Stokes' theorem: Difference between revisions
→First step of the proof (defining the pullback): minor fix for punctuation and reference mix up |
more of the same. Also the wording in this article is pretty atrocious ... |
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* |
*For a more conventional discussion see '''[[Stokes' theorem#Kelvin.E2.80.93Stokes theorem|Stokes Theorem.]]''' |
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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> |
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> |
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[[Nagayoshi Iwahori]], et.al:"Bi-Bun-Seki-Bun-Gaku" [[:ja:裳華房|Sho-Ka-Bou]](jp) 1983/12 ISBN978-4-7853-1039-4 |
[[Nagayoshi Iwahori]], et.al:"Bi-Bun-Seki-Bun-Gaku" [[:ja:裳華房|Sho-Ka-Bou]](jp) 1983/12 ISBN978-4-7853-1039-4 |
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[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)" |
[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)" |
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[[: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 {{math|'''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]]. |
[[: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 {{math|'''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 {{math|'''R'''<sup>3</sup>}} can be considered as a [[differential form|1-form]] in which case curl is the [[exterior derivative]]. |
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==Theorem== |
==Theorem== |
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Let {{math|''γ'': [''a'', ''b''] → '''R'''<sup>2</sup>}} be a [[Piecewise smooth]] [[Jordan curve|Jordan plane curve]], that bounds the domain {{math|''D'' ⊂ '''R'''<sup>2</sup>}}.<ref group="note" name=JC>The [[Jordan curve theorem]] implies that the [[Jordan curve]] divides {{math|'''R'''<sup>2</sup>}} into two components, a [[compact space|compact]] one (the bounded area) and another is non-compact.</ref> Suppose {{math|''ψ'' |
Let {{math|''γ'': [''a'', ''b''] → '''R'''<sup>2</sup>}} be a [[Piecewise smooth]] [[Jordan curve|Jordan plane curve]], that bounds the domain {{math|''D'' ⊂ '''R'''<sup>2</sup>}}.<ref group="note" name=JC>The [[Jordan curve theorem]] implies that the [[Jordan curve]] divides {{math|'''R'''<sup>2</sup>}} into two components, a [[compact space|compact]] one (the bounded area) and another is non-compact.</ref> Suppose {{math|''ψ'': ''D'' → '''R'''<sup>3</sup>}} is smooth, with {{math|''S'' :{{=}} ''ψ''(''D'')}}, and {{math|Γ}} is the [[space curve]] defined by {{math|Γ(''t'') {{=}} ''ψ''(''γ''(''t''))}}.<ref group="note" name=cgamma> {{mvar|γ}} and {{math|Γ}} are both loops, however, {{math|Γ}} is not necessarily a [[Jordan curve]]</ref> If {{math|'''F'''}} a smooth vector field on {{math|'''R'''<sup>3</sup>}}, then<ref name="Jame"/><ref name="bath"/><ref name="proofwik"/> |
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:<math>\oint_\Gamma \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS </math> |
:<math>\oint_\Gamma \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS </math> |
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==Proof== |
==Proof== |
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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>), |
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 {{math|'''A''' {{=}} (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>)}}, |
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:<math>\mathbf{A} = \omega_{\mathbf{A}} = a_1 dx_1+a_2\,dx_2+a_3\,dx_3 </math> |
:<math>\mathbf{A} = \omega_{\mathbf{A}} = a_1 dx_1+a_2\,dx_2+a_3\,dx_3 </math> |
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:<math>\mathbf{A} = {}^{*} \omega_{\mathbf{A}} = a_1\,dx_2 \wedge dx_3+a_2 \,dx_3 \wedge dx_1+a_3 \,dx_1 \wedge dx_2</math> |
:<math>\mathbf{A} = {}^{*} \omega_{\mathbf{A}} = a_1\,dx_2 \wedge dx_3+a_2 \,dx_3 \wedge dx_1+a_3 \,dx_1 \wedge dx_2</math> |
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the proof here is similar to the proof using the pull-back of {{math|''ω''<sub>'''F'''</sub>}}. In actual, under the identification of {{math|''ω''<sub>'''F'''</sub> {{=}} '''F'''}} following equations are satisfied. |
the proof here is similar to the proof using the pull-back of {{math|''ω''<sub>'''F'''</sub>}}. In actual, under the identification of {{math|''ω''<sub>'''F'''</sub> {{=}} '''F'''}} following equations are satisfied. |
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\end{align}</math> |
\end{align}</math> |
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Here, ''d'' stands for [[Exterior derivative]] of [[Differential form]], ψ |
