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Madhava's sine table
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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 R3. 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 R3 can be considered as a 1-form in which case curl is the exterior derivative.
The proof of the Theorem consists of 4 steps [2][3][note 4] The proof below does not require background information on differential form, and may be helpful for understanding the notion of differential form, especially pull-back of differential form.
First Step of Proof (Defining the Pullback)
Define
so that P is the pull-back[note 4] of F, and that P(u, v) is R2-valued function, depends on two parameter u, v. In order to do so we define P1 and P2 as follows.
On the other hand, according to the definition of surface integral,
So, we obtain
Fourth Step of Proof (Reduction to Green's Theorem)
According to the result of 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
strongly increase function that,
for all piece wise sooth path c:[a,b]→R3, 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
Definition 2-1 (Lamellar vector field). A smooth vector field, F on an openU ⊆ R3 is called a Lamellar vector field if ∇ × F = 0.
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 7]
Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics).[7] and see p142 of Fujimoto[5]
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].
Then,
Some textbooks such as Lawrence[7] 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 toward (stronger than) typical definitions of “Homotope” or “Homotopy”.[note 8]
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 9]
Proof of the Theorem
Hereinafter, the ⊕ stands for joining paths
[note 10]
the stands for backwards of curve
[note 11]
Let D = [0, 1] × [0, 1]. By our assumption, c1 and c2 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,
that, line integral along
and line integral along
are compensated each other[note 11]
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.[7][8] 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].
Then,
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 ⊆ 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 contenious,
[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, 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 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, to fill the gap, following resources may be helpful for you.
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.
Theorem 2-2.[7][8] Let U ⊆ R3 be a simply connected and open with a Lamellar vector field F. For all piecewise smooth loops, c : [0, 1] → U we have:
Kelvin–Stokes theorem on Singular 2-cube and Cube subdivisionable sphere
Singular 2-cube and boundary
Definition 3-1 (Singular 2-cube)[10] 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.
Given ,
we define the notarization map of sngler two cube
here, the I:=[0,1] and I2 stands for .
Above mentioned is strongly increase function (that means
(for
all ) that,
following lemma is satisfied.
Lemma 3-1(Notarization map of sngler two cube).
Set D = [a1, b1]× [a2, b2] ⊆ R2 and let U be a non-empty opensubset of R3.
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 I2 under a piecewise smooth map ,
be a singular 2-cube.then,
For all ,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, I2.
In order to facilitate the discussion of boundary, we define
by
γ1, ..., γ4 are the one-dimensional edges of the image of I2.Hereinafter, the ⊕ stands for joining paths[note 10] and,
the stands for backwards of curve
.[note 11]
Cube subdivision
Definition 3-2(Cube subdivisionable sphere).(see Iwahori[4] p399)
Let S ⊆ R3 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
such that
and then abovementioned
are said to be a Cube subdivision of the S.
Definitions 3-3(Boundary of ).(see Iwahori[4] p399)
Let S ⊆ R3 be a "Cube subdivisionable sphere" and,
Let be a Cube subdivision of the S.,then
(1)The are said to be an edge of if satisfies
"then,"
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 iff
"There is only one only one c and only one j such that,
"
(2)Boundary of is a collection of
edges in the sense of "(1)". means the boundary of
(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.
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References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
↑ 1.01.1James Stewart;"Essential Calculus: Early Transcendentals" Cole Pub Co (2010)[1]
↑ 2.02.12.2This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) [2], please refer the [3]
↑ 8.08.18.28.38.48.5John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218) "Springer (2002/9/23) [7][8]
↑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 [9])[10]
↑Michael Spivak:"Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus" Westview Press, 1971 [11]
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