# Kelvin–Stokes theorem

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*For a more conventional discussion see***Stokes Theorem**.

The **Kelvin–Stokes theorem**,^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}
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, the vector field on **R**^{3} can be considered as a 1-form in which case curl is the exterior derivative.

## Theorem

Let γ : [*a*, *b*] → **R**^{2} be a Piecewise smooth Jordan plane curve, that bounds the domain *D* ⊂ **R**^{2}.^{[note 1]} Suppose ψ : *D* → **R**^{3} is smooth, with *S* := ψ[*D*]
^{[note 2]}
, and Γ is the space curve defined by Γ(*t*) = ψ(γ(*t*)).^{[note 3]} If **F** a smooth vector field on **R**^{3}, then^{[1]}^{[2]}^{[3]}

## Proof

The proof of the Theorem consists of 4 steps.^{[2]}^{[3]}^{[note 4]} We assume that the Green's theorem is known, so what is of concern is "how to boil down the three-dimensional complicated problem (Kelvin–Stokes theorem) to a two-dimensional rudimentary problem (Green's theorem)". In an ordinary, mathematicians use the differential form, especially "pull-back^{[note 4]} of differential form" is very powerful tool for this situation, however, learning differential form needs too much background knowledge. So, the proof below does not require background information on differential form, and may be helpful for understanding the notion of differential form.

### First step of the proof (defining the pullback)

Define

so that **P** is the pull-back^{[note 4]} of **F**, and that **P**(*u*, *v*) is **R**^{2}-valued function, depends on two parameter *u*, *v*. In order to do so we define *P*_{1} and *P*_{2} as follows.

Where, is the normal inner product of **R**^{3} and
hereinafter, stands for bilinear form according to matrix **A** ^{[note 5]}
.^{[note 6]}

### Second step of the proof (first equation)

According to the definition of line integral,

where, *J*ψ stands for the Jacobian matrix of ψ. Hence,^{[note 5]}^{[note 6]}

So, we obtain following equation

### Third step of the proof (second equation)

First, calculate the partial derivatives, using Leibniz rule of inner product

So,^{[note 5]} ^{[note 6]}
^{[note 7]}

On the other hand, according to the definition of surface integral,

So, we obtain

### Fourth step of the proof (reduction to Green's theorem)

According to the result of the second step, and according to the result of Third step, and further considering the Green's theorem, subjected equation is proved.

## Application for conservative force 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.

Above-mentioned is a strongly increasing function that, for all piece wise smooth paths *c*:[*a*,*b*] → **R**^{3}, for all smooth vector field *F*, domain of which includes (image of [*a*,*b*] under *c*.), following equation is satisfied.

So, we can unify the domain of the curve from the beginning to [0,1].

### The Lamellar vector field

In mechanics a lamellar vector field is called a conservative force; in fluid dynamics, it is called a Vortex-free vector field. So, lamellar vector field, conservative force, and vortex-free vector field are the same notion.

### Helmholtz's theorems

In this section, we will introduce a theorem that is derived from the Kelvin–Stokes theorem and characterizes vortex-free vector fields. In fluid dynamics it is called Helmholtz's theorems,.^{[note 8]}

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 2-1 as “homotope”and the function *H* : [0, 1] × [0, 1] → *U* as “homotopy between *c*_{0} and *c*_{1}”.

However, “homotope” or “homotopy” in above-mentioned sense are different toward (stronger than) typical definitions of “homotope” or “homotopy”.^{[note 9]}

So there are no appropriate terminology which can discriminate between homotopy in typical sense and sense of Theorem 2-1. So, in this article, to discriminate between them, we say “Theorem 2-1 sense homotopy as **tube-like-homotopy** and, we say “Theorem 2-1 sense homotope” as **tube-like homotope**.^{[note 10]}

### Proof of the theorem

Hereinafter, the ⊕ stands for joining paths
^{[note 11]}
the stands for backwards of curve
^{[note 12]}

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,

and, H is Tubeler-Homotopy that,

that, line integral along
and line integral along
are compensated each other^{[note 12]}
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 2-2, which is a corollary of and a special case of Helmholtz's theorem.

Lemma 2-2, obviously follows from Theorem 2-1. In Lemma 2-2, the existence of *H* satisfying [SC0] to [SC3]" is crucial. It is a well-known 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 2-2 and Definition 2-2 is same.

So, someone may think that, the issue, "when the Conservative Force, the work done in changing an object's position is path independent" is elucidated. However **there are very large gap 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, to fill the gap, 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 above-mentioned fact and Lemma 2-2, we will obtain following theorem. That theorem is anser for subjecting issue.

## Kelvin–Stokes theorem on singular 2-cube and cube subdivisionable sphere

### Singular 2-cube and boundary

Given , we define the notarization map of sngler two cube

here, the I:=[0,1] and I^{2} stands for .

Above-mentioned is strongly increase function (that means (for all ) that, following lemma is satisfied.

Above-mentioned lemma is obverse that, we neglects the proof. Acceding to the above-mentioned lemma, hereinafter, we consider that, domain of all singular 2-cube are notarized (that means, hereinafter, we consider that domain of all singular 2-cube are from the beginning, *I*^{2}.

