MATH 2423 #15: QUESTION 5

#15: 5a) ANSWER: 0

SOLUTION:

• From problem 4,
• the projection of the hemisphere on the xy-plane is r = 2 (in polar coordinates) and
• dS = [ 2< x , y , z > / ( 2z ) ] dA _xy = [ < x , y , z > / z ] dA _xy
• F • dS = < y , x ² + z , xyz > • [ < x , y , z > / z ] dA _xy =
  [ ( xy + x ² y + yz + xyz )/z ] dA _xy = [ y + xy + xy ( 1 + x )/z ] dA _xy =
  [ rsin0 + r ² cos0 sin0 + r ² cos0 sin0 ( 1 + rsin0 )/( 4 - r ² ) ] r dr d0
• 

© 2000: Roxanne M. Byrne