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| The ''' Kelvin–Stokes theorem''',<ref name="Jame">James Stewart;"Essential Calculus: Early Transcendentals" Cole Pub Co (2010)[http://books.google.co.jp/books?id=btIhvKZCkTsC&pg=PA786&lpg=PA786&dq=Stoke%E2%80%99s+theorem&source=bl&ots=T2zQR8Qg28&sig=q5riBZK0mPQRq8MQmD6mraAG6xI&hl=ja&sa=X&ei=G6QrUO_kAYqemQWEooGQCA&ved=0CFEQ6AEwBQ#v=onepage&q=Stoke%E2%80%99s%20theorem&f=false]</ref><ref name="bath">This proof is based on the Lecture Notes given by Prof. Robert Scheichl ([[University of Bath]], U.K) [http://www.maths.bath.ac.uk/~masrs/ma20010/], please refer the [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf]</ref><ref name="proofwik">[http://www.proofwiki.org/wiki/Classical_Stokes'_Theorem This proof is also same to the proof shown in]</ref><ref name=iwahori>
| |
| [[Nagayoshi Iwahori]], et.al:"Bi-Bun-Seki-Bun-Gaku" [[:ja:裳華房|Sho-Ka-Bou]](jp) 1983/12 ISBN978-4-7853-1039-4
| |
| [http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1039-4.htm](Written in Japanese)</ref><ref name=fujimno>Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
| |
| [[:ja:培風館|Bai-Fu-Kan]](jp)(1979/01) ISBN 978-4563004415 [http://books.google.co.jp/books/about/%E7%8F%BE%E4%BB%A3%E6%95%B0%E5%AD%A6%E3%83%AC%E3%82%AF%E3%83%81%E3%83%A3%E3%83%BC%E3%82%BA.html?id=nXhDywAACAAJ&redir_esc=y](Written in Japanese)</ref>
| |
| also known as the '''curl theorem''',<ref name="wolf">http://mathworld.wolfram.com/CurlTheorem.html</ref> is a theorem in [[vector calculus]] on '''R'''<sup>3</sup>. Given a [[vector field]], the theorem relates the [[Surface integral|integral]] of the [[Curl (mathematics)|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]].”<ref name="DTPO">Lawrence Conlon; "Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11) [http://books.google.co.jp/books?id=r2K31Pz5EGcC&pg=PA194&lpg=PA194&dq=Piecewise+Smooth+Homotopy&source=bl&ots=UxiEdS2Zs7&sig=Hyxm5iPebJ_sEKz1IGfKO5Zs130&hl=ja#v=onepage&q=Piecewise%20Smooth%20Homotopy&f=false]</ref><ref name=lee>John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218) "Springer (2002/9/23) [http://books.google.co.jp/books/about/Introduction_to_Smooth_Manifolds.html?id=eqfgZtjQceYC&redir_esc=y] [http://books.google.co.jp/books?id=xygVcKGPsNwC&pg=PA421&lpg=PA421&dq=Piecewise+smooth+Homotopy&source=bl&ots=j_LrUZYbST&sig=Nd-LKN2brxvZxy9NaK2Im1UBpZw&hl=ja&sa=X&ei=5q7rUJ30GYrVkQWyl4HIAg&ved=0CEYQ6AEwAg#v=onepage&q=Piecewise%20smooth%20Homotopy&f=false]</ref> In particular, the vector field on '''R'''<sup>3</sup> can be considered as a [[differential form|1-form]] in which case curl is the [[exterior derivative]].
| |
| | |
| ==The Theorem==
| |
| <div align=left> | |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Kelvin–Stokes Theorem'''.<ref name="Jame"/><ref name="bath"/><ref name="proofwik"/> Let γ : [''a'', ''b''] → '''R'''<sup>2</sup> be a [[Piecewise smooth]] [[Jordan curve|Jordan plane curve]], that bounds the domain ''D'' ⊂ '''R'''<sup>2</sup>.<ref group="note" name=JC>The [[Jordan curve theorem]] implies that the [[Jordan curve]] divides '''R'''<sup>2</sup> into two components, a [[compact space|compact]] one (the bounded area) and another is non-compact.</ref> Suppose ψ : ''D'' → '''R'''<sup>3</sup> is smooth, with ''S'' := ψ[''D'']
| |
| <ref group="note" name=map>
| |
| When ψ is a [[Map (mathematics)|mapping]] and
| |
| D is a subset of the [[Domain of a function|domain]] of ψ,
| |
| ψ[''D''] stands for the [[Image (mathematics)|image]] of D under ψ.
