"그린 정리"의 두 판 사이의 차이

수학노트
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1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
  
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* [[스토크스 정리]]의 특수한 경우<br><math>\iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, {d}A=\oint_{\partial D} (P\, {d}x + Q\, {d}y)</math>
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* [[스토크스 정리]]의 특수한 경우:<math>\iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, {d}A=\oint_{\partial D} (P\, {d}x + Q\, {d}y)</math>
  
  
 
==폐곡선에 둘러싸인 영역의 넓이==
 
==폐곡선에 둘러싸인 영역의 넓이==
 
* 폐곡선 C에 둘러싸인 영역의 넓이는 다음 공식으로 주어진다 :<math>A=\oint_{C} x dy = \oint_{C} - y dx =\frac{1}{2}\oint_{C} x dy-y dx</math>
 
* 폐곡선 C에 둘러싸인 영역의 넓이는 다음 공식으로 주어진다 :<math>A=\oint_{C} x dy = \oint_{C} - y dx =\frac{1}{2}\oint_{C} x dy-y dx</math>
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===증명===
 
===증명===
17번째 줄: 18번째 줄:
  
 
==꼭지점이 주어진 다각형의 넓이==
 
==꼭지점이 주어진 다각형의 넓이==
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* 꼭지점이 $\{(x_i,y_i)| i=0,1,\cdots, n-1 \}$로 주어진 n-각형의 넓이 $A$는 다음으로 주어진다 $$A=\frac{1}{2}\sum_{i=0}^{n-1}x_iy_{i+1}-y_ix_{i+1}$$ 이 때, $(x_{n+1},y_{n+1})=(x_{0},y_{0})$
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* 평면위의 점 <math>P_i=(x_i,y_i), i=0,1,\cdots, n-1</math>을 꼭지점으로 갖는 n-각형 <math>\overline{P_0P_1\cdots P_{n-1}}</math>의 넓이 <math>A</math>는 다음으로 주어진다 :<math>A=\frac{1}{2}\sum_{i=0}^{n-1}x_iy_{i+1}-y_ix_{i+1}</math> 이 때, <math>(x_{n},y_{n})=(x_{0},y_{0}).</math> 이다
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==역사==
 
==역사==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[수학사연표 (역사)|수학사연표]]
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* [[수학사 연표]]
 
*   
 
*   
  
35번째 줄: 37번째 줄:
 
* [http://www.youtube.com/watch?v=pvGuGaImTek Digital planimeter demonstration ]
 
* [http://www.youtube.com/watch?v=pvGuGaImTek Digital planimeter demonstration ]
 
* [http://www.mathematik.com/Planimeter/explanation.html How does the planimeter work?]
 
* [http://www.mathematik.com/Planimeter/explanation.html How does the planimeter work?]
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==매스매티카 파일 및 계산 리소스==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxZThyY3Rtbk9BMFE/edit?usp=drivesdk
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==관련된 항목들==
 
==관련된 항목들==
41번째 줄: 48번째 줄:
 
* [[선적분]]
 
* [[선적분]]
 
* [[타원의 넓이]]
 
* [[타원의 넓이]]
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* [[픽의 정리(Pick's Theorem)]]
  
  
 
==사전 형태의 자료==
 
==사전 형태의 자료==
  
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* [http://ko.wikipedia.org/wiki/%EA%B7%B8%EB%A6%B0%EC%A0%95%EB%A6%AC http://ko.wikipedia.org/wiki/그린정리]
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* http://ko.wikipedia.org/wiki/그린정리
 
* http://en.wikipedia.org/wiki/Green_theorem
 
* http://en.wikipedia.org/wiki/Green_theorem
  
54번째 줄: 62번째 줄:
 
==관련논문==
 
==관련논문==
  
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* [http://www.jstor.org/stable/2689760 Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years]<br>
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* [http://www.jstor.org/stable/2689760 Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years]
 
** Thomas Archibald, , Math. Mag. 62 (1989), 219-232
 
** Thomas Archibald, , Math. Mag. 62 (1989), 219-232
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.ams.org/mathscinet
 
