induced by current everywhere in space Homework Equations stokes -...
  7. Y

    How can Stoke's theorem be applied to vector fields?

    Homework Statement \nabla \times f \vec{v} = f (\nabla \times \vec{v}) + ( \nabla f) \times \vec{v} Use with Stoke's theorem \oint _C \vec{A} . \vec{dr} = \int \int _S (\nabla \times \vec{A}) . \vec{dS} to show that \oint _c f \vec{dr} = \int \int _S \vec{dS} \times \nabla...
  8. R

    Stokes theorem and line integral

    Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...
  9. F

    How Do You Apply Stokes' Theorem to Evaluate a Line Integral?

    Homework Statement Use Stokes's Theorem to evaluate \int F · dr In this case, C (the curve) is oriented counterclockwise as viewed from above. Homework Equations F(x,y,z) = xyzi + yj + zk, x2 + y2 ≤ a2 S: the first-octant portion of z = x2 over x2 + y2 = a2 The Attempt at a...
  10. S

    Stokes Theorem in cylindrical coordinates

    Homework Statement A vector field A is in cylindrical coordinates is given. A circle S of radius ρ is defined. The line integral \intA∙dl and the surface integral \int∇×A.dS are different. Homework Equations Field: A = ρcos(φ/2)uρ+ρ2 sin(φ/4) uφ+(1+z)uz (1) The Attempt at...
  11. J

    Why is \hat{a} \bullet \vec{ds} = ds? Explaining Stokes Theorem.

    For stokes theorem, can someone tell me why \hat{a} \bullet \vec{ds} = ds? My notes say it's because they are parallel, but I'm not sure what that means. Also to get things clear, Stokes theorem is the generalized equation of Green's theorem. The purpose of Stokes theorem is to provide a...
  12. C

    Solving Basic Stokes Theorem Homework on Ellipse

    Homework Statement Use the surface integral in stokes theorem to find circulation of field F around the curve C. F=x^2i+2xj+z^2k C: the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from above Homework Equations stokes theroem: cirlulation=double integral of nabla...
  13. R

    Calculating Surface Integral with Stokes' Theorem on a Cube?

    Homework Statement I have to use stokes' theorem and calculate the surface integral, where the function F = <xy,2yz,3zx> and the surface is the cube bounded by the points (2,0,0), (0,2,0),(0,0,2),(0,2,2),(2,0,2),(2,2,0),(2,2,2). The back side of the cube is open. [/B] Homework...
  14. T

    Surface integral, grad, and stokes theorem

    Hi I'm practicing for my exam but I totally suck at the vector fields stuff. I have three questions: 1. Compute the surface integral \int_{}^{} F \cdot dS F vector is=(x,y,z) dS is the area differential Calculate the integral over a hemispherical cap defined by x ^{2}+y ^{2}+z...
  15. T

    Evaluating Surface Integral with Stokes Theorem

    use the stokes theorem to evaluate the surface integral [curl F dot dS] where F=(x^2+y^2; x; 2xyz) and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane. I don't see however how to take F dot dr. What is this closed curve dr bounding...
  16. A

    Stokes Theorem Problem: Surface Integral on Ellipse with Curl and Normal Vector

    Homework Statement F = xi + x3y2j + zk; C the boundary of the semi-ellispoid z = (4 - 4x2 - y2)1/2 in the plane z = 0Homework Equations (don't know how to write integrals on here, sorry) double integral (curl F) . n dsThe Attempt at a Solution curl F = 3y2x2k n = k curl F . n = 3y2x2 So I...
  17. B

    Stokes' Theorem for Line Integrals on Closed Curves: A Problem Solution

    Homework Statement Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also. Find \int y dx + z dy + x dz over the closed curve C which is the intersection of the surfaces whose...
  18. S

    Understanding Stokes Theorem: Solving Boundary Curve Dilemmas in Vector Calculus

    Homework Statement This is a question about stokes theorem in general, not about a specific problem. Directly from lecture: "If S has no boundry (eg. if S is the boundry of a solid region) then \int\int_{S}Curl(\stackrel{\rightarrow}{F})\bullet ds = 0 " because apparently "no boundry C...
  19. T

    Stokes theorem under covariant derivaties?

    in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this... So, is it true that: \int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x if so, how do I make sense of the d^{n-1}x term? would...
  20. L

