CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND  nˆ 3 2z   0 dS dA dA for z 6     F n   xy2 1 yz  F  nˆ dS  dA dA xy2zyz 3z xy2 yz S 3 S S The projection of S on the xy- plane is the region R bounded by the circle . Then 9, 0 x  y  z  2 2 xy2 yz S R z dxdy Converting to spherical polar coordinates; 2   3sin cos,  3cos x z  ; 0  ,   3sin sin 0 y  2  So that x, y  , and we have   9sin 2 2 1  sin cos  6sin sin  3cos d d 3 2 0 0  2  2 2 0 2 0 3 4  2 sin   Example 5.13 0 z and Evaluate where 3y F nˆ dS F  ziˆ  xˆj z  kˆ  2 S S and is the surface of the cylinder 16 x  y  5  z  2 2 included in the first octant between
CHAPTER 5 nˆ  3 VECTOR INTEGRATION AND INTEGRAL THEOREMS dS  dA  6 2z dA for z 0 F  n    1   xy2 yz F  nˆ dS  S S xy2 yz 3  3z dA xy2zyz dA S The projection of S on the xy- plane is the region R bounded by the circle x 2  y 2  9, z  0 . Then xy2 yz S R z dxdy Converting to spherical polar coordinates; x  3sin  cos, y  3sin  sin , z  3cos 0   2 ; 0   2 So that x, y  , and we have 1 3   2 2 0 0 2  sin cos  6sin sin  3cos 9sin d d  2 0 sin 2  2   2 2 0 3 4  Example 5.13 Evaluate F  nˆ dS where F  ziˆ  xˆj 3y 2 z  kˆ S and S is the surface of the cylinder x 2  y 2  16 included in the first octant between z  0 and z  5