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Quiz 3 Solution - Math 3010 Quiz 3 Nov 14 2012 1(5 points...

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Math 3010 Quiz 3 Nov. 14, 2012 1. (5 points) Evaluate the integral of the function f ( x, y, z ) = z + 6 over the surface S given by Φ ( u, v ) = p u, v 3 , v P , u [0 , 2] , v [0 , 3] . Solution: V T u = (1 , 0 , 0) and V T v = ( 0 , 1 3 , 1 ) V T u × V T v = ± 0 , - 1 , 1 3 ² , v v v V T u × V T v v v v = 10 3 ii S f dS = i 3 0 i 2 0 f (Φ ( u, v )) v v v V T u × V T v v v v du dv = i 3 0 i 2 0 ( v + 6) 10 3 du dv = 10 3 ±i 2 0 du ²±i 3 0 ( v + 6) dv ² = 10 3 ( u | 2 0 ) ³ v 2 2 + 6 v V V V V 3 0 ´ = 15 10 2. (5 points) Let D be a region for which Green’s theorem holds. Suppose f is harmonic; that is 2 f ∂x 2 + 2 f ∂y 2 = 0 on D . Prove that i ∂D ∂f ∂y dx - ∂x dy = 0 . Solution: Green’s theorem states that i ∂D + P dx + Q dy = D ± ∂Q - ∂P ² dx dy.
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