Kelvin–Stokes theorem
Part of a series of articles about  
Calculus  



Specialized 

The Kelvin–Stokes theorem^{[1]}^{[2]}^{[3]}^{[4]}^{[5]} (named for Lord Kelvin and George Stokes), also known as the curl theorem,^{[6]} is a theorem in vector calculus on R^{3}. 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. The Kelvin–Stokes theorem is a special case of the “generalized Stokes' theorem.”^{[7]}^{[8]} In particular, a vector field on R^{3} can be considered as a 1form in which case curl is the exterior derivative.
Theorem
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 that is bounded by γ and suppose ψ: D → R^{3} is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))^{[note 1]} and F is a smooth vector field on R^{3}, then:^{[1]}^{[2]}^{[3]}
Proof
The proof of the theorem consists of 4 steps.^{[2]}^{[3]}^{[note 2]} 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). When proving this theorem, mathematicians normally use the differential form. The "pullback^{[note 2]} 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.
First step of the proof (defining the pullback)
Define
so that P is the pullback^{[note 2]} of F, and that P(u, v) is R^{2}valued function, dependent on two parameters u, v. In order to do so we define P_{1} and P_{2} as follows.
Where is the normal inner product (for Euclidean vectors, the dot product; see Braket notation) of R^{3} and hereinafter, stands for the bilinear form according to matrix A.^{[note 3]}
Second step of the proof (first equation)
According to the definition of a line integral,
where, Jψ stands for the Jacobian matrix of ψ, and the clear circle denotes function composition. Hence,^{[note 3]}
So, we obtain the following equation
Third step of the proof (second equation)
First, calculate the partial derivatives, using the Leibniz rule (product rule):
So,^{[note 3]}^{[note 4]}
On the other hand, according to the definition of a surface integral,
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.
Application for conservative vector fields and scalar potential
In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.
First, we define the notarization map as follows.
is a strictly increasing function. For all piecewise smooth paths c: [a, b] → R^{3} and all smooth vector fields F, the domain of which includes c([a, b]), one has:
So, we can assume the domain of the curve to be [0, 1].
The Lamellar vector field
In fluid dynamics, it is often referred to as a vortexfree or irrotational vector field. Furthermore, if the domain of F is simply connected, then in mechanics, it can be identified as a conservative force.
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.^{[note 5]}
That theorem is also important in the area of Homotopy theorem.^{[7]}
Some textbooks such as Lawrence^{[7]} call the relationship between c_{0} and c_{1} stated in Theorem 21 as “homotope” and the function H: [0, 1] × [0, 1] → U as “homotopy between c_{0} and c_{1}”. However, “homotope” or “homotopy” in abovementioned sense are different (stronger than) typical definitions of “homotope” or “homotopy”.^{[note 6]} So from now on we refer to homotopy (homotope) in the sense of Theorem 21 as tubelikehomotopy (homotope).^{[note 7]}
Proof of the theorem
Hereinafter, the ⊕ stands for joining paths^{[note 8]} the stands for backwards of curve^{[note 9]}
Let D = [0, 1] × [0, 1]. By our assumption, c_{1} and c_{2} are piecewise smooth homotopic, there are the piecewise smooth homogony H: D → M
And, let S be the image of D under H. Then,
will be obvious according to the Theorem 1 and, F is Lamellar vector field that, right side of that equation is zero, so,
Here,
 ^{[note 8]}
and, H is TubelerHomotopy that,
that, line integral along Γ_{2}(s) and line integral along Γ_{4}(s) are compensated each other^{[note 9]} so,
On the other hand,
that, subjected equation is proved.
Application for conservative force
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, obviously follows from Theorem 21. In Lemma 22, the existence of H satisfying [SC0] to [SC3] is crucial. It is a wellknown fact that, if U is simply connected, such H exists. The definition of Simply connected space follows:
You will find that, the [SC1] to [SC3] of both Lemma 22 and Definition 22 is same.
