Unformatted text preview: h 4 / 5 ,3 / 5 , i . (b) We have an inﬁnite number of solutions u = v /  v  where v = h 3 , 4 ,t i or v = h3 ,4 ,t i and t is any real number. 7. A normal to the plane z = f ( x,y ) is h f x ,f y ,1 i = h 4 ,3 ,1 i and so the plane is given by 0 = h 4 ,3 ,1 i · h x2 ,y1 ,z3 i = 4 x3 yz2 . 8. Z 2 π Z 2 r 3 cos 2 θ dr dθ . 9. Z 1 Z 1 f ( x,y ) dxdy + Z 2 1 Z 1 y1 f ( x,y ) dxdy . 10. Z e Z 1 /y ye xy dxdy = Z e e xy ﬂ ﬂ ﬂ x =1 /y x =0 = Z e ( e1) dy = ( e1) e . 11. Solving the equations for x and y , we have x = u + v 2 and y = vu 2 . The Jacobian is x u y vx v y u = 1 / 2. The domain is ' ( u,v ) ﬂ ﬂ ≤ u ≤ 1 ,1 ≤ v ≤ 1 “ . Thus the integral becomes Z 11 Z 1 u 2( v + 2) dudv ....
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This note was uploaded on 08/14/2011 for the course MATH 20 C taught by Professor Ronevans during the Spring '08 term at UCSD.
 Spring '08
 RONEVANS
 Math, Multivariable Calculus

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