| |
| </ref>
| |
| , and Γ is the [[space curve]] defined by Γ(''t'') = ψ(γ(''t'')).<ref group="note" name=cgamma>γ and Γ are both loops, however, Γ is not necessarily a [[Jordan curve]]</ref> If '''F''' a smooth vector field on '''R'''<sup>3</sup>, then | |
| <div align=center>
| |
| <math>\oint_{\Gamma} \mathbf{F}\, d\Gamma = \iint_{S} \nabla\times\mathbf{F}\, dS </math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| ==Proof==
| |
| The proof of the Theorem consists of 4 steps <ref name="bath"/><ref name="proofwik"/><ref group="note" name="f1">. If you know the [[differential form]], when we considering following identification of the vector field '''A''' = (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>),
| |
| :<math>\mathbf{A} = \omega_{\mathbf{A}} = a_1 dx_1+{a}_{2}d{x}_{2}+a_3dx_{3} </math>
| |
| :<math>\mathbf{A} = {}^{*} \omega_{\mathbf{A}}= {a}_{1}d{x}_{2} \wedge d{x}_{3}+{a}_{2}d{x}_{3} \wedge d{x}_{1}+{a}_{3}d{x}_{1} \wedge d{x}_{2}</math>,
| |
| the proof here is similar to the proof using the pull-back of ω<sub>'''F'''</sub>. In actual, under the identification of ω<sub>'''F'''</sub> = '''F''' following equations are satisfied.
| |
| :<math>\begin{align}
| |
| \nabla\times\mathbf{F} &= d\omega_{\mathbf{F}} \\
| |
| \psi^{*}\omega_{\mathbf{F}} &= P_1du +P_2dv \\
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| \psi^{*}(d \omega_{\mathbf{F}}) &= \left (\frac{\partial P_2}{\partial u} - \frac{\partial P_1}{\partial v} \right ) du\wedge dv
| |
| \end{align}</math>
| |
| | |
| Here, ''d'' stands for [[Exterior derivative]] of [[Differential form]], ψ* stands for [[pull back]] by ψ and, ''P''<sub>1</sub> and ''P''<sub>2</sub> of above mentuoned are same as ''P''<sub>1</sub> and ''P''<sub>2</sub> of the body text of this article respectively.</ref> The proof below does not require background information on [[differential form]], and may be helpful for understanding the notion of [[differential form]], especially [[Pullback (differential geometry)|pull-back]] of differential form.
| |
| | |
| === First Step of Proof (Defining the Pullback)===
| |
| Define
| |
| :<math>\mathbf{P}(u,v) = (P_1(u,v), P_2(u,v))</math>
| |
| so that '''P''' is the pull-back<ref group="note" name="f1"/> of '''F''', and that '''P'''(''u'', ''v'') is '''R'''<sup>2</sup>-valued function, depends on two parameter ''u'', ''v''. In order to do so we define ''P''<sub>1</sub> and ''P''<sub>2</sub> as follows.
| |
| :<math>P_1(u,v)=\left\langle \mathbf{F}(\psi(u,v)) \bigg| \frac{\partial \psi}{\partial u} \right\rangle, \qquad P_2(u,v)=\left\langle \mathbf{F}(\psi(u,v)) \bigg| \frac{\partial \psi}{\partial v} \right\rangle </math>
| |
| | |
| Where, <math>\langle \ |\ \rangle</math> is the normal [[inner product]] of '''R'''<sup>3</sup> and
| |
| hereinafter, <math>\langle \ |A|\ \rangle</math> stands for [[bilinear form]] according to matrix '''A''' <ref group="note" name="bil">Given a ''n'' × ''m'' matrix ''A'' we define a [[bilinear form]]:
| |
| :<math> \mathbf{x} \in \mathbf{R}^m, \mathbf{y} \in \mathbf{R}^n \ : \qquad \left\langle \mathbf{x} |A| \mathbf{y} \right\rangle ={}^{t}\mathbf{x}A\mathbf{y} </math>
| |
| which also satisfies:
| |
| :<math> \left \langle \mathbf{x} |A| \mathbf{y} \right \rangle= \left \langle \mathbf{y} |{}^{t}A| \mathbf{x} \right \rangle.</math>
| |
| :<math> \left \langle \mathbf{x} |A| \mathbf{y} \right \rangle + \left \langle \mathbf{x} |B| \mathbf{y}\right \rangle = \left \langle \mathbf{x} |A+B|\mathbf{y}\right \rangle</math></ref>
| |
| .<ref group="note" name=trans>Given a ''n'' × ''m'' matrix ''A'' , ''<sup>t</sup>A'' stands for [[transposed matrix]] of ''A''.</ref>
| |
| | |
| ===Second Step of Proof (First Equation)===
| |
| According to the definition of [[line integral]],
| |
| | |
| :<math>\begin{align}
| |
| \oint_{\Gamma} \mathbf{F} d\Gamma &=\int_{a}^{b} \left\langle (\mathbf{F}\circ \psi (t))\bigg|\frac{d\Gamma}{dt}(t) \right\rangle dt \\
| |
| &= \int_{a}^{b} \left\langle (\mathbf{F}\circ \psi (t))\bigg|\frac{d(\psi\circ\gamma)}{dt}(t) \right\rangle dt \\
| |
| &= \int_{a}^{b} \left\langle (\mathbf{F}\circ \psi (t))\bigg|(J\psi)_{\gamma(t)}\cdot \frac{d\gamma}{dt}(t) \right\rangle dt
| |
| \end{align}</math>
| |
| | |
| where, ''J''ψ stands for the [[Jacobian matrix and determinant|Jacobian matrix]] of ψ. Hence,<ref group="note" name="bil"/><ref group="note" name=trans/>
