What is Stokes theorem: Definition and 80 Discussions

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.,








Ω


ω
=



Ω


d
ω

.


{\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:







Γ



F




d


Γ


=



S



×

F




d

S



{\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






(




F

x







F

y







F

z





)



d
Γ


F

x



d
x
+

F

y



d
y
+

F

z



d
z


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






×


(




F

x







F

y







F

z





)



d

S

=


(






y



F

z






z



F

y









z



F

x






x



F

z









x



F

y






y



F

x





)



d

S




{\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 }





d
(

F

x



d
x
+

F

y



d
y
+

F

z



d
z
)
=
(



y



F

z






z



F

y


)

d
y

d
z
+
(



z



F

x






x



F

z


)

d
z

d
x
+
(



x



F

y






y



F

x


)

d
x

d
y


{\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).

View More On Wikipedia.org
  1. WMDhamnekar

    Computing line integral using Stokes' theorem

    ##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...
  2. M

    I Stokes Theorem: Vector Integral Identity Proof

    Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...
  3. beefbrisket

    I Sign mistake when computing integral with differential forms

    The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
  4. Morbidly_Green

    Using Stoke's theorem on an off-centre sphere

    Homework Statement Homework Equations Stokes theorem $$\int_C \textbf{F} . \textbf{dr} = \int_S \nabla \times \textbf{F} . \textbf{ds}$$ The Attempt at a Solution I have the answer to the problem but mine is missing a factor of$$\sqrt 2 $$ I can't seem to find my error
  5. yecko

    Navier Stokes Thm Homework: Equations & Solutions

    Homework Statement [/B] Homework Equations Navier strokes theorem The Attempt at a Solution May I ask why would there suddenly a "h" in the highlighted part? "h" wasnt existed in the previous steps, which C2=0 shouldn't add height of the liquid as a constant in the formula... thanks
  6. C

    Faraday's Law and Stokes Theorem

    Homework Statement Homework Equations Stokes Theorem The Attempt at a Solution I'm having a tough time "cancelling" out integrals from both sides of an equation (if possible). On the right hand of the equation, we know since it is a closed curve, that Stoke's Theorem applies and we can...
  7. M

    MHB Calculate integral using Stokes Theorem

    Hey! :o I want to calculate $\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )$ using the fomula of Stokes, when $\sigma$ is the curve that is defined by the relations $x^2+y^2=1$ and $x+y+z=1$. Is the curve not closed? Because we have an integral of the form $\int_{\sigma}$ and not of the form...
  8. S

    I Understanding De Rham's Period and Stokes Theorem

    Hello! I am reading this paper and on page 9 it defines the De Rham's period as ##\int_C \omega = <C,\omega>##, where C is a cycle and ##\omega## is a closed one form i.e. ##d\omega = 0##. The author says that ##<C,\omega>:\Omega^p(M) \times C_p(M) \to R##. I am a bit confused by this, as...
  9. E

    Determining between direct evaluation or vector theorems

    So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...
  10. C

    I Proofs of Stokes Theorem without Differential Forms

    Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...
  11. R

    B Breaking Stokes theorem

    Let's say there is a 5 sided cube that is missing the bottom face. Obviously, there is a boundary curve at the middle of this cube that goes around the 4 sides, front, right, back, and left. This boundary curve forms the boundary of the top half of the cube with the 5 faces and the bottom...
  12. ShayanJ

    A Non-Abelian Stokes theorem and variation of the EL action

    Today I heard the claim that its wrong to use Stokes(more specificly divergence/Gauss) theorem when trying to get the Einstein equations from the Einstein-Hilbert action and the correct way is using the non-Abelian stokes theorem. I can't give any reference because it was in a talk. It was the...
  13. ifidamas

    Orientability of Null Submanifold w/ Boundary - Stokes' Theorem

    I have this question: Is it possible to define an orientation for a null submanifold with boundary? In that case, is possible to use Stokes' theorem? In particular, there is a way to define a volume form on that submanifold?
  14. O

    Potential magnetic field lines and Stokes theorem....

    Hi, A potential magnetic field has no curl. According to the "curl theorem" or stokes theorem, a vector field with no curl does not close. Yet, Maxwell's equation tell us we shall not have magnetic monopoles, so the loops have to close... ? What am I missing to remove this apparent paradox of a...
  15. A

    Stokes theorom question with a line

    Homework Statement F[/B]=(y + yz- z, 5x+zx, 2y+xy ) use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1) 2. The attempt at a solution I was thinking that I could decide...
  16. B

    Spherical coordinates path integral and stokes theorem

    Homework Statement Homework Equations The path integral equation, Stokes Theorem, the curl The Attempt at a Solution [/B] sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...
  17. H

    MHB Understanding the proof of a lemma concerning Stokes Theorem

    I have problems understanding the proof of this lemma: $$\lambda \in \Lambda (m, n), \ \ \text{this means that it is an increasing function} \ \ \lambda: \{1,2,...,m\} \rightarrow \{1,...,n\}, \ \ \text{so} \ \ \lambda(1) < ... < \lambda(m)$$ $$p_{\lambda} : \mathbb{R}^m \ni (x_1, ..., x_m)...
  18. K

    Boundary of a chain, Stokes' theorem.

