그린 정리

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

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

메모

• Digital planimeter demonstration
• How does the planimeter work?

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxZThyY3Rtbk9BMFE/edit?usp=drivesdk

관련된 항목들

• 각원소벡터장
• 선적분
• 타원의 넓이
• 픽의 정리(Pick's Theorem)

사전 형태의 자료

• http://ko.wikipedia.org/wiki/그린정리
• http://en.wikipedia.org/wiki/Green_theorem

관련논문

• Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years
  • Thomas Archibald, , Math. Mag. 62 (1989), 219-232
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

노트

말뭉치

1. Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).
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.
3. Are you ready to use Green's theorem?
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.
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.
6. Note that Green's Theorem applies to regions in the xy-plane.
7. We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.
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.
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.
10. We can also write Green's Theorem in vector form.
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.
12. The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.
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.
14. The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.

메타데이터

위키데이터

• ID : Q321237

Spacy 패턴 목록

• [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}]