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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