assignment6

assignment6 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Assignment 6 Due date: 2010 December 3 (Friday), 5:00pm. 1. Let F = M ( x,y ) i + N ( x,y ) j . The divergence theorem shows that the flux of F across a closed curve C (i.e., H C F · n ds ) is equal to RR D ( M x + N y ) dA , where D is the region bounded by C . Use Green’s theorem to establish a similar result involving H C F · T ds , the circulation of F along C . 2. Let F be a vector field in R 2 defined by F = - y i + x j x 2 + y 2 (a) Verify Green’s theorem over the annular region D = { ( x,y ) | a 2 x 2 + y 2 1 } . (See figure at right.) (b) Now let D be the unit disk. Does the formula of Green’s theorem hold for D ? Can you explain why? (c) Suppose C is any simple closed curve lying outside the circle C a = { ( x,y ) | x 2 + y 2 = a 2 } (see figure at right). Argue that if
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This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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assignment6 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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