    Using Stokes Theorem to $\int_{L}^{} y dx + z dy + x dx$

    Homework Statement Use Stokes Theorem to compute \int_{L}^{} y dx + z dy + x dx where L is the circle x2 + y2 + z2 = a2, x + y + z = 0 The Attempt at a Solution I feel like this problem shouldn't be that hard but I can't get the right answer: (pi)a2/3. I calculated the curl of F as...
  21. A

    Finding Area of L using Stokes Theorem

    Homework Statement Hey. I need to find the circulation of F through out the line L. I know I need to use stokes theorem, the problem is, how do I find the area of L? I mean, I know the intersection line of the sphere and the plot looks like an ellipse on the XY surface, but how do I find...
  22. D

    Stokes theorem equivalent for cross product line integral

    "Stokes theorem" equivalent for cross product line integral Homework Statement I am aware that the vector path integral of a closed curve under certain conditions is equivalent to the flux of of the curl of the vector field through any surface bound by the closed path. In other words, Stokes...
  23. R

    Vector Calculus question Div and Stokes Theorem

    If you start with the two dimensional green's theorem, and you want to extend this three dimensions. F=<P,Q> Closed line integral = Surface Integral of the partials (dP/dx + dQ/dy) da seems to leads the divergence theorem, When the space is extended to three dimensions. On the...
  24. Simfish

    Stokes Theorem Problem: Evaluating Line Integral with Vector Field

    Homework Statement Let C be the closed curve that goes along straight lines from (0,0,0) to (1,0,1) to (1,1,1) to (0,1,0) and back to (0,0,0). Let F be the vector field F = (x^2 + y^2 + z^2) i + (xy + xz + yz) j + (x + y + z)k. Find \int F \cdot dr By Stokes Theorem, I know that I can...
  25. S

    Proving \int curl A.n dS = 0 w/ Stokes Theorem

    (1) using stokes theorem and cutting the surface into 2 parts how can we prove that \int curl A.n dS = 0 assume the surface "S" to be smooth and closed, and "n" is the unit outward normal as usual. (2) How can you prove \int curl A.n dS = 0 using the divergence theorem?
  26. haushofer

    Understanding Stokes Theorem and is the variation of the metric a tensor?

    Hi, I have 2 little questions and hope to find some clarity here. It concerns some mathematics. 1) Is the variation of the metric again a tensor? I have the suspicion that it's not, because i would say that it doesn't transform like one. How can i get a sensible expression then for the...
  27. T

    Proving Stokes Theorem: The Intuition and Application

    I was wondering as to how to prove stokes theorem in its general and smexy form.Also what is the intuition behind it(more important) aside from the fact that its a more general form of the other theorems from vector calculus?
  28. J

    Stokes Theorem for Surface S: Parametrization, Flux and Integral

    For the surface S (helicoid or spiral ramp) swept out by the line segment joining the point (2t, cost, sint) to (2t,0,0) where 0 is less than or equal to t less than or equal to pi. (a) Find a parametrisation for this surface S and of the boundary A of this surface. I can only guess that...
  29. S

    Help stokes theorem - integral problem

    Hello all, http://img244.imageshack.us/img244/218/picture8ce5.png I am completely new to this stokes theorem bussiness..what i have got so far is the nabla x F part, but i am unsure of how to find N (the unit normal field i think its called). any suggestions people? i get that nabla...
  30. D

    Solving Stokes Theorem Problem: F(x,y,z)

    Hi, i can't seem to figure out how stokes theorem works. I've run through a lot of examples but i still am not having any luck. Anyway, some advice on a particular problem would be greatly appreciated. The problem is: F(x,y,z) = <2y,3z,-2x>. The surface is the part of the unit...
  31. Cyrus

    How Is Stokes Theorem Applied to Partial Derivatives?

    I can't figure this out, help me NOW! :-p Just kidding. So anyways, here's the question: Part of stokes theorem has the following in it: \frac {\partial } {\partial x} ( Q + R \frac{\partial z}{\partial y} ) Which is written as: \frac { \partial Q }{\partial x} + \frac {...
  32. K

    Directionality in Stokes Theorem for Volumes

    I'm not sure if this post should go here or in the Calc setion, but I figure more knowledgeable people browse this form. This question is relating to 'directionality' of doing closed loop integrals. If you have some 2D wire structure, let's image it looks like a square wave, or a square well...
  33. M

    Is Stokes Theorem Easier Than It Seems?

    i thought stokes theorem (green's thm) was hard after reading it in spivak, who calls it trivial nonetheless. however lang showed it is indeed trivial in his analysis I. the same proof occurs in courant. I.e. the point is that the theorem is easy for a rectangle, where it follows...
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