So, someone may think that, "for a conservative force, the work done in changing an object's position is path independent" is elucidated. However, there are very large gaps between following two:
 There are continuous H such that it satisfies [SC1] to [SC3]
 There are piecewise smooth H such that it satisfies [SC1] to [SC3]
To fill that gap, the deep knowledge of Homotopy Theorem is required. For example, the following resources may be helpful for you.
 Lee teaches Whitney approximation theorem (^{[8]} page 136) and "How to use that theorem to this isuue" (^{[8]} page 421).
 More general statements appear in^{[9]} (see Theorems 7 and 8).
Considering abovementioned fact and Lemma 22, we will obtain following theorem.
Kelvin–Stokes theorem on singular 2cube and cube subdivisionable sphere
Singular 2cube and boundary
We omit the proof of the lemma. Using the lemma from now we consider all singular 2cubes to be notarized. In other words, we assume that the domain of all singular 2cubes is I × I.
In order to facilitate the discussion of boundary, we define
γ_{1}, ..., γ_{4} are the onedimensional edges of the image of I × I.Hereinafter, the ⊕ stands for joining paths^{[note 8]} and, the stands for backwards of curve.^{[note 9]}
Cube subdivision
The definition of the boundary of the Definitions 33 is apparently depends on the cube subdivision. However, considering the following fact, the boundary is not depends on the cube subdivision.
Therefore, the following Definition is welldefined:
Notes
 ↑ γ and Γ are both loops, however, Γ is not necessarily a Jordan curve
 1 2 3 Knowledge of differential forms and identification of the vector field A = (a_{1}, a_{2}, a_{3}),
 1 2 3 Given a n × m matrix A we define a bilinear form:
 ↑ We prove following (★0).
 (★0)
 (★1)
 (★21)
 (★22)
 (★23)
 (★3)
 (★4)
 ↑ There are a number of theorems with the same name, however they are not necessarily the same.
 ↑ Typical definition of homotopy and homotope are as follows.
 ↑ In some textbooks such as Conlon, Lawrence (2008). Differentiable Manifolds. Modern Birkhauser Classics. Boston: Birkhaeuser. use the term of homotopy and homotope in Theorem 21 sense. homotopy and homotope in Theorem 21 sense Indeed, it is convenience to adopt such sense to discuss conservative force. However, homotopy in Theorem 21 sense and homotope in Theorem 21 sense are different from and stronger than homotopy in typical sense and homotope in typical sense. So there are no appropriate terminology which can discriminate between homotopy in typical sense and sense of Theorem 21. In this article, to avoid ambiguity and to discriminate between them, we will define two “justintime term”, tubelike homotopy and tubelike homotope as follows.
 1 2 3 If the two curves α: [a_{1}, b_{1}] → M, β: [a_{2}, b_{2}] → M, satisfy α(b_{1}) = β(a_{2}) then, we can define new curve α ⊕ β so that, for all smooth vector field F (if domain of which includes image of α ⊕ β)
 1 2 3 Given curve on M, α: [a_{1}, b_{1}] → M, we can define new curve α so that, for all smooth vector field F (if domain of which includes image of α)
References
 1 2 Stewart, James (2010). Essential Calculus: Early Transcendentals. Cole.
 1 2 3 This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) , please refer the
 1 2 3 This proof is also same to the proof shown in
 1 2 3 4 5 Nagayoshi Iwahori, et.al:"BiBunSekiBunGaku" ShoKaBou(jp) 1983/12 ISBN 9784785310394 (Written in Japanese)
 1 2 Atsuo Fujimoto;"VectorKaiSeki Gendai sugaku rekucha zu. C(1)" BaiFuKan(jp)(1979/01) ISBN 9784563004415 (Written in Japanese)
 ↑ http://mathworld.wolfram.com/CurlTheorem.html
 1 2 3 4 5 6 7 Lawrence Conlon; "Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11)
 1 2 3 4 5 6 John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218) "Springer (2002/9/23)
 ↑ 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. MR 0115178 (22 #5980 )
 ↑ Spivak, Michael (1971). Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus. Westview Press.