    Hi, I'm studying multivariable analysis using Spivak's book "calculus on manifolds" When I see this book, one strange problem arouse. Thank you for seeing this. Here is the problem. c0 , c1 : [0,1] → ℝ2 - {0} c : [0,1]2 → ℝ2 - {0} given by c0(s) = (cos2πs,sin2πs) : a circle of radius 1 c1(s) =...
  19. AwesomeTrains

    Line Integral - Stokes theorem

    Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...
  20. M

    When using stokes theorem to remove integrals

    hey pf! i had a question. namely, in the continuity equation we see that \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S} and we may use the divergence theorem to have: \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho...
  21. G

    Stokes theorem, parametrizing composite curves

    Homework Statement Calculate the line integral: F = <xz, (xy2 + 2z), (xy + z)> along the curve given by: 1) x = 0, y2 + z2 = 1, z > 0, y: -1 → 1 2) z = 0, x + y = 1, y: 1→0 3) z = 0, x-y = 1, y: 0 → -1 Homework Equations The Attempt at a Solution I don't think the...
  22. F

    Flux integral of surface using Stokes theorem

    Homework Statement I have to control stokes theorem( I have to calculate line-and surface integral. I have a vectorfield a=(3y,xz,yz^2). And surface S is a paraboloid 2z=x^2+y^2. And it is limited by plane z=2. For line integral the line is a circle C: x^2+y^2=4 on the plane z=2. Vector n is...
  23. P

    Use Stokes Theorem to show a relationship

    Homework Statement Use the Stokes' Theorem to show that \intf(∇ X A) dS = \int(A X ∇f) dS + \ointf A dl Homework Equations Use vector calculus identities. Hint given : Start with the last integral in the above relation. The Attempt at a Solution To be honest, I really don't know...
  24. M

    Finding the n in stokes theorem.

    Homework Statement Hey guys, I'm having trouble finding the n in stokes theorem. For example, F(x,y,z)= z2i+2xj-y3; C is the circle x2 + y2=1 in the xy-plane with counterclockwise orientation looking down the positive z-axis. ∫∫CurlF*n I know the curl is -3y2i+2zj+2k The...
  25. R

    Stokes theorem in a cylindrical co-ordinates, vector field

    Homework Statement given a vector field v[/B=]Kθ/s θ (which is a two dimensional vector field in the direction of the angle, θ with a distance s from the origin) find the curl of the field and verify stokes theorem applies to this field, using a circle of radius R around the origin Homework...
  26. G

    Stokes Theorem cone oriented downwards

    Homework Statement Verify stokes theorem where F(xyz) = -yi+xj-2k and s is the cone z^2 = x^2 + y^2 , 0≤ Z ≤ 4 oriented downwards Homework Equations \oint_{c} F.dr = \int\int_{s} (curlF).dS The Attempt at a Solution Firstly the image of the widest part of the cone on the xy plane is the...
  27. G

    Use Stokes Theorem to evaluate the integral

    Homework Statement Use Stokes Theorem to evaluate the integral\oint_{C} F.dr where F(x,y,z) = e^{-x} i + e^x j + e^z k and C is the boundary of that part of the plane 2x+y+2z=2 in the first octant Homework Equations \oint_{C} F.dr = \int\int curlF . dS The Attempt at a Solution So first...
  28. G

    Stokes Theorem paraboloid intersecting with cylinder

    Homework Statement Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards Homework Equations The Attempt at a...
  29. T

    Parameterizing Z-Value in Line Integral for Cylinder (Stokes Thm)

    I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...
  30. G

    Proving Stokes Theorem w/ Homework Equations

    Homework Statement Prove that ## \oint_{\partial S} ||\vec{F}||^2 d\vec{F} = -\int\int_S 2 \vec{F}\times d\vec{A} ## Homework Equations Identities: ##\nabla \times (||\vec{F}||^2 \vec{k}) = 2\vec{F} \times \vec{k} ## For ##\vec{k} ## constant i.e. ## \nabla \times \vec{k} = 0 ## Stokes...
  31. H