* http://www.ams.org/mathscinet
 
* http://dx.doi.org/
 
* http://dx.doi.org/
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[[분류:미적분학]]
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== 노트 ==
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===말뭉치===
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# Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).<ref name="ref_bb3b58b6">[https://mathinsight.org/greens_theorem_idea#:~:text=Green's%20theorem%20says%20that%20if,the%20macroscopic%20circulation%20around%20C. The idea behind Green's theorem]</ref>
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# Green's theorem and other fundamental theorems Green's theorem is one of the four fundamental theorems of vector calculus all of which are closely linked.<ref name="ref_bb3b58b6" />
 +
# Are you ready to use Green's theorem?<ref name="ref_bb3b58b6" />
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# Make sure you understand when you are allowed to use Green's theorem, check out some other ways of writing Green's theorem, then investigate some examples.<ref name="ref_bb3b58b6" />
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# Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses.<ref name="ref_512e2c21">[https://brilliant.org/wiki/greens-theorem/ Brilliant Math & Science Wiki]</ref>
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# Note that Green's Theorem applies to regions in the xy-plane.<ref name="ref_bdba4266">[http://www.math.lsa.umich.edu/~glarose/classes/calcIII/web/17_4/ green's theorem]</ref>
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# We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.<ref name="ref_bdba4266" />
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# The term Green's theorem is applied to a collection of results that are really just restatements of the fundamental theorem of calculus in higher dimensional problems.<ref name="ref_728ef97a">[https://wiki.seg.org/wiki/Green%27s_theorem Green's theorem]</ref>
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# The various forms of Green's theorem includes the Divergence Theorem which is called by physicists Gauss's Law, or the Gauss-Ostrogradski law.<ref name="ref_728ef97a" />
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# We can also write Green's Theorem in vector form.<ref name="ref_07694272">[https://math24.net/greens-theorem.html Green’s Theorem]</ref>
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# Subsection 12.7.2 Green's Theorem So far in this section, we have restricted ourselves to relatively nice closed curves when thinking about circulation.<ref name="ref_a71c3b35">[https://activecalculus.org/vector/S_Vector_GreensTheorem.html Green's Theorem]</ref>
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# The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.<ref name="ref_a71c3b35" />
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# } \end{equation*} In Activity 12.7.2, we showed that Green's Theorem holds when the region \(R\) is a rectangle with sides parallel to the coordinate axes.<ref name="ref_a71c3b35" />
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# The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.<ref name="ref_a71c3b35" />
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q321237 Q321237]
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===Spacy 패턴 목록===
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* [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}]

2022년 8월 11일 (목) 21:34 기준 최신판

개요

  • 스토크스 정리의 특수한 경우\[\iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, {d}A=\oint_{\partial D} (P\, {d}x + Q\, {d}y)\]


폐곡선에 둘러싸인 영역의 넓이

  • 폐곡선 C에 둘러싸인 영역의 넓이는 다음 공식으로 주어진다 \[A=\oint_{C} x dy = \oint_{C} - y dx =\frac{1}{2}\oint_{C} x dy-y dx\]


증명

면적은 \(A= \iint_{D} 1 \, {d}A\)으로 주어지므로, 그린 정리를 이용하여 다음 각각의 경우 \(P,Q\) 가 \(\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)=1\)을 만족함을 보이면 된다.

  • \(P=0,Q=x\)
  • \(P=-y,Q=0\)
  • \(P=-y/2,Q=x/2\)


꼭지점이 주어진 다각형의 넓이

  • 평면위의 점 \(P_i=(x_i,y_i), i=0,1,\cdots, n-1\)을 꼭지점으로 갖는 n-각형 \(\overline{P_0P_1\cdots P_{n-1}}\)의 넓이 \(A\)는 다음으로 주어진다 \[A=\frac{1}{2}\sum_{i=0}^{n-1}x_iy_{i+1}-y_ix_{i+1}\] 이 때, \((x_{n},y_{n})=(x_{0},y_{0}).\) 이다


역사



메모


매스매티카 파일 및 계산 리소스


관련된 항목들


사전 형태의 자료



관련논문

노트

말뭉치

  1. Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).[1]
  2. Green's theorem and other fundamental theorems Green's theorem is one of the four fundamental theorems of vector calculus all of which are closely linked.[1]
  3. Are you ready to use Green's theorem?[1]
  4. Make sure you understand when you are allowed to use Green's theorem, check out some other ways of writing Green's theorem, then investigate some examples.[1]
  5. Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses.[2]
  6. Note that Green's Theorem applies to regions in the xy-plane.[3]
  7. We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.[3]
  8. The term Green's theorem is applied to a collection of results that are really just restatements of the fundamental theorem of calculus in higher dimensional problems.[4]
  9. The various forms of Green's theorem includes the Divergence Theorem which is called by physicists Gauss's Law, or the Gauss-Ostrogradski law.[4]
  10. We can also write Green's Theorem in vector form.[5]
  11. Subsection 12.7.2 Green's Theorem So far in this section, we have restricted ourselves to relatively nice closed curves when thinking about circulation.[6]
  12. The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.[6]
  13. } \end{equation*} In Activity 12.7.2, we showed that Green's Theorem holds when the region \(R\) is a rectangle with sides parallel to the coordinate axes.[6]
  14. The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.[6]

소스

  1. ↑ 1.0 1.1 1.2 1.3 The idea behind Green's theorem
  2. ↑ Brilliant Math & Science Wiki
  3. ↑ 3.0 3.1 green's theorem
  4. ↑ 4.0 4.1 Green's theorem
  5. ↑ Green’s Theorem
  6. ↑ 6.0 6.1 6.2 6.3 Green's Theorem

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=그린_정리&oldid=53067"