Here, ''d'' stands for [[Exterior derivative]] of [[Differential form]], {{mvar|ψ<sup>∗</sup>}} stands for [[pull back]] by {{mvar|ψ}} and, ''P''<sub>1</sub> and ''P''<sub>2</sub> are same as ''P''<sub>1</sub> and ''P''<sub>2</sub> of the body text of this article respectively.</ref> 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). In an ordinary, mathematicians use the [[differential form]], especially "[[Pullback (differential geometry)|pull-back]]<ref group="note" name="f1"/> 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]]. |
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=== First step of the proof (defining the pullback)=== |
=== First step of the proof (defining the pullback)=== |
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:<math>\mathbf{P}(u,v) = (P_1(u,v), P_2(u,v))</math> |
:<math>\mathbf{P}(u,v) = (P_1(u,v), P_2(u,v))</math> |
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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. |
so that {{math|'''P'''}} is the pull-back<ref group="note" name="f1"/> of {{math|'''F'''}}, and that {{math|'''P'''(''u'', ''v'')}} is {{math|'''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. |
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:<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> |
:<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> |
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So, we obtain following equation |
So, we obtain following equation |
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:<math>\oint_{\Gamma} \mathbf{F} d\Gamma = \oint_{\gamma} \mathbf{P} d\gamma</math> |
:<math>\oint_{\Gamma} \mathbf{F} d\Gamma = \oint_{\gamma} \mathbf{P} d\gamma</math> |
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\end{align}</math> |
\end{align}</math> |
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So,<ref group="note" name="bil"/> <ref group="note" name=trans/><ref group="note" name="gaiseki"> |
So,<ref group="note" name="bil"/> <ref group="note" name=trans/><ref group="note" name="gaiseki">We prove following (★0). |
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We prove following (★0). |
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:<math>({(J\mathbf{F})}_{\psi(u,v)} - {}^{t}{(J\mathbf{F})}_{\psi(u,v)}) \mathbf{x} =(\nabla\times\mathbf{F})\times \mathbf{x}, \quad \text{for all}\, \mathbf{x}\in\mathbf{R}^{3}</math> (★0) |
:<math>({(J\mathbf{F})}_{\psi(u,v)} - {}^{t}{(J\mathbf{F})}_{\psi(u,v)}) \mathbf{x} =(\nabla\times\mathbf{F})\times \mathbf{x}, \quad \text{for all}\, \mathbf{x}\in\mathbf{R}^{3}</math> (★0) |
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First, we focus attention on the linearity of the algebraic operator "'''a'''×" and, we obtain the matrix representation thereof. |
First, we focus attention on the linearity of the algebraic operator "'''a'''×" and, we obtain the matrix representation thereof. |
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The method how to obtain the matrix representation of Linear map are described in for example [http://math.kennesaw.edu/~sellerme/sfehtml/classes/math3260/sec44outline.pdf].Let both of '''a''' and '''x''' |
The method how to obtain the matrix representation of Linear map are described in for example [http://math.kennesaw.edu/~sellerme/sfehtml/classes/math3260/sec44outline.pdf]. Let both of {{math|'''a'''}} and {{math|'''x'''}} be three dimensional [[Column vector]] represented as follows, |
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:<math> \mathbf{a}= \begin{pmatrix} a_1\\ a_2\\ a_3 \end{pmatrix}, \quad \mathbf{x}= \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}</math> |
:<math> \mathbf{a}= \begin{pmatrix} a_1\\ a_2\\ a_3 \end{pmatrix}, \quad \mathbf{x}= \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}</math> |
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Then, according to the definition of [[Cross product]] ” ×”, '''a'''×'''x''' are represented as follows. |
Then, according to the definition of [[Cross product]] ” ×”, '''a'''×'''x''' are represented as follows. |
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:<math> \mathbf{a}\times\mathbf{x}= \begin{pmatrix} a_2 x_3-a_3 x_2\\ a_3 x_1-a_1 x_3\\ a_1 x_2-a_2 x_1\end{pmatrix}</math> |
:<math> \mathbf{a}\times\mathbf{x}= \begin{pmatrix} a_2 x_3-a_3 x_2\\ a_3 x_1-a_1 x_3\\ a_1 x_2-a_2 x_1\end{pmatrix}</math> |
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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). |
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). |
