MATH 2423 #15: PROBLEM
4
#15: 4a) ANSWER: Yes, - 4 pi.
SOLUTION: The projection of the hemisphere on the
xy-plane is the circle x 2 + y 2 = 4 or r = 2 if
you use polar coordinates.
- GRAD ( x 2 + y 2 + z 2 ) = < 2x , 2y ,
2z > = 2< x , y , z >
- ( x 2 + y 2 + z 2 ) z = 2z
- dS = [ 2< x , y , z > / ( 2z ) ] dA xy = [ < x , y , z > / z ]
dA xy
- CURL F = < ( xyz ) y - ( x 2 + z ) z , ( y
) z - ( xyz ) x , ( x 2 + z ) x - ( y )
y > =
< xz - 1 , - yz , 2x - 1 >
- CURL F • dS = < xz - 1 , - yz , 2x - 1 > • [ < x , y ,
z > / z ] dA xy =
[( x 2 z - x - y 2 z + 2xz - z )/ z] dA xy
- Let z = ( 4 - x 2 - y 2 ) ½ = ( 4 - r
2 ) ½, x = rcos
0 and y = rsin0.
- [( x 2 z - x - y 2 z + 2xz - z )/ z] dA xy = [ x
2 - x /z - y 2 + 2x - 1 ] dA xy =
[ r 2 cos 2 0 - rcos0/( 4 - r
2 ) ½ - r 2 sin 2
0 + 2rcos0 - 1] r dr
d0
© 2000: Roxanne M. Byrne