| |
| | |
| :<math>\begin{align}
| |
| \left\langle (\mathbf{F}\circ \Gamma(t))\bigg|(J\psi)_{\gamma(t)}\frac{d\gamma}{dt}(t) \right\rangle &=
| |
| \left\langle (\mathbf{F}\circ \Gamma (t))\bigg|(J\psi)_{\gamma(t)} \bigg|\frac{d\gamma}{dt}(t) \right\rangle \\
| |
| &= \left\langle ({}^{t}\mathbf{F}\circ \Gamma (t))\cdot(J\psi)_{\gamma(t)}\ \bigg|\ \frac{d\gamma}{dt}(t) \right\rangle \\
| |
| &= \left\langle \left( \left\langle (\mathbf{F}(\psi(\gamma(t))))\bigg|\frac{\partial\psi}{\partial u}(\gamma(t)) \right\rangle , \left\langle (\mathbf{F}(\psi(\gamma(t))))\bigg |\frac{\partial\psi}{\partial v}(\gamma(t)) \right\rangle \right) \bigg|\frac{d\gamma}{dt}(t)\right\rangle \\
| |
| &= \left\langle (P_1(u,v) , P_2(u,v))\bigg|\frac{d\gamma}{dt}(t)\right\rangle\\
| |
| &= \left\langle \mathbf{P}(u,v)\ \bigg|\frac{d\gamma}{dt}(t)\right\rangle
| |
| \end{align}</math>
| |
| | |
| So, we obtain following equation
| |
| :<math>\oint_{\Gamma} \mathbf{F} d\Gamma = \oint_{\gamma} \mathbf{P} d\gamma</math>
| |
| | |
| === Third Step of Proof (Second Equation)===
| |
| First, calculate the partial derivatives, using [[General Leibniz rule|Leibniz rule]] of inner product
| |
| | |
| :<math>\begin{align}
| |
| \frac{\partial P_1}{\partial v} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial v} \bigg | \frac{\partial \psi}{\partial u} \right\rangle + \left\langle \mathbf{F}\circ \psi \bigg | \frac{\partial^2 \psi}{ \partial v \partial u} \right\rangle \\
| |
| \frac{\partial P_2}{\partial u} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial u} \bigg | \frac{\partial \psi}{\partial v} \right\rangle + \left\langle \mathbf{F}\circ \psi \bigg | \frac{\partial^2 \psi}{\partial u \partial v} \right\rangle
| |
| \end{align}</math>
| |
| | |
| So,<ref group="note" name="bil"/> <ref group="note" name=trans/>
| |
| | |
| :<math>\begin{align}
| |
| \frac{\partial P_1}{\partial v} - \frac{\partial P_2}{\partial u} &= \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial v} \bigg| \frac{\partial \psi}{\partial u} \right\rangle - \left\langle \frac{\partial (\mathbf{F}\circ \psi)}{\partial u} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\
| |
| &= \left\langle (J\mathbf{F})_{\psi(u,v)}\cdot \frac{\partial \psi}{\partial v} \bigg |\frac{\partial \psi}{\partial u} \right\rangle - \left\langle (J\mathbf{F})_{\psi(u,v)}\cdot \frac{\partial \psi}{\partial u} \bigg|\frac{\partial \psi}{\partial v} \right\rangle && \text{ Chain Rule}\\
| |
| &= \left\langle \frac{\partial \psi}{\partial u} \bigg|(J\mathbf{F})_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle - \left\langle \frac{\partial \psi}{\partial u} \bigg |{}^{t}(J\mathbf{F})_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\
| |
| &= \left\langle \frac{\partial \psi}{\partial u}\bigg |(J\mathbf{F})_{\psi(u,v)} - {}^{t}{(J\mathbf{F})}_{\psi(u,v)} \bigg| \frac{\partial \psi}{\partial v} \right\rangle \\
| |
| &= \left\langle \frac{\partial \psi}{\partial u}\bigg |\left ((J\mathbf{F})_{\psi(u,v)} - {}^{t} (J\mathbf{F})_{\psi(u,v)} \right )\cdot \frac{\partial \psi}{\partial v} \right\rangle \\
| |
| &= \left\langle \frac{\partial \psi}{\partial u}\bigg |(\nabla\times\mathbf{F})\times\frac{\partial \psi}{\partial v} \right\rangle && \left ( (J\mathbf{F})_{\psi(u,v)} - {}^{t} (J\mathbf{F})_{\psi(u,v)} \right ) \cdot \mathbf{x} = (\nabla\times\mathbf{F})\times \mathbf{x} \\
| |
| &=\det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ] && \text{ Scalar Triple Product}
| |
| \end{align}</math>
| |
| | |
| On the other hand, according to the definition of [[surface integral]],
| |
| | |
| :<math>\begin{align}
| |
| \iint_S (\nabla\times\mathbf{F}) dS &=\iint_D \left\langle (\nabla\times\mathbf{F})(\psi(u,v)) \bigg |\frac{\partial\psi}{\partial u}(u,v)\times \frac{\partial\psi}{\partial v}(u,v)\right\rangle dudv\\
| |
| &= \iint_D \det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ] du dv && \text{ Scalar Triple Product}
| |
| \end{align}</math>
| |
| | |
| So, we obtain
| |
| :<math> \iint_S (\nabla\times\mathbf{F}) dS =\iint_{D} \left( \frac{\partial P_2}{\partial u} - \frac{\partial P_1}{\partial v} \right) dudv </math>
| |
| | |
| ===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,
| |
| <math>{\theta}_{[a,b]}:[0,1]\to[a,b]</math> as follows.