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:<math>\mathbf{a}\times\mathbf{e} |
:<math>\mathbf{a}\times\mathbf{e}_1 = \begin{pmatrix} 0 \\ a_3\\ -a_2 \end{pmatrix}, \quad \mathbf{a}\times \mathbf{e}_{2}= \begin{pmatrix} -a_3\\ 0\\ a_1 \end{pmatrix}, \quad \mathbf{a}\times\mathbf{e}_{3}= \begin{pmatrix} a_2\\ -a_1\\ 0 \end{pmatrix}</math> |
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:<math>\mathbf{a}\times\mathbf{x}= \begin{pmatrix} 0&-a_3&a_2\\ a_3&0&-a_1\\ -a_2&a_1&0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}</math> (★1) |
:<math>\mathbf{a}\times\mathbf{x}= \begin{pmatrix} 0&-a_3&a_2\\ a_3&0&-a_1\\ -a_2&a_1&0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}</math> (★1) |
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Seconds, let <math>A=(a_{ij})</math> is a {{math|3 × 3}} matrix and using the following substitution, (a<sub>1</sub>, ... , a<sub>3</sub> are components of |
Seconds, let <math>A=(a_{ij})</math> is a {{math|3 × 3}} matrix and using the following substitution, (a<sub>1</sub>, ... , a<sub>3</sub> are components of {{math|'''a'''}}), |
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:<math>{a}_{1}={a}_{32}-{a}_{23}</math> (★2-1) |
:<math>{a}_{1}={a}_{32}-{a}_{23}</math> (★2-1) |
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:<math>{a}_{2}={a}_{13}-{a}_{31}</math> (★2-2) |
:<math>{a}_{2}={a}_{13}-{a}_{31}</math> (★2-2) |
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===Fourth step of the proof (reduction to Green's theorem)=== |
===Fourth step of the proof (reduction to Green's theorem)=== |
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Combining the second and third steps, and then applying [[Green's theorem]] completes the proof. |
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==Application for conservative force and scalar potential== |
==Application for conservative force and scalar potential== |
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:<math> \begin{cases} \theta_{[a,b]}:[0,1]\to[a,b] \\ \theta_{[a,b]}=s(b-a)+a \end{cases}</math> |
:<math> \begin{cases} \theta_{[a,b]}:[0,1]\to[a,b] \\ \theta_{[a,b]}=s(b-a)+a \end{cases}</math> |
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Above-mentioned <math>\theta_{[a,b]}</math> is a strongly increasing function that, for all piece wise smooth paths {{math|''c'': [''a'', ''b''] → '''R'''<sup>3</sup>}}, for all smooth vector field ''F'', domain of which includes |
Above-mentioned <math>\theta_{[a,b]}</math> is a strongly increasing function that, for all piece wise smooth paths {{math|''c'': [''a'', ''b''] → '''R'''<sup>3</sup>}}, for all smooth vector field ''F'', domain of which includes {{math|''c''([''a'', ''b''])}}, following equation is satisfied. |
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:<math>\int_c \mathbf{F} dc =\int_{c\circ\theta_{[a,b]}}\ \mathbf{F}\ d(c\circ\theta_{[a,b]})</math> |
:<math>\int_c \mathbf{F} dc =\int_{c\circ\theta_{[a,b]}}\ \mathbf{F}\ d(c\circ\theta_{[a,b]})</math> |
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</div> |
</div> |
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In mechanics a [[lamellar vector field]] is called a [[conservative force]]; |
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. |
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=== Helmholtz's theorems=== |
=== Helmholtz's theorems=== |
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<div class="messagebox standard-talk NavFrame"> |
<div class="messagebox standard-talk NavFrame"> |
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<div align=left> '''Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics).'''<ref name="DTPO"/> and see p142 of Fujimoto<ref name=fujimno/><br> |
<div align=left> '''Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics).'''<ref name="DTPO"/> and see p142 of Fujimoto<ref name=fujimno/><br> |
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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> |
Let ''U'' ⊆ '''R'''<sup>3</sup> be an [[open set|open]] [[subset]] with a Lamellar vector field {{math|'''F'''}}, and piecewise smooth loops {{math|''c''<sub>0</sub>, ''c''<sub>1</sub>: [0, 1] → ''U''}}. If there is a function {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that |
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* '''[TLH0]''' ''H'' is piecewise smooth, |
* '''[TLH0]''' ''H'' is piecewise smooth, |
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* '''[TLH1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1], |
* '''[TLH1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1], |
