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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 27 Simply Connected Regions April 14 Reading Material: From Lecture Notes : V5 and V6. Last time: Greens Theorem: why it works. Examples. Today: Simply Connected Regions. 2 Generalized (or Extended) Greens Theorem Consider a region R with n holes. Call C the outside boundary and C 1 ,.....,C n the inside boundaries of the holes. Orient the boundaries so that the region R is always on the left while wolking on these curves. Also assume that these curves are simple and closed: Assume that a vector filed ~ F is continuously differentiable on R . Then the Greens theorem says that n X i =0 I C i ~ F d~ r = ZZ R N x- M y dA. Proof: Lets consider for simplicity a region with two holes. We cut through them and we split the region R into two new regions R 1 and R 2 that are now without holes. Lets call their boundaries B 1 and B 2 respectively: 1 For each of the two regions we apply the usual Greens theorem I B 1 ~ F d~ r + I B 2 ~ F d~ r = ZZ R 1 N x- M y dA + ZZ R 2 N x- M y dA = ZZ R N x- M y dA. On the other hand, in the left hand side, since the line integral on the segment introduced in the cutting appears twice but in opposite directions, we have that the only contribution remaining is relative only to the outside boundary C and the boundaries of the two holes C 1 and C 2 . this concludes the proof. 3 Simply connected regions Consider the two vector fields ~ G =- y x 2 + y 2 i + x x 2 + y 2 j. (3.1) ~ F =- y x 2 i + x x 2 j. (3.2) Question: Are these two vectors fields conservative in their domains?Are these two vectors fields conservative in their domains?...
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- Spring '06
- Multivariable Calculus