| |
| :<math>{\theta}_{[a,b]}=s(b-a)+a</math>
| |
| | |
| Above mentioned <math>{\theta}_{[a,b]}</math>is
| |
| strongly increase function that,
| |
| for all piece wise sooth path c:[a,b]→R<sup>3</sup>, for all smooth vector field F,
| |
| domain of which includes <math>c[[a,b]]</math> (image of [a,b] under c.), following equation is satisfied.
| |
| :<math>\int_{c} \mathbf{F}\ dc\
| |
| =\int_{c\circ{\theta}_{[a,b]}}\ \mathbf{F}\ d(c\circ{\theta}_{[a,b]})</math>
| |
| | |
| So, we can unify the domain of the curve from the beginning
| |
| to [0,1].
| |
| | |
| === The Lamellar vector field ===
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition 2-1 (Lamellar vector field).''' A smooth vector field, '''F''' on an [[open set|open]] ''U'' ⊆ '''R'''<sup>3</sup> is called a '''Lamellar vector field''' if ∇ × '''F''' = 0.
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| 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]],.<ref group="note" name=hhz>There are a number of theorems with the same name, however they are not necessarily the same.</ref>
| |
| | |
| That theorem is also important in the area of [[Homoropy thorem]].<ref name="DTPO"/>
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left> '''Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics).'''<ref name="DTPO"/> and see p142 of Fujimoto<ref name=fujimno/><br>
| |
| Let ''U'' ⊆ '''R'''<sup>3</sup> be an [[open set|open]] [[subset]] with a Lamellar vector field '''F''', and piecewise smooth loops ''c''<sub>0</sub>, ''c''<sub>1</sub> : [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) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1],
| |
| * '''[TLH2]''' ''H''(''t'', 1) = ''c''<sub>1</sub>(''t'') for all ''t'' ∈ [0, 1],
| |
| * '''[TLH3]''' ''H''(0, ''s'') = ''H''(1, ''s'') for all ''s'' ∈ [0, 1].
| |
| Then,
| |
| <div align=center>
| |
| <math>\int_{c_0} \mathbf{F} d{c}_{0}=\int_{c_1} \mathbf{F} dc_{1}</math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| Some textbooks such as Lawrence<ref name="DTPO"/> call the relationship between ''c''<sub>0</sub> and ''c''<sub>1</sub> stated in Theorem 2-1 as “homotope”and the function ''H'' : [0, 1] × [0, 1] → ''U'' as “Homotopy between ''c''<sub>0</sub> and ''c''<sub>1</sub>”.
| |
| | |
| However, “Homotope” or “Homotopy” in above mentioned sense are different toward (stronger than) typical definitions of “Homotope” or “Homotopy”.<ref group="note" name="typHomoto">Typical definition of homotopy and homotope are as follows.
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition (Homotopy and Homotope).''' Suppose ''Z'' and ''W'' are topological spaces, with continuous maps ''f''<sub>0</sub>, ''f''<sub>1</sub> : ''Z'' → ''W''.
| |
| | |
| (1) The continuous map ''H'' : ''Z'' × [0, 1] → ''W'' is said to be a "Homotopy between ''f''<sub>0</sub> and ''f''<sub>1</sub>" if
| |
| *[H1] ''H''(''t'', 0) = ''f''<sub>0</sub>(''t'') for all ''t'' ∈ ''Z'',
| |
| *[H2] ''H''(''t'', 1) = ''f''<sub>1</sub>(''t'') for all ''t'' ∈ ''Z''.
| |
| | |
| (2) If there is a homotopy between ''f''<sub>0</sub> and ''f''<sub>1</sub>", ''f''<sub>0</sub> and ''f''<sub>1</sub>" are said to be homotope.
| |
| | |
| (3) Suppose ''f''<sub>0</sub> and ''f''<sub>1</sub> are homotope and ''H'' is a homotopy between them. ''f''<sub>0</sub> and ''f''<sub>1</sub> are said to be piecewise homotope, if ''f''<sub>0</sub>, ''f''<sub>1</sub>, and ''H'' are piecewise smooth. ''H'' is then said to be the piecewise homotopy between ''f''<sub>0</sub> and ''f''<sub>1</sub>.
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </ref>
| |
| | |
| 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'''.<ref group="note" name="TLH">In Some textbooks such as Lawrence Conlon;"Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11)[http://books.google.co.jp/books?id=r2K31Pz5EGcC&pg=PA194&lpg=PA194&dq=Piecewise+Smooth+Homotopy&source=bl&ots=UxiEdS2Zs7&sig=Hyxm5iPebJ_sEKz1IGfKO5Zs130&hl=ja#v=onepage&q=Piecewise%20Smooth%20Homotopy&f=false] 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 sence. 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.