In order to facilitate the discussion of boundary, we define by

γ_{1}, ..., γ_{4} are the one-dimensional edges of the image of *I*^{2}.Hereinafter, the ⊕ stands for joining paths^{[note 11]} and,
the stands for backwards of curve
.^{[note 12]}

### Cube subdivision

The definition of the boundary of the Definitions 3-3 is apparently depends on the cube subdevision. However, considering the following fact, the boundary is not depends on the cube subdevision.

So, considering the above-mentioned fact, following "Definition3-4" is well-defined.

## Notes and references

### Notes

- ↑ The Jordan curve theorem implies that the Jordan curve divides
**R**^{2}into two components, a compact one (the bounded area) and another is non-compact. - ↑
When ψ is a mapping and
D is a subset of the domain of ψ,
ψ[
*D*] stands for the image of D under ψ. - ↑ γ and Γ are both loops, however, Γ is not necessarily a Jordan curve
- ↑
^{4.0}^{4.1}^{4.2}If you know the differential form, when we considering following identification of the vector field**A**= (*a*_{1},*a*_{2},*a*_{3}), the proof here is similar to the proof using the pull-back of ω_{F}. In actual, under the identification of ω_{F}=**F**following equations are satisfied. Here,*d*stands for Exterior derivative of Differential form, ψ* stands for pull back by ψ and,*P*_{1}and*P*_{2}of above mentioned are same as*P*_{1}and*P*_{2}of the body text of this article respectively. - ↑
^{5.0}^{5.1}^{5.2}Given a*n*×*m*matrix*A*we define a bilinear form: which also satisfies: - ↑
^{6.0}^{6.1}^{6.2}Given a*n*×*m*matrix*A*,stands for transposed matrix of^{t}A*A*. - ↑
We prove following (★0).
First, we focus attention on the linearity of the algebraic operator "
**a**×" and, we obtain the matrix representation thereof. The method how to obtain the matrix representation of Linear map are described in for example [1].Let both of**a**and**x**are the three dimensional Column vector and are represented as follows, Then, according to the definition of Cross product ” ×”,**a**×**x**are represented as follows. Therefore, when we act the operator ”**a**×” to each of the standard basis, we obtain followings, therefore, we obtain the matrix representation of**a**×” as shown in (★1). Seconds, let is a 3×3 matrix and using the following substitution,( a_{1}, ... , a_{3}are components of**a**), Then we obtain following (★3) from the (★1). Third we substitute the to abovementioned A, under the substitution of (★2-1), (★2-2), and (★2-3), we obtain the following (★4) The (★0) is obvious from (★3) and (★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 Lawrence Conlon;"Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11)[2] use the term of homotopy and homotope in Theorem 2-1 sense. homotopy and homotope in Theorem 2-1 sense Indeed, it is convenience to adopt such sense to discuss conservative force. However, homotopy in Theorem 2-1 sense and homotope in Theorem 2-1 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 2-1. In this article, to avoid ambiguity and to discriminate between them, we will define two “just-in-time term”,
**tube-like homotopy**and**tube-like homotope**as follows. - ↑
^{11.0}^{11.1}^{11.2}Given two curves α: [a_{1}, b_{1}] →*M*, β: [a_{2}, b_{2}, ] →*M*, if α and β 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 ) which is also used when we define the fundamental group. To do so, accurate definition of the “joint of paths” is as follows. - ↑
^{12.0}^{12.1}^{12.2}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 ) which is also used when we define fundamental group. To do so, accurate definition of the “backwards of curve” is as follows.And, given two curves on

*M*, α: [a_{1}, b_{1}] →*M*β: [a_{2}, b_{2}] →*M*which satisfies*α*(*b*_{1}=*β*(*b*_{2}) (that means*α*(*b*_{1}) =*β*(*a*_{2}), we can define as following manner.

### References

- ↑
^{1.0}^{1.1}James Stewart;"Essential Calculus: Early Transcendentals" Cole Pub Co (2010)[3] - ↑
^{2.0}^{2.1}^{2.2}This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) [4], please refer the [5] - ↑
^{3.0}^{3.1}^{3.2}This proof is also same to the proof shown in - ↑
^{4.0}^{4.1}^{4.2}^{4.3}^{4.4}Nagayoshi Iwahori, et.al:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 ISBN978-4-7853-1039-4 [6](Written in Japanese) - ↑
^{5.0}^{5.1}Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" Bai-Fu-Kan(jp)(1979/01) ISBN 978-4563004415 [7](Written in Japanese) - ↑ http://mathworld.wolfram.com/CurlTheorem.html
- ↑
^{7.0}^{7.1}^{7.2}^{7.3}^{7.4}^{7.5}^{7.6}Lawrence Conlon; "Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11) [8] - ↑
^{8.0}^{8.1}^{8.2}^{8.3}^{8.4}^{8.5}John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218) "Springer (2002/9/23) [9] [10] - ↑ 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 [11])[12]
- ↑ Michael Spivak:"Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus" Westview Press, 1971 [13]