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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.
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]}
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.
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, \langle \ |\ \rangle is the normal inner product of R^{3} and hereinafter, \langle \ |A|\ \rangle stands for bilinear form according to matrix A ^{[note 5]} .^{[note 6]}
According to the definition of line integral,
where, Jψ stands for the Jacobian matrix of ψ. Hence,^{[note 5]}^{[note 6]}
So, we obtain following 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
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.
In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.
First, we define the notarization map, \theta_:[0,1]\to[a,b] as follows.
Above-mentioned \theta_ 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 c (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].
Definition 2-1 (Lamellar vector field). A smooth vector field, F on an open U ⊆ R^{3} is called a Lamellar vector field if ∇ × F = 0.
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.
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]}
Let U ⊆ R^{3} be an open subset with a Lamellar vector field F, and piecewise smooth loops c_{0}, c_{1} : [0, 1] → U. If there is a function H : [0, 1] × [0, 1] → U such that
Then,
\int_{c_0} \mathbf{F} \, dc_0=\int_{c_1} \mathbf{F} \, dc_1
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]}
Hereinafter, the ⊕ stands for joining paths ^{[note 11]} the \ominus 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 \Gamma_{2}(s) and line integral along \Gamma_{4}(s) are compensated each other^{[note 12]} so,
On the other hand,
that, subjected equation is proved.
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.^{[7]}^{[8]} Let U ⊆ R^{3} be an open subset, with a Lamellar vector field F and a piecewise smooth loop c_{0} : [0, 1] → U. Fix a point p ∈ U, if there is a homotopy (tube-like-homotopy) H : [0, 1] × [0, 1] → U such that
\int_{c_0} \mathbf{F} \, dc_0=0
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).^{[7]}^{[8]} Let M ⊆ R^{n} 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
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.
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.
Considering above-mentioned fact and Lemma 2-2, we will obtain following theorem. That theorem is anser for subjecting issue.
Theorem 2-2.^{[7]}^{[8]} Let U ⊆ R^{3} be a simply connected and open with a Lamellar vector field F. For all piecewise smooth loops, c : [0, 1] → U we have:
Definition 3-1 (Singular 2-cube)^{[10]} Set D = [a_{1}, b_{1}] × [a_{2}, b_{2}] ⊆ R^{2} and let U be a non-empty open subset of R^{3}. The image of D under a piecewise smooth map ψ : D → U is called a singular 2-cube.
Given D:=\times, we define the notarization map of sngler two cube {\theta}_{D}:{I}^{2}\to D
here, the I:=[0,1] and I^{2} stands for {I}^{2}=I\times I.
Above-mentioned is strongly increase function (that means det(J(\theta_{D})_{({u}_{1},{u}_{2})})>0 (for all ({u}_{1},{u}_{2})\in\mathbb{R}^{3} ) that, following lemma is satisfied.
Lemma 3-1(Notarization map of singular two cube). Set D = [a_{1}, b_{1}]× [a_{2}, b_{2}] ⊆ R^{2} and let U be a non-empty open subset of R^{3}. Let the image of D under a piecewise smooth map ψ : D → U, S:= ψ[D] be a singular 2-cube. Let the image of I^{2} under a piecewise smooth map \varphi\circ{\theta}_{D}, \tilde{S}:=\varphi\circ{\theta}_{D} be a singular 2-cube.then, For all \mathbf{F},smooth vector field on U,
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 \delta_:\mathbb{R}^k \to \mathbb{R}^{k+1} 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 \ominus stands for backwards of curve .^{[note 12]}
Definition 3-2(Cube subdivisionable sphere).(see Iwahori^{[4]} p399) Let S ⊆ R^{3} be a non empty subset then, that S is said to be a "Cube subdivisionable sphere" when there are at least one Indexed family of singular 2-cube \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} such that
and then abovementioned \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} are said to be a Cube subdivision of the S.
Definitions 3-3(Boundary of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} ).(see Iwahori^{[4]} p399)
Let S ⊆ R^{3} be a "Cube subdivisionable sphere" and, Let \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} be a Cube subdivision of the S.,then
(1)The {\varphi}_\circ\delta_} are said to be an edge of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} if {\varphi}_\circ\delta_}[I] satisfies
that means "although not line contact even if the point contact with other ridge line" and above-mentioned "=" stands for equal as a set. That means, l is said to be an edge of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} iff "There is only one \lambda only one c and only one j such that, l={\varphi}_\circ\delta_}"
(2)Boundary of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} is a collection of edges in the sense of "(1)". \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} means the boundary of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}
(3)If l is an edge in the sense of "(1)", then, we described as follows.
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.
Fact (boundary of \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} ).(see Iwahori^{[4]} p399)
Let S ⊆ R^{3} be a "Cube subdivisionable sphere" and, Let both \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda} and \{({I}^{2},{\psi}_{\mu},{L}_{\mu})\}_{\mu\in M} be a Cube subdivision of the S, then
that means, the definition of boundary is not depends on the cube subdivision.
So, considering the above-mentioned fact, following "Definition3-4" is well-defined.
Definitions 3-4(Boundary of Surface(see Iwahori^{[4]} p399) Let S ⊆ R^{3} be a "Cube subdivisionable sphere" and, Let \{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}, then
(1)
(2)If
then
and then such "l" are said to be an edge of S.
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