The proof of the theorem consists of 4 steps.[note 2] We assume Green's theorem, 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). When proving this theorem, mathematicians normally use the differential form. The "pull-back[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)
so that P is the pull-back[note 2] of F, and that P(u, v) is R2-valued function, dependent on two parameters u, v. In order to do so we define P1 and P2 as follows.
Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics). and see p142 of Fujimoto
Let U ⊆ R3 be an opensubset with a Lamellar vector field F, and piecewise smooth loops c0, c1: [0, 1] → U. If there is a function H: [0, 1] × [0, 1] → U such that
[TLH0]H is piecewise smooth,
[TLH1]H(t, 0) = c0(t) for all t ∈ [0, 1],
[TLH2]H(t, 1) = c1(t) for all t ∈ [0, 1],
[TLH3]H(0, s) = H(1, s) for all s ∈ [0, 1].
Some textbooks such as Lawrence call the relationship between c0 and c1 stated in Theorem 2-1 as “homotope” and the function H: [0, 1] × [0, 1] → U as “homotopy between c0 and c1”. However, “homotope” or “homotopy” in above-mentioned 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 2-1 as tube-like-homotopy (homotope).[note 7]
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. Let U ⊆ R3 be an opensubset, with a Lamellar vector field F and a piecewise smooth loop c0: [0, 1] → U. Fix a point p ∈ U, if there is a homotopy (tube-like-homotopy) H: [0, 1] × [0, 1] → U such that
[SC0]H is piecewise smooth,
[SC1]H(t, 0) = c0(t) for all t ∈ [0, 1],
[SC2]H(t, 1) = p for all t ∈ [0, 1],
[SC3]H(0, s) = H(1, s) = p for all s ∈ [0, 1].
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:
Definition 2-2 (Simply Connected Space). Let M ⊆ Rn be non-empty, connected and path-connected. M is called simply connected if and only if for any continuous loop, c: [0, 1] → M there exists H: [0, 1] × [0, 1] → M such that
[SC0']H is continuous,
[SC1]H(t, 0) = c(t) for all t ∈ [0, 1],
[SC2]H(t, 1) = p for all t ∈ [0, 1],
[SC3]H(0, s) = H(1, s) = p for all s ∈ [0, 1].
You will find that, the [SC1] to [SC3] of both Lemma 2-2 and Definition 2-2 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 continuousH such that it satisfies [SC1] to [SC3]
There are piecewise smoothH 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.
Definition 3-1 (Singular 2-cube) Set D = [a1, b1] × [a2, b2] ⊆ R2 and let U be a non-empty opensubset of R3. The image of D under a piecewise smooth map ψ: D → U is called a singular 2-cube. Moreover, we define the notarization map of D
where I = [0, 1]. Then θD has the following property:
Lemma 3-1 (Notarization map of singular two cube). Let D be a singular 2-cube with map ψ and U ⊆ R3 open and non-empty. Suppose the image of I × I under a piecewise smooth map be a singular 2-cube. If F is a smooth vector field on U we have:
We omit the proof of the lemma. Using the lemma from now we consider all singular 2-cubes to be notarized. In other words, we assume that the domain of all singular 2-cubes is I × I.
In order to facilitate the discussion of boundary, we define
γ1, ..., γ4 are the one-dimensional edges of the image of I × I.Hereinafter, the ⊕ stands for joining paths[note 8] and, the stands for backwards of curve.[note 9]
Definition 3-2 (Cube subdivisionable sphere). (see Iwahori p. 399) A non-empty subset S ⊆ R3 is said to be a "Cube subdivisionable sphere" when there are at least one Indexed family of singular 2-cube
First, we note the linearity of the algebraic operator a × and we obtain its matrix representation (see linear map). Let both of a and x be 3-dimensional column vectors, represented as follows,
Then, according to the definition of the cross product, a × x are represented as follows.
Therefore, when we apply the operator a × to each of the standard basis, we obtain following, and thus, the matrix representation of a × as shown in (★1).
Next, let is a 3 × 3 matrix and using the following substitution, (a1, ... , a3 are components of a),
Then we obtain following (★3) from the (★1).
We substitute the JF to above mentioned 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.
Definition (Homotopy and Homotope). Suppose Z and W are topological spaces, with continuous maps f0, f1: Z → W.
(1) The continuous map H: Z × [0, 1] → W is said to be a "Homotopy between f0 and f1" if
[H1] H(t, 0) = f0(t) for all t ∈ Z,
[H2] H(t, 1) = f1(t) for all t ∈ Z.
(2) If there is a homotopy between f0 and f1", f0 and f1" are said to be homotope.
(3) Suppose f0 and f1 are homotope and H is a homotopy between them. f0 and f1 are said to be piecewise homotope, if f0, f1, and H are piecewise smooth. H is then said to be the piecewise homotopy between f0 and f1.
^In some textbooks such as Conlon, Lawrence (2008). Differentiable Manifolds. Modern Birkhauser Classics. Boston: Birkhaeuser. 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.
Definition (tube-like homotopy and tube-homotope). Suppose c0, c1 satisfy the following:
[B] The domain of c0: [0, 1] → M and c1 : [0, 1] → M are the same,
[C] Both c0, and c1 are continuous curves.
(1) Tube-Like-Homotopy: A homotopy H: [0, 1] × [0, 1] → M is "Tube-Like", if
[TLH0]H is continues
[TLH1]H(t, 0) = c0(t)
[TLH2]H(t, 1) = c1(t)
[TLH3]H(0, s) = H(1, s) for all s ∈ [0, 1]
(2) Tube homotope:c0, and c1 are "Tube Homotope" if and only if "there are H such that there is a Tube-like-Homotopy between c0 and c1.
(3) Tube like and piecewise smooth homotopy: The homotopy H of (1) is Tube like and piecewise smooth homotopy when that H is piecewise smooth. And the relation of (1) is “Piecewise smooth Tube Homotope” when that H is piecewise smooth (so, it is “Piecewise smooth Tube Homotope”).
^ abcIf the two curves α: [a1, b1] → M, β: [a2, b2] → M, satisfy α(b1) = β(a2) 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.
Definition (Joint of paths). Let M be a topological space and α: [a1, b1] → M, β: [a2, b2] → M, be two paths on M. If α and β satisfy α(b1) = β(a2) then we can join them at this common point to produce new curve α ⊕ β : [a1, b1+(b2-a2)] → M defined by:
^ abcGiven curve on M, α: [a1, b1] → 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.
Definition (Backward of curve). Let M be a topological space and α : [a1, b1] → M, be path on M. We can define backward thereof, α : [a1, b1] → M by:
And, given two curves on M, α: [a1, b1] → M, β: [a2, b2] → M, which satisfy α(b1 = β(b2) (that means α(b1) = β(a2), we can define as following manner.
^ abcdefJohn 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 )