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</div> |
</div> |
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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'' |
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 {{math|''H'': [0, 1] × [0, 1] → ''U''}} as “homotopy between ''c''<sub>0</sub> and ''c''<sub>1</sub>”. |
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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. |
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. |
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<div class="messagebox standard-talk NavFrame"> |
<div class="messagebox standard-talk NavFrame"> |
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<div align=left> |
<div align=left> |
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'''Definition (Homotopy and Homotope).''' Suppose ''Z'' and ''W'' are topological spaces, with continuous maps ''f''<sub>0</sub>, ''f''<sub>1</sub> |
'''Definition (Homotopy and Homotope).''' Suppose ''Z'' and ''W'' are topological spaces, with continuous maps ''f''<sub>0</sub>, ''f''<sub>1</sub>: ''Z'' → ''W''. |
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(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 |
(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 |
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Then, |
Then, |
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'''(1) Tube-Like-Homotopy:''' A homotopy |
'''(1) Tube-Like-Homotopy:''' A homotopy {{math|''H'': [0, 1] × [0, 1] → ''M''}} is "Tube-Like", if |
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* '''[TLH0]''' ''H'' is continues |
* '''[TLH0]''' ''H'' is continues |
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* '''[TLH1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') |
* '''[TLH1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') |
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[[File:Domain of singular 2 cube.jpg|thumb| The definitions of γ<sub>1</sub>, ..., γ<sub>4</sub>]] |
[[File:Domain of singular 2 cube.jpg|thumb| The definitions of γ<sub>1</sub>, ..., γ<sub>4</sub>]] |
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Hereinafter, the ⊕ stands for joining paths<ref group="note" name=vee>If the two curves {{math|''α'': [''a''<sub>1</sub>, ''b''<sub>1</sub>] → ''M'', ''β'': [''a''<sub>2</sub>, ''b''<sub>2</sub>] → ''M''}}, satisfy α(b<sub>1</sub>) = β(a<sub>2</sub>) then, we can define new curve α ⊕ β |
Hereinafter, the ⊕ stands for joining paths<ref group="note" name=vee>If the two curves {{math|''α'': [''a''<sub>1</sub>, ''b''<sub>1</sub>] → ''M'', ''β'': [''a''<sub>2</sub>, ''b''<sub>2</sub>] → ''M''}}, satisfy {{math|''α''(''b''<sub>1</sub>) {{=}} ''β''(''a''<sub>2</sub>)}} then, we can define new curve {{math|''α'' ⊕ ''β''}} so that, for all smooth vector field {{math|'''F'''}} (if domain of which includes image of {{math|''α'' ⊕ ''β''}}) |
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:<math>\int_{\alpha\oplus \beta} \mathbf{F} \, d(\alpha\oplus \beta)= \int_\alpha \mathbf{F} \, d\alpha + \int_\beta \mathbf{F}\, d\beta</math> |
:<math>\int_{\alpha\oplus \beta} \mathbf{F} \, d(\alpha\oplus \beta)= \int_\alpha \mathbf{F} \, d\alpha + \int_\beta \mathbf{F}\, d\beta</math> |
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'''Definition (Joint of paths).''' Let ''M'' be a topological space and {{math|''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: |
'''Definition (Joint of paths).''' Let ''M'' be a topological space and {{math|''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 {{math|''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: |
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:<math>(\alpha\oplus \beta) |
:<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> |
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\alpha(t) & a_1 \le t \le \ b_1, \\ |
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\beta(t+(a_2-b_1)) & b_1 < t \le b_1+(b_2-a_2). |
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\end{cases}</math> |
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</ref> |
</ref> |
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the <math>\ominus</math> stands for backwards of curve<ref group="note" name=omin>Given curve on |
the <math>\ominus</math> stands for backwards of curve<ref group="note" name=omin>Given curve on {{mvar|M}}, {{math|''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) |
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:<math>\int_{\ominus\alpha} \mathbf{F} \, d(\ominus\alpha)= -\int_{\alpha}\mathbf{F} \, d\alpha, |