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition (Tube-like-Homotopy and Tube-Homotope).''' Suppose ''c''<sub>0</sub>, ''c''<sub>1</sub> satisfy the following:
| |
| * '''[A]''' ''M'' is [[differentiable manifold]],
| |
| * '''[B]''' The domain of ''c''<sub>0</sub> : [0, 1] → ''M'' and ''c''<sub>1</sub> : [0, 1] → ''M'' are the same,
| |
| * '''[C]''' Both ''c''<sub>0</sub>, and ''c''<sub>1</sub> are continuous curves.
| |
| | |
| Then,
| |
| | |
| '''(1) Tube-Like-Homotopy:''' A homotopy "H" : [0, 1] × [0, 1] → ''M'' is "Tube-Like", if
| |
| * '''[TLH0]''' ''H'' is continues
| |
| * '''[TLH1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'')
| |
| * '''[TLH2]''' ''H''(''t'', 1) = ''c''<sub>1</sub>(''t'')
| |
| * '''[TLH3]''' ''H''(0, ''s'') = ''H''(1, ''s'') for all ''s'' ∈ [0, 1]
| |
| | |
| '''(2) Tube Homotope:''' ''c''<sub>0</sub>, and ''c''<sub>1</sub> are "Tube Homotope" if and only if "there are ''H'' such that there is a Tube-like-Homotopy between ''c''<sub>0</sub> and ''c''<sub>1</sub>.
| |
| | |
| '''(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”).
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </ref>
| |
| | |
| ===Proof of the Theorem===
| |
| [[File:Domain of singular 2 cube 2.jpg|thumb| The definitions of γ<sub>1</sub>, ..., γ<sub>4</sub>]]
| |
| | |
| Hereinafter, the ⊕ stands for joining paths
| |
| <ref group="note" name=vee>
| |
| Given two curves
| |
| α: [a<sub>1</sub>, b<sub>1</sub>] → ''M'' ,
| |
| β: [a<sub>2</sub>, b<sub>2</sub>, ] → ''M'',
| |
| if α and β satisfy α(b<sub>1</sub>) = β(a<sub>2</sub>) 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 )
| |
| | |
| :<math>{\int}_{\alpha\oplus \beta} \mathbf{F} d(\alpha\oplus \beta)=
| |
| {\int}_{\alpha}\mathbf{F} d\alpha
| |
| +{\int}_{\beta} \mathbf{F} d\beta
| |
| </math>
| |
| | |
| ,which is also used when we define [[Fundamental group]].
| |
| To do so, accurate definition of the “Joint of paths” is as follows.
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition (Joint of paths).''' Let ''M'' be a topological space and
| |
| α: [a<sub>1</sub>, b<sub>1</sub>] → ''M'' ,
| |
| β: [a<sub>2</sub>, b<sub>2</sub>] → ''M'',
| |
| be two paths on ''M''.
| |
| If α and β satisfy α and β satisfy α(b<sub>1</sub>) = β(a<sub>2</sub>) then
| |
| we can join them at this common point to produce new curve
| |
| α ⊕ β : [a<sub>1</sub>, b<sub>1</sub>+(b<sub>2</sub>-a<sub>2</sub>)] → ''M'' defined by:
| |
| :<math>(\alpha\oplus \beta) (t) =
| |
| \begin{cases}
| |
| \alpha(t) & {a}_{1}\le t \le \ {b}_{1}, \\
| |
| \beta(t+({a}_{2}-{b}_{1})) & {b}_{1} < t \le {b}_{1}+({b}_{2}-{a}_{2}).
| |
| \end{cases}</math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </ref>
| |
| the <math>\ominus</math> stands for backwards of curve
| |
| <ref group="note" name=omin>
| |
| Given curve on ''M'',
| |
| α: [a<sub>1</sub>, b<sub>1</sub>] → ''M'' ,
| |
| we can define new curve <math>\ominus</math>α
| |
| so that, for all smooth vector field ''F'' (if domain of which includes image of α )
| |
| | |
| :<math>{\int}_{\ominus\alpha} \mathbf{F} d(\ominus\alpha)=
| |
| -{\int}_{\alpha}\mathbf{F} d\alpha
| |
| </math>
| |
| | |
| ,which is also used when we define [[Fundamental group]].
| |
| To do so, accurate definition of the “Backwards of curve” is as follows.
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition (Backward of curve).''' Let ''M'' be a topological space and
| |
| α: [a<sub>1</sub>, b<sub>1</sub>] → ''M'' ,
| |
| be path on ''M''.
| |
| we can define backward thereof,
| |
| <math>\ominus</math>α : [a<sub>1</sub>, b<sub>1</sub>] → ''M'' defined by:
| |
| :<math>\ominus\alpha(t)=\alpha({b}_{1}+{a}_{1}-t)</math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| And, given two curves on ''M'',
| |
| α: [a<sub>1</sub>, b<sub>1</sub>] → ''M''
| |
| β: [a<sub>2</sub>, b<sub>2</sub>] → ''M''
| |
| which satisfies α(b<sub>1</sub>)=β(b<sub>2</sub>) (that means
| |
| α(b<sub>1</sub>)=<math>\ominus</math>β(a<sub>2</sub>),
| |
| we can define <math>\alpha\ominus\beta</math> as following manner.
