# sol3 - Math 264 Advanced Calculus Winter 2008 Assignment 3...

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Unformatted text preview: Math 264: Advanced Calculus Winter 2008 Assignment 3 Solutions Problem 1 (Adams, Β§ 16.2 # 14). Verify the identity β β’ ( f ( β g Γ β h )) = β f β’ ( β g Γ β h ) for smooth functions f,g and h . Solution: We shall use properties stated in Theorem 3 on page 859 in Adams. By the property (b), we have β β’ ( f ( β g Γ β h )) = β f β’ ( β g Γ β h ) + f β β’ ( β g Γ β h ) . By the property (d), the last term can be rewritten as f [ β Γ β g ) β’ β h- β g β’ ( β Γ β h )] = , where in the last equality we have used the property (h) in Theorem 3 in Adams. The result follows. Problem 2 (Adams, Β§ 16.3 # 4). Evaluate Z C x 2 y dx- xy 2 dy, where C is the clockwise boundary of the region 0 β€ y β€ β 9- x 2 . Solution: Denote the region by R . By Greenβs theorem (note that the boundary is negatively oriented!) the boundary integral is equal to- Z Z R β’ β βx (- xy 2 )- β βy ( x 2 y ) β dxdy = Z Z R ( x 2 + y 2 ) dxdy. In polar coordinates, we get Z 3 r =0 Z Ο ΞΈ =0 r 2 Β· rdrdΞΈ = Ο Z 3 r 3 dr = 81 Ο 4 ....
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sol3 - Math 264 Advanced Calculus Winter 2008 Assignment 3...

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