Kelvin–Stokes theorem
The Kelvin–Stokes theorem,[1][2] named after Lord Kelvin and George Stokes, also known as the Stokes' theorem,[3] the fundamental theorem for curls or simply the curl theorem,[4] is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface.
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Specialized 

If a vector field is defined in a region with smooth oriented surface and has first order continuous partial derivatives then:
where is boundary of region with smooth surface .
The Kelvin–Stokes theorem is a special case of the “generalized Stokes' theorem.”[5][6] In particular, a vector field on can be considered as a 1form in which case curl is the exterior derivative.
Theorem
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are wellknown not to exhibit a Riemannintegrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a nonLipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for fulldimensional subsets of ℝ^{2}.
Let γ: [a, b] → R^{2} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R^{2} into two components, a compact one and another that is noncompact. Let D denote the compact part; then D is bounded by γ. It now suffices to transfer this notion of boundary along a continuous map to our surface in ℝ^{3}. But we already have such a map: the parametrization of Σ.
Suppose ψ: D → R^{3} is smooth, with Σ = ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))[note 1], then we call Γ the boundary of Σ, written ∂Σ.
With the above notation, if F is any smooth vector field on R^{3}, then[7][8]
Proof
The proof of the theorem consists of 4 steps. We assume Green's theorem, so what is of concern is how to boil down the threedimensional complicated problem (Kelvin–Stokes theorem) to a twodimensional rudimentary problem (Green's theorem).[9] When proving this theorem, mathematicians normally use the differential form. The "pullback of a differential form" is a very powerful tool for this situation, but learning differential forms requires substantial background knowledge. So, the proof below does not require knowledge of differential forms, and may be helpful for understanding the notion of differential forms.[8]
Elementary proof
First step of the proof (parametrization of integral)
As in Theorem § Notes, we reduce the dimension by using the natural parametrization of the surface. Let ψ and γ be as in that section, and note that by change of variables
where Jψ stands for the Jacobian matrix of ψ.
Now let {e_{u},e_{v}} be an orthonormal basis in the coordinate directions of ℝ^{2}. Recognizing that the columns of J_{y}ψ are precisely the partial derivatives of ψ at y , we can expand the previous equation in coordinates as
Second step in the proof (defining the pullback)
The previous step suggests we define the function
This is the pullback of F along ψ , and, by the above, it satisfies
We have successfully reduced one side of Stokes' theorem to a 2dimensional formula; we now turn to the other side.
Third step of the proof (second equation)
First, calculate the partial derivatives appearing in Green's theorem, via the product rule:
Conveniently, the second term vanishes in the difference, by equality of mixed partials. So,
But now consider the matrix in that quadratic form—that is, . We claim this matrix in fact describes a cross product.
To be precise, let be an arbitrary 3 × 3 matrix and let
Note that x ↦ a × x is linear, so it is determined by its action on basis elements. But by direct calculation
Thus (AA^{T}) x = a × x for any x . Substituting J F for A, we obtain
We can now recognize the difference of partials as a (scalar) triple product:
On the other hand, the definition of a surface integral also includes a triple product — the very same one!
So, we obtain
Fourth step of the proof (reduction to Green's theorem)
Combining the second and third steps, and then applying Green's theorem completes the proof.
Proof via differential forms
ℝ→ℝ^{3} can be identified with the degree1 one differential forms on ℝ^{3} via the map
 .
Write the differential 1form associated to a function F as ω_{F}. Then one can calculate that
where ★ is the Hodge star. Thus, by generalized Stokes' theorem, [10]
Applications
In fluid dynamics
In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.
Irrotational fields
If the domain of F is simply connected, then F is a conservative vector field.
Helmholtz's theorems
In this section, we will introduce a theorem that is derived from the Kelvin–Stokes theorem and characterizes vortexfree vector fields. In fluid dynamics it is called Helmholtz's theorems.
Some textbooks such as Lawrence[5] call the relationship between c_{0} and c_{1} stated in Theorem 21 as “homotopic” and the function H: [0, 1] × [0, 1] → U as “homotopy between c_{0} and c_{1}”. However, “homotopic” or “homotopy” in abovementioned sense are different (stronger than) typical definitions of “homotopic” or “homotopy”; the latter omit condition [TLH3]. So from now on we refer to homotopy (homotope) in the sense of Theorem 21 as a tubular homotopy (resp. tubularhomotopic).[note 2]
Proof of the theorem
In what follows, we abuse notation and use "+" for concatenation of paths in the fundamental groupoid and "" for reversing the orientation of a path.
Let D = [0, 1] × [0, 1], and split ∂D into 4 line segments γ_{j}.
By our assumption that c_{1} and c_{2} are piecewise smooth homotopic, there is a piecewise smooth homotopy H: D → M
Let S be the image of D under H. That
follows immediately from the Kelvin–Stokes theorem. F is lamellar, so the left side vanishes, i.e.
As H is tubular, Γ_{2}=Γ_{4}. Thus the line integrals along Γ_{2}(s) and Γ_{4}(s) cancel, leaving
On the other hand, c_{1}=Γ_{1} and c_{3}=Γ_{3}, so that the desired equality follows almost immediately.
Conservative forces
Helmholtz's theorem, gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 22, which is a corollary of and a special case of Helmholtz's theorem.
Lemma 22 follows from Theorem 21. In Lemma 22, the existence of H satisfying [SC0] to [SC3] is crucial. If U is simply connected, such H exists. The definition of Simply connected space follows:
The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately. But recall that simpleconnection only guarantees the existence of a continuous homotopy satisfiying [SC13]; we seek a piecewise smooth hoomotopy satisfying those conditions instead.
However, the gap in regularity is resolved by the Whitney approximation theorem.[6]^{:136,421}[12] We thus obtain the following theorem.
Notes
 Γ may not be a Jordan curve, if the loop γ interacts poorly with ψ. Nonetheless, Γ is always a loop, and topologically a connected sum of countablymany Jordan curves, so that the integrals are welldefined.
 There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of Theorem 21.[11] Indeed, this is very convenient for the specific problem of conservative forces. However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that end.
References
 Nagayoshi Iwahori, et.al:"BiBunSekiBunGaku" ShoKaBou(jp) 1983/12 ISBN 9784785310394 (Written in Japanese)
 Atsuo Fujimoto;"VectorKaiSeki Gendai sugaku rekucha zu. C(1)" BaiFuKan(jp)(1979/01) ISBN 9784563004415 (Written in Japanese)
 Stewart, James (2012). Calculus  Early Transcendentals (7th ed.). Brooks/Cole Cengage Learning. p. 1122. ISBN 9780538497909.
 Griffiths, David (2013). Introduction to Electrodynamics. Pearson. p. 34. ISBN 9780321856562.
 Lawrence Conlon; "Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11)
 John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218) "Springer (2002/9/23)
 Stewart, James (2010). Essential Calculus: Early Transcendentals. Cole.
 Robert Scheichl, lecture notes for University of Bath mathematics course
 Colley, Susan Jane (2002). Vector Calculus (4th ed.). Boston: Pearson. pp. 500–3.
 Edwards, Harold M. Advanced Calculus: A Differential Forms Approach. Springer.
 Conlon, Lawrence (2008). Differentiable Manifolds. Modern Birkhauser Classics. Boston: Birkhaeuser.
 L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American Mathematical Society Translations, Ser. 2, Vol. 11, American Mathematical Society, Providence, R.I., 1959, pp. 1–114. MR0115178 (22 #5980 ). See theorems 7 & 8.