:<math>\int_{\ominus\alpha} \mathbf{F} \, d(\ominus\alpha)= -\int_{\alpha}\mathbf{F} \, d\alpha, |
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'''Definition (Backward of curve).''' Let |
'''Definition (Backward of curve).''' Let {{mvar|M}} be a topological space and ''α'' : [''a''<sub>1</sub>, ''b''<sub>1</sub>] → ''M'' , |
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be path on |
be path on {{mvar|M}}. We can define backward thereof, <math>\ominus</math>''α'' : [''a''<sub>1</sub>, ''b''<sub>1</sub>] → ''M'' by: |
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:<math>\ominus\alpha(t)=\alpha(b_1+a_1-t)</math> |
:<math>\ominus\alpha(t)=\alpha(b_1+a_1-t)</math> |
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</div> |
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And, given two curves on |
And, given two curves on {{mvar|M}}, ''α'': [a<sub>1</sub>, b<sub>1</sub>] → ''M'', ''β'': [a<sub>2</sub>, b<sub>2</sub>] → ''M'', which satisfy ''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. |
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:<math>\alpha\ominus\beta:=\alpha\oplus(\ominus\beta)</math> |
:<math>\alpha\ominus\beta:=\alpha\oplus(\ominus\beta)</math> |
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</ref> |
</ref> |
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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'' |
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 {{math|''H'': ''D'' → ''M''}} |
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:<math>\begin{align} |
:<math>\begin{align} |
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\begin{cases} |
\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} \\ |
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\begin{cases} \gamma_3:[0, |
\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} |
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\end{align}</math> |
\end{align}</math> |
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:<math>\gamma(t):= (\gamma_1 \oplus \gamma_2 \oplus \gamma_3 \oplus \gamma_4)(t) </math> |
:<math>\gamma(t):= (\gamma_1 \oplus \gamma_2 \oplus \gamma_3 \oplus \gamma_4)(t) </math> |
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And, let ''S'' be the image of {{mvar|D}} under ''H''. Then, |
And, let ''S'' be the image of {{mvar|D}} under ''H''. Then, |
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:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS </math> |
:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma = \iint_S \nabla\times\mathbf{F}\, dS </math> |
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will be obvious according to the Theorem 1 and, '''F''' is Lamellar vector field that, right side of that equation is zero, so, |
will be obvious according to the Theorem 1 and, {{math|'''F'''}} is Lamellar vector field that, right side of that equation is zero, so, |
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:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =0</math> |
:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =0</math> |
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Here, |
Here, |
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:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =\sum_{i=1}^{4} \oint_{\Gamma_i} \mathbf{F} d\Gamma </math> <ref group="note" name=vee/> |
:<math>\oint_{\Gamma} \mathbf{F}\, d\Gamma =\sum_{i=1}^{4} \oint_{\Gamma_i} \mathbf{F} d\Gamma </math> <ref group="note" name=vee/> |
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On the other hand, |
On the other hand, |
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:<math>c_{1}(t)=H(t,0)=H({\gamma}_{1}(t))={\Gamma}_{1}(t)</math> |
:<math>c_{1}(t)=H(t,0)=H({\gamma}_{1}(t))={\Gamma}_{1}(t)</math> |
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:<math>c_{2}(t)=H(t,1)=H(\ominus{\gamma}_{3}(t))=\ominus{\Gamma}_{3}(t) |
:<math>c_{2}(t)=H(t,1)=H(\ominus{\gamma}_{3}(t))=\ominus{\Gamma}_{3}(t)</math> |
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</math> |
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that, subjected equation is proved. |
that, subjected equation is proved. |
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'''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> |
'''Lemma 2-2.'''<ref name="DTPO"/><ref name=lee/> Let {{math|''U'' ⊆ '''R'''<sup>3</sup>}} be an [[open set|open]] [[subset]], with a Lamellar vector field '''F''' and a piecewise smooth loop {{math|''c''<sub>0</sub>: [0, 1] → ''U''}}. Fix a point '''p''' ∈ ''U'', if there is a homotopy (tube-like-homotopy) {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that |