| |
| :<math>\alpha\ominus\beta:=\alpha\oplus(\ominus\beta)</math>
| |
| </ref>
| |
| | |
| Let ''D'' = [0, 1] × [0, 1]. By our assumption, ''c''<sub>1</sub> and ''c''<sub>2</sub> are piecewise smooth homotopic, there are the piecewise smooth homogony ''H'' : ''D'' → ''M''
| |
| :<math>\begin{align}
| |
| \begin{cases} \gamma_{1}:[0, 1]\to D \\ \gamma_{1}(t) := (t,0) \end{cases}, \qquad &\begin{cases}\gamma_{2}:[0,1] \to D \\ \gamma_{2}(s) := (1, s) \end{cases} \\
| |
| \begin{cases} \gamma_{3}:[0, 1] \to D \\ \gamma_{3}(t) := (-t+0+1, 1)\end{cases}, \qquad &\begin{cases}\gamma_{4}:[0,1] \to D \\ \gamma_{4}(s) := (0, 1-s)\end{cases}
| |
| \end{align}</math>
| |
| | |
| :<math>\gamma(t):= (\gamma_{1} \oplus \gamma_{2} \oplus \gamma_{3} \oplus \gamma_{4})(t) </math>
| |
| :<math>\Gamma_{i}(t):= H(\gamma_{i}(t)), \qquad i=1, 2, 3, 4</math>
| |
| :<math>\Gamma(t):= H(\gamma(t))
| |
| =(\Gamma_{1} \oplus \Gamma_{2} \oplus \Gamma_{3} \oplus \Gamma_{4})(t)
| |
| </math>
| |
| | |
| And, let ''S'' be the image of ''D'' under ''H''. Then,
| |
| :<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS </math>
| |
| | |
| will be obvious according to the Theorem 1 and, '''F''' is Lamellar vector field that, right side of that equation is zero, so,
| |
| :<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =0</math>
| |
| | |
| Here,
| |
| :<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =\sum_{i=1}^{4} \oint_{\Gamma_i} \mathbf{F} d\Gamma </math> <ref group="note" name=vee/>
| |
| | |
| and, H is Tubeler-Homotopy that,
| |
| :<math>\Gamma_{2}(s)= {\Gamma}_{4}(1-s)=\ominus{\Gamma}_{4}(s)</math>
| |
| that, line integral along <math>\Gamma_{2}(s)</math>
| |
| and line integral along <math>\Gamma_{4}(s)</math>
| |
| are compensated each other<ref group="note" name=omin/>
| |
| so,
| |
| :<math>\oint_{{\Gamma}_{1}} \mathbf{F} d\Gamma +\oint_{\Gamma_3} \mathbf{F} d\Gamma =0</math>
| |
| | |
| On the other hand,
| |
| :<math>c_{1}(t)=H(t,0)=H({\gamma}_{1}(t))={\Gamma}_{1}(t)</math>
| |
| :<math>c_{2}(t)=H(t,1)=H(\ominus{\gamma}_{3}(t))=\ominus{\Gamma}_{3}(t)
| |
| </math>
| |
| 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.
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Lemma 2-2.'''<ref name="DTPO"/><ref name=lee/> Let ''U'' ⊆ '''R'''<sup>3</sup> be an [[open set|open]] [[subset]], with a Lamellar vector field '''F''' and a piecewise smooth loop ''c''<sub>0</sub> : [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) = ''c''<sub>0</sub>(''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,
| |
| <div align=center>
| |
| <math>\int_{c_0} \mathbf{F} dc_0=0</math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| 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:
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition 2-2 (Simply Connected Space).'''<ref name="DTPO"/><ref name=lee/> Let ''M'' ⊆ '''R'''<sup>''n''</sup> be non-empty, [[connected space|connected]] and [[Connected_space#Path_connectedness|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].
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| 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]] (<ref name=lee/> page 136) and "How to use that theorem to this isuue" (<ref name=lee/> page 421).
| |
| *More general statements appear in<ref name="ptr">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 [http://www.ams.org/mathscinet-getitem?mr=0115178])[http://www.math.rochester.edu/u/faculty/doug/otherpapers/pont4.pdf]</ref> (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.
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Theorem 2-2.'''<ref name="DTPO"/><ref name=lee/> Let ''U'' ⊆ '''R'''<sup>3</sup> be a simply connected and [[open set|open]] with a Lamellar vector field '''F'''. For all piecewise smooth loops, ''c'' : [0, 1] → ''U'' we have:
| |
| <div align=center>
| |
| <math>\int_{c_0} \mathbf{F} dc_0=0</math>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| ==Kelvin–Stokes theorem on Singular 2-cube and Cube subdivisionable sphere==
| |
| | |
| ===Singular 2-cube and boundary===
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition 3-1 (Singular 2-cube)'''<ref>[[Michael Spivak]]:"Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus" Westview Press, 1971 [http://books.google.co.jp/books/about/Calculus_On_Manifolds.html?id=POIJJJcCyUkC&redir_esc=y]</ref> Set ''D'' = [''a''<sub>1</sub>, ''b''<sub>1</sub>] × [''a''<sub>2</sub>, ''b''<sub>2</sub>] ⊆ '''R'''<sup>2</sup> and let ''U'' be a non-empty [[open set|open]] [[subset]] of '''R'''<sup>3</sup>. The image of ''D'' under a piecewise smooth map ψ : ''D'' → ''U'' is called a singular 2-cube.