    Trouble with the unit normal vector for stokes theorem

    We're given x^2+2*y^2=1. so x^2=1-2y^2 now using distance formula d^2=x^2+y^2 since x^2=1-2y^2, substituting it in the distance formula we get: d^2=1-2y^2+y^2=1-y^2; differentiating and then setting the eq to 0 we get; 0=-4y or y=0. now x^2=1-2y^2=1 so x=+-1 so point having...
  32. H

    Stokes theorem over a hemisphere

    Homework Statement The vector field F is defined in 3-D Cartesian space as F = y(z^2−a^2)i + x(a^2− z^2)j, where i and j are unit vectors in the x and y directions respectively, and a is a real constant. Evaluate the integral  Integral:(∇ ×F)·dS, where S is the open surface of the...
  33. M

    Help doing an integral using stokes theorem?

    Homework Statement F= xi + x3y2j + zk C is the boundary of the semi-ellipsoid z=√(4-4x2-y2) in the plane z=0 Homework Equations Stokes theorem states: ∫∫(curlF ° n)dS The Attempt at a Solution I found the curl of the F to be 3x2y2k I found that the dot product of CurlF and n =...
  34. M

    Concept question about stokes theorem?

    Homework Statement This is not actually a homework question, just a question I ran into while studying for my math final. When I am using stokes theorem: ∫∫(curlF ° n)dS I have listed in my notes from lecture that there are time when it is applicable to replace dS with an easier...
  35. A

    When is not necesary to do surface ordering in the non-Abelian Stokes theorem?

    Acording to the non-Abelian stokes thoerem http://arxiv.org/abs/math-ph/0012035 I can transform a path ordered exponential to a surface ordered one. P e\oint\tilde{A}= P e∫F where F is some twisted curvature;F=U-1FU, and U is a path dependet operator.So, I have a system where every element...
  36. S

    [Calc 3] Verifying Stokes Theorem

    Homework Statement V.Field F(x,y,z)=<x^2 z, xy^2, z^2> where S is part of the plane x+y+z=1 inside cylinder x2 + y2 =9 Homework Equations Line integrals, Stokes Theorem, Parametrizing intersections... The Attempt at a Solution I found the answer to be 81pi/2 using stoke's theorem...
  37. B

    Stokes theorem application

    Homework Statement Let F(x, y, z) = \left ( e^{-y^2} + y^{1+x^2} +cos(z), -z, y \right) Let s Be the portion of the paraboloid y^2+z^2=4(x+1) for 0 \leq x \leq 3 and the portion of the sphere x^2 + y^2 +z^2 = 4 for x \leq 0 Find \iint\limits_s curl(\vec{F}) d \vec{s} Homework...
  38. C

    Verify Stokes theorem for the given Surface

    Homework Statement verify Stokes theorem for the given Surface and VECTOR FIELD x2 + y2+z2=4, z≤0 oriented by a downward normal. F=(2y-z)i+(x+y2-z)j+(4y-3x)k Homework Equations ∫∫S Δ χ F dS=∫ ∂SF.ds the triangle is supposed to be upside down. The Attempt at a Solution myΔχF =...
  39. S

    Question about Stokes Theorem?

    So I'm self teaching myself Multivariable Calculus from UCBerkeley's Youtube series and an online textbook. I'm up to Stokes Theorem and I'm getting conflicting definitions. UCBerkeley Youtube series says that Stokes Theorem is defined by: \int {(Curl f)} {ds} And then the textbook says that...
  40. Z

    Proving Stokes Theorem for Vector Field E on Given Contour and Surface

    Homework Statement for the vector field E=x(xy)-y(x^2 +2y^2) find E.dl along the contour find (nabla)xE along the surface x=0 and x=1 y=0 and y=1 Homework Equations The Attempt at a Solution i tried the second question (nabla)xE over the surface by finding the...
  41. P

    Solving Double Integral Using Stokes Theorem for Curl

    Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...
  42. jegues

    Troublesome Stokes Theorem Problem

    Homework Statement See figure attached for problem statement Homework Equations The Attempt at a Solution See figure attached for my attempt. I found this problem to be a little long and drawn out, which leads me to believe I made a mistake somewhere. Is this the case? Or...
  43. jegues

    Stokes Theorem Question

    Homework Statement Use Stoke's theorem to evaluate the line integral \oint y^{3}zdx - x^{3}zdy + 4dz where C is the curve of intersection of the paraboloid z = 2 + x^{2} + y^{2} and the plane z=5, directed clockwise as viewed from the point (0,0,7). Homework Equations The...
  44. 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...
  45. P

    Special case of Stokes theorem

    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.
  46. 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
Back
Top