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* '''[SC0]''' ''H'' is '''piecewise smooth''', |
* '''[SC0]''' ''H'' is '''piecewise smooth''', |
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* '''[SC1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1], |
* '''[SC1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1], |
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'''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]]. |
'''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]]. {{mvar|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 |
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* '''[SC0']''' ''H'' is '''contenious''', |
* '''[SC0']''' ''H'' is '''contenious''', |
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* '''[SC1]''' ''H''(''t'', 0) = ''c''(''t'') for all ''t'' ∈ [0, 1], |
* '''[SC1]''' ''H''(''t'', 0) = ''c''(''t'') for all ''t'' ∈ [0, 1], |
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here, the {{math|''I'' {{=}} [0, 1]}} and {{math|''I''<sup>2</sup> {{=}} ''I'' × ''I''}}. |
here, the {{math|''I'' {{=}} [0, 1]}} and {{math|''I''<sup>2</sup> {{=}} ''I'' × ''I''}}. |
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Above-mentioned is strongly increase function (that means <math>det(J(\theta_{D})_{( |
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\mathbf{R}^{3} </math>) that, following lemma is satisfied. |
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'''Lemma 3-1(Notarization map of singular two cube).''' |
'''Lemma 3-1(Notarization map of singular two cube).''' |
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Set {{math|''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 |
Set {{math|''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 {{math|'''R'''<sup>3</sup>}}. Let the image of {{mvar|D}} under a piecewise smooth map {{math|''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, \tilde{S}:=\varphi\circ \theta_D (I^2)</math> be a singular 2-cube. Then, For all {{math|'''F'''}}, smooth vector field on {{mvar|U}}, |
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<math>\tilde{S}:=\varphi\circ{\theta}_{D}[{I}^{2}]</math> be a singular 2-cube. Then, For all {{math|'''F'''}},smooth vector field on {{mvar|U}}, |
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:<math>\int_S \mathbf{F} \, dS =\int_{\tilde{S}} \mathbf{F} \, d\tilde{S} </math> |
:<math>\int_S \mathbf{F} \, dS =\int_{\tilde{S}} \mathbf{F} \, d\tilde{S} </math> |
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\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} |
\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} |
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\end{align}</math> |
\end{align}</math> |
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:<math>\gamma(t):= (\gamma_{1} \oplus \gamma_{2} \oplus \gamma_{3} \oplus \gamma_{4})(t) </math> |
:<math>\gamma(t):= (\gamma_{1} \oplus \gamma_{2} \oplus \gamma_{3} \oplus \gamma_{4})(t) </math> |
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:<math>\Gamma_{i}(t):= \varphi(\gamma_{i}(t)), \qquad i=1, 2, 3, 4</math> |
:<math>\Gamma_{i}(t):= \varphi(\gamma_{i}(t)), \qquad i=1, 2, 3, 4</math> |
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'''Definition 3-2(Cube subdivisionable sphere).'''(see Iwahori<ref name=iwahori/> p399) |
'''Definition 3-2 (Cube subdivisionable sphere).'''(see Iwahori<ref name=iwahori/> p399) A non-empty subset {{math|''S'' ⊆ '''R'''<sup>3</sup>}} is said to be a "Cube subdivisionable sphere" when there are at least one [[Indexed family]] of singular 2-cube |
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⚫ | |||
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 |
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<math>\ |
*[CSS0] For all <math>\lambda\in\Lambda</math>, <math>(I^2, \varphi_\lambda, S_\lambda)</math> are Legular that means, |
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⚫ | |||
⚫ | |||
*[CSS0] For all <math>\lambda\in\Lambda</math>, <math>({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})</math> are Legular that means, |
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⚫ | |||
⚫ | |||
⚫ | |||
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*[CSS1]<math>S = \bigcup_{\lambda \in \Lambda} S_{\lambda}</math> |
*[CSS1]<math>S = \bigcup_{\lambda \in \Lambda} S_{\lambda}</math> |
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*[CSS2]{{math|''λ''<sub>1</sub> ≠ ''λ''<sub>2</sub>}} implies <math>\varphi_{\lambda_1}[Int(I^2)]\cap \varphi_{\lambda_2}[Int(I^2)]=\varnothing</math> |
