그린 정리
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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
- 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
- Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).[1]
- 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]
- Are you ready to use Green's theorem?[1]
- 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]
- 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]
- Note that Green's Theorem applies to regions in the xy-plane.[3]
- We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.[3]
- 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]
- 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]
- We can also write Green's Theorem in vector form.[5]
- 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]
- The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.[6]
- } \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]
- The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.[6]
소스
메타데이터
위키데이터
- ID : Q321237
Spacy 패턴 목록
- [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}]