MIT
18 02 -
Lecture 24
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Lecture 24 Vector Field F F r curl F A curl M N N M Divergence F n s div f A div M N M N A puzzle V curl V Line integral parameterize C cost sint r sint cost t V r sint cost sint cost dt sin t cos t t 2 but we were expecting 0 Why Because the vector eld is not de ned and continuous for all x y therefore we cannot apply Green s theorem De nition A region in R is simply connected if every closed curve C bounds a region R also in Theorem If R is simply connected and curl V 0 on and V de ned on then there is a function f on so that V Vf Proof Must show V r 0 for and closed curve C but any such C bounds a region R where curl V 0 and Green applies Triple integral of Region R within R and function f f V where dV x y z M z z y x outer integral
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