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*[CSS2]<math>{\lambda}_{1}\neq{\lambda}_{2}</math> implies <math> |
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⚫ | *[CSS3]If <math>c_1, c_2\ =0\ or\ 1, 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] = {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I]</math> |
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{\varphi}_{{\lambda}_{1}}[Int(I^2)]\cap{\varphi}_{{\lambda}_{2}}[Int(I^2)]=\varnothing</math> |
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⚫ | *[CSS3]If <math> |
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<math>\{(I^2,\varphi_\lambda, S_{\lambda})\}_{\lambda\in\Lambda}</math> is called a Cube subdivision of the ''S''. |
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'''Definitions 3-3(Boundary of |
'''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" |
Let ''S'' ⊆ '''R'''<sup>3</sup> be a "Cube subdivisionable sphere" with cube subdivision: <math>\{(I^2,{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>. Then |
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⚫ | |||
(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 |
(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>\varphi_{\lambda_1}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I]= {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I] |
:<math>\varphi_{\lambda_1}\circ\delta_{[1,{j}_{1},{c}_{1}]}[I]= {\varphi}_{{\lambda}_{2}}\circ\delta_{[1,{j}_{2},{c}_{2}]}[I] |
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</math> |
</math> |
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then, <math>({\lambda}_{1}, j_1, c_1)=({\lambda}_{2},{j}_{2},{c}_{2})</math>. That means "although not line contact even if the point contact with other ridge line" and above-mentioned "=" stands for equal as a set. |
then, <math>({\lambda}_{1}, j_1, c_1)=({\lambda}_{2},{j}_{2},{c}_{2})</math>. That means "although not line contact even if the point contact with other ridge line" and above-mentioned "=" stands for equal as a set. |
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⚫ | |||
"There is only one {{mvar|λ}} only one c and only one j such that, <math>l={\varphi}_{{\lambda}}\circ\delta_{[1,{j},{c}]}</math>" |
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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 there is only one {{mvar|λ}} only one {{mvar|c}} and only one j such that, <math>l={\varphi}_{{\lambda}}\circ\delta_{[1,{j},{c}]}</math>. |
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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> |
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. |
(3) If l is an edge in the sense of "(1)", then, we described as follows. |
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:<math>l \prec \partial\{( |
:<math>l \prec \partial\{( I^2, \varphi_\lambda, S_\lambda)\}_{\lambda\in\Lambda}</math> |
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'''Fact (boundary of |
'''Fact (boundary of <math>\{(I^2, {\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> ).'''(see Iwahori<ref name=iwahori/> p399) Let {{math|''S'' ⊆ '''R'''<sup>3</sup>}} be a cube subdivisionable sphere with cube subdivisions: <math>\{(I^2, \varphi_\lambda, S_\lambda)\}_{\lambda\in\Lambda}</math> and <math>\{(I^2, \psi_\mu, L_\mu)\}_{\mu\in M}</math>, then |
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Let ''S'' ⊆ '''R'''<sup>3</sup> be a "Cube subdivisionable sphere" and, let both <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> and <math>\{({I}^{2},{\psi}_{\mu},{L}_{\mu})\}_{\mu\in M}</math> be a cube subdivision of ''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> |
:<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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'''Definitions 3-4(Boundary of Surface'''(see Iwahori<ref name=iwahori/> p399) Let ''S'' ⊆ '''R'''<sup>3</sup> be a |
'''Definitions 3-4(Boundary of Surface'''(see Iwahori<ref name=iwahori/> p399) Let ''S'' ⊆ '''R'''<sup>3</sup> be a cube subdivisionable sphere and, <math>\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math>, then |
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(1) :<math>\partial S:= \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> |
(1) :<math>\partial S:= \partial\{({I}^{2},{\varphi}_{\lambda},{S}_{\lambda})\}_{\lambda\in\Lambda}</math> |
Revision as of 01:14, 12 February 2015
It has been suggested that this article be merged into Stokes' Theorem. (Discuss) Proposed since November 2014. |
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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 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.
