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Unformatted text preview: 1 Lecture XXVIII Measures; Irrotational fields Circulation and ux measures Let us first see what the theorems from the past lectures say about the circula tion and ux measures. From Greens theorem, we get that, if F is a C 1 vector field on D in E 2 , then R F ( R ) = rot F dA for every regular region R of D , where F is the circulation measure given by F . In E 3 , from the divergence theorem we have that f F ( R ) = divF dV, R f F is the ux measure for every regular region R of D , where D is in E 3 and given by F . From Stokess theorem, we have F , ( S ) = curl d c F S is the circulation measure given by F . Remember that in E 2 , if where c F F = L i + M j , then L rot = M F x y F j + N In E 3 , if = L i + M k , then M N div = L + + F x y z F curl = i j k z y x L M N 1 2 Irrotational fields Definition 1 A vector field F on D in E 2 such that rotF = 0 everywhere is called an...
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This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.
 Two '04
 Duorg
 Math

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