| |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| Given <math>D:=[{a}_{1},{b}_{1}]\times[{a}_{2},{b}_{2}]</math>,
| |
| we define the notarization map of sngler two cube
| |
| <math>{\theta}_{D}:{I}^{2}\to D</math>
| |
| :<math>{\theta}_{D}({u}_{1},{u}_{2})=
| |
| \left(
| |
| \begin{array}{c}
| |
| {u}_{1}({b}_{1}-{a}_{1}) +{a}_{1}\\
| |
| {u}_{2}({b}_{2}-{a}_{2}) +{a}_{2}
| |
| \end{array}
| |
| \right)
| |
| </math>
| |
| here, the I:=[0,1] and I<sup>2</sup> stands for <math>{I}^{2}=I\times I</math>.
| |
| | |
| Above mentioned is strongly increase function (that means
| |
| <math>det(J(\theta_{D})_{({u}_{1},{u}_{2})})>0</math> (for
| |
| all <math>({u}_{1},{u}_{2})\in\mathbb{R}^{3} </math>) that,
| |
| following lemma is satisfied.
| |
| | |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Lemma 3-1(Notarization map of sngler two cube).'''
| |
| Set ''D'' = [''a''<sub>1</sub>, ''b''<sub>1</sub>]× [''a''<sub>2</sub>, ''b''<sub>2</sub>] ⊆ '''R'''<sup>2</sup> and let ''U'' be a non-empty [[open set|open]] [[subset]] of '''R'''<sup>3</sup>.
| |
| 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''<sup>2</sup> under a piecewise smooth map <math>\varphi\circ{\theta}_{D}</math>,
| |
| <math>\tilde{S}:=\varphi\circ{\theta}_{D}[{I}^{2}]</math> be a singular 2-cube.then,
| |
| For all <math>\mathbf{F}</math>,smooth vector field on U,
| |
| | |
| :<math>{\int}_{S} \mathbf{F} dS ={\int}_{\tilde{S}} \mathbf{F} d\tilde{S} </math>
| |
| | |
| </div>
| |
| </div>
| |
| </div>
| |
| | |
| 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''<sup>2</sup>.
| |
| | |
| In order to facilitate the discussion of boundary, we define
| |
| <math>{\delta}_{[k,j,c]}:\mathbb{R}^{k}\to \mathbb{R}^{k+1}</math> by
| |
| :<math>{\delta}_{[k,j,c]}({t}_{1},\cdots,{t}_{k}):=
| |
| ({t}_{1},\cdots,{t}_{j-1},c,{t}_{j+1},\cdots ,{t}_{k})
| |
| </math>
| |
| | |
| γ<sub>1</sub>, ..., γ<sub>4</sub> are the one-dimensional [[edge (geometry)|edges]] of the image of ''I''<sup>2</sup>.Hereinafter, the ⊕ stands for joining paths<ref group="note" name=vee/> and,
| |
| the <math>\ominus</math> stands for backwards of curve
| |
| .<ref group="note" name=omin/>
| |
| | |
| :<math>\begin{align}
| |
| \begin{cases} \gamma_{1}:[0, 1]\to {I}^{2} \\ \gamma_{1}(t) := {\delta}_{[1,2,0]}(t)=(t,0) \end{cases}, \qquad &\begin{cases}\gamma_{2}:[0,1] \to { I }^{2} \\ \gamma_{2}(t) :={\delta}_{[1,1,1]}(t)=(1, t) \end{cases} \\
| |
| \begin{cases} \gamma_{3}:[0, 1] \to {I}^{2} \\ \gamma_{3}(t) := \ominus{\delta}_{[1,2,1]}(t)= (-t+0+1, 1)\end{cases}, \qquad &\begin{cases}\gamma_{4}:[0,1] \to {I}^{2} \\ \gamma_{4}(t) := \ominus{\delta}_{[1,1,0]}(t) =(0, 1-t)\end{cases}
| |
| \end{align}</math>
| |
| | |
| :<math>\gamma(t):= (\gamma_{1} \oplus \gamma_{2} \oplus \gamma_{3} \oplus \gamma_{4})(t) </math>
| |
| :<math>\Gamma_{i}(t):= \varphi(\gamma_{i}(t)), \qquad i=1, 2, 3, 4</math>
| |
| :<math>\Gamma(t):= \varphi(\gamma(t))
| |
| =(\Gamma_{1} \oplus \Gamma_{2} \oplus \Gamma_{3} \oplus \Gamma_{4})(t)
| |
| </math>
| |
| | |
| ===Cube subdivision===
| |
| <div align=left>
| |
| <div class="messagebox standard-talk NavFrame">
| |
| <div align=left>
| |
| '''Definition 3-2(Cube subdivisionable sphere).'''(see Iwahori<ref name=iwahori/> p399)
| |
| Let ''S'' ⊆ '''R'''<sup>3</sup> 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
| |
| <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> such that
| |
| | |
| *[CSS0] For all <math>\lambda\in\Lambda</math>, <math>({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})</math> are Legular that means,
| |
| **<math>\Lambda</math> is [[finite set]].