Theorem
Let γ: [a, b] → R2 be a Piecewise smooth Jordan plane curve, that bounds the domain D ⊂ R2.[note 1] Suppose ψ: D → R3 is smooth, with S := ψ(D), and Γ is the space curve defined by Γ(t) = ψ(γ(t)).[note 2] If F a smooth vector field on R3, then[1][2][3]
Proof
The proof of the Theorem consists of 4 steps.[2][3][note 3] 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). In an ordinary, mathematicians use the differential form, especially "pull-back[note 3] 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 3] 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.
Where, is the normal inner product of R3 and hereinafter, stands for bilinear form according to matrix A.[note 4][note 5]
Second step of the proof (first equation)
According to the definition of line integral,
where, Jψ stands for the Jacobian matrix of ψ. Hence,[note 4][note 5]
So, we obtain following equation
Third step of the proof (second equation)
First, calculate the partial derivatives, using Leibniz rule of inner product
On the other hand, according to the definition of surface integral,
So, we obtain
Fourth step of the proof (reduction to Green's theorem)
Combining the second and third steps, and then applying Green's theorem completes the proof.
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] → R3, for all smooth vector field F, domain of which includes c([a, b]), 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 7]
That theorem is also important in the area of Homotopy theorem.[7]
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,
Here,
and, H is Tubeler-Homotopy that,
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, 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 D
here, the I = [0, 1] and I2 = I × I.
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, I2.
In order to facilitate the discussion of boundary, we define
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
The definition of the boundary of the Definitions 3-3 is apparently depends on the cube subdivision. However, considering the following fact, the boundary is not depends on the cube subdevision.
So, considering the above-mentioned fact, following "Definition 3-4" is well-defined.
This section needs expansion. You can help by adding to it. (January 2013) |
Notes and references
Notes
- ^ The Jordan curve theorem implies that the Jordan curve divides R2 into two components, a compact one (the bounded area) and another is non-compact.
- ^ γ and Γ are both loops, however, Γ is not necessarily a Jordan curve
- ^ a b c If you know the differential form, when we considering following identification of the vector field A = (a1, a2, a3),
- ^ a b c Given a n × m matrix A we define a bilinear form:
- ^ a b c Given a n × m matrix A, tA stands for transposed matrix of A.
- ^ We prove following (★0).
- (★0)
- (★1)
- (★2-1)
- (★2-2)
- (★2-3)
- (★3)
- (★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.
- ^ a b c If 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)
- ^ a b c Given 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)
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.
References
- ^ a b James Stewart;"Essential Calculus: Early Transcendentals" Cole Pub Co (2010)[3]
- ^ a b c This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) [4], please refer the [5]
- ^ a b c This proof is also same to the proof shown in
- ^ a b c d e Nagayoshi Iwahori, et.al:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 ISBN978-4-7853-1039-4 [6](Written in Japanese)
- ^ a b 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
- ^ a b c d e f g Lawrence Conlon; "Differentiable Manifolds (Modern Birkhauser Classics)" Birkhaeuser Boston (2008/1/11) [8]
- ^ a b c d e f 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]