| |
| **<math>{\varphi}_{\lambda}</math> are [[Injective function]] on <math>{I}^{2}</math> and,
| |
| **for almost all <math>({u}_{1},{u}_{2})\in Int({I}^{2})</math>, <math> det((J\varphi)_{({u}_{1},{u}_{2})})\neq 0</math>
| |
| *[CSS1]<math>S = \bigcup_{\lambda \in \Lambda} S_{\lambda}</math>
| |
| *[CSS2]<math>{\lambda}_{1}\neq{\lambda}_{2}</math> ⇒ <math>
| |
| {\varphi}_{{\lambda}_{1}}[Int({I}^{2})]\cap{\varphi}_{{\lambda}_{2}}[Int({I}^{2})]=\varnothing</math>
| |
| *[CSS3]If <math>{c}_{1},{c}_{2}\ =0\ or\ 1</math>, <math>{j}_{1},{j}_{2} =1\ or\ 2</math> and, <math>
| |
| {\varphi}_{{\lambda}_{1}}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I]\cap
| |
| {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I]\neq \varnothing
| |
| </math>then,<math>{\varphi}_{{\lambda}_{1}}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I] =
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| {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I]</math>
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| | |
| and then abovementioned <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>
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| are said to be a Cube subdivision of the ''S''.
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| </div>
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| </div>
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| </div>
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| | |
| <div align=left>
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| <div class="messagebox standard-talk NavFrame">
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| <div align=left>
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| '''Definitions 3-3(Boundary of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> ).'''(see Iwahori<ref name=iwahori/> p399)
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| | |
| Let ''S'' ⊆ '''R'''<sup>3</sup> be a "Cube subdivisionable sphere" and,
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| Let <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> be a Cube subdivision of the ''S''.,then
| |
| | |
| (1)The <math>{\varphi}_{{\lambda}_{1}}\circ\delta_{[1,{j}_{1},{c}_{1}]}</math> are said to be an edge of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> if <math>{\varphi}_{{\lambda}_{1}}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I]</math> satisfies
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| :"<math>
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| {\varphi}_{{\lambda}_{1}}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I]=
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| {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I]
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| </math>then,<math>({\lambda}_{1},{j}_{1},{c}_{1})=({\lambda}_{2},{j}_{2},{c}_{2})</math>"
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| that means "although not line contact even if the point contact with other ridge line" and above mentioned "=" stands for equal as a set.<br>
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| That means, l is said to be an edge of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> iff
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| "There is only one <math>\lambda</math> only one c and only one j such that,
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| <math>l={\varphi}_{{\lambda}}\circ\delta_{[1,{j},{c}]}</math>"
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| | |
| (2)Boundary of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> is a collection of
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| edges in the sense of "(1)". <math>\partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> means the boundary of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>
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| | |
| (3)If l is an edge in the sense of "(1)", then, we described as follows.
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| :<math>l \prec \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>
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| | |
| </div>
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| </div>
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| </div>
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| | |
| The definition of the boundary of the Definitions 3-3 is apparently depends on the cube subdevision.
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| However, considering the following fact, the boundary is not depends on the cube subdevision.
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| | |
| <div align=left>
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| <div class="messagebox standard-talk NavFrame">
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| <div align=left>
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| '''Fact(Boundary of <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> ).'''(see Iwahori<ref name=iwahori/> p399)
| |
| | |
| Let ''S'' ⊆ '''R'''<sup>3</sup> be a "Cube subdivisionable sphere" and,
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| Let both <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> and
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| <math>\{({I}^{2},{\psi}_{\mu},{L}_{\mu})\}_{\mu\in M}</math>
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| be a Cube subdivision of the ''S'', then
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| | |
| :<math>\partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}=\partial\{({I}^{2},{\psi}_{\mu},{L}_{\mu})\}_{\mu\in M}</math>
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| | |
| that means, the definition of boundary is not depends on the cube subdivision.
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| </div>
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| </div>
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| </div>
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| So, considerting above mentiond fact, following "Definition3-4" is well-defined.
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| <div align=left>
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| <div class="messagebox standard-talk NavFrame">
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| <div align=left>
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| '''Definitions 3-4(Boundary of Surface'''(see Iwahori<ref name=iwahori/> p399)
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| Let ''S'' ⊆ '''R'''<sup>3</sup> be a "Cube subdivisionable sphere" and,
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| Let <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>, then
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| | |
| (1)
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| :<math>\partial S:=
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| \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>
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| | |
| (2)If
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| :<math>l \prec \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>
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| then
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| :<math>l \prec \partial S</math>
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| and then such "l" are said to be an edge of S.
| |
| </div>
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| </div>
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| </div>
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| | |
| {{Expand section|date=January 2013}}
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| | |
| ==Notes and References==
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| | |
| ===Notes===
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| {{Reflist|group=note}}
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| | |
| ===References===
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| {{Reflist}}
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| | |
| {{DEFAULTSORT:Kelvin-Stokes theorem}}
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| [[Category:Electromagnetism]]
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| [[Category:Mechanics]]
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| [[Category:Vectors]]
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| [[Category:Vector calculus]]
| |