CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Example 5.11 S Evaluate , where and is that part of the plane 18ziˆ 12 3ykˆ F nˆ dS F    ˆj S which is located in the first octant. 2x  3y  6z  12 If S is part of the level surface G(x,y,z)=C then may be taken to be one of the two nˆ vectors    nˆ or nˆ depending on which vector gives the preferred direction. Thus let  2x  3y  6z G 2iˆ  3  6kˆ ˆj    nˆ 7  kˆ  dS 7 6 dA  S, On the surface the value is given by the formula F nˆ  ˆ 6z 12 2x 3y  where Thus the flux of outward through S is F 36z 36 7 18 y Rxy x-axis; y=0, z=0 0 6. Now, along the and we integrate from to From the given surface on the xy-plane, z=0 so that along the y-axis, we integrate from 0 to 12 2x 3 2 6 3 Example 5.12 A vector field exists over a surface S defined by 2 9       F yiˆ ˆj kˆ x y z 2 2 2 bounded by in the first octant. 0, 0, 0 x  y  z  Evaluate over the indicated surface. F  d S nˆ Z Y 3 p ; x 2  y 2  9, z  0  X 3 : 9 0 S x  y  z  2 2 2  2xiˆ 2 2zkˆ    yˆj
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Example 5.11 Evaluate F  nˆ dS , where F  18ziˆ 12 ˆj  3ykˆ and S is that part of the plane S 2x  3y  6z  12 which is located in the first octant. If S is part of the level surface G(x,y,z)=C then nˆ may be taken to be one of the two vectors nˆ   or nˆ  depending on which vector gives the preferred direction. Thus let G  2x  3y  6z nˆ    2iˆ  3 ˆj  6kˆ 7 dS   kˆ dA  7 6 On the surface S, the value F  nˆ is given by the formula ˆ Thus the flux of F outward through S is 36z 36 18 y 7 Rxy where 6z  12 2x 3y Now, along the x-axis; y=0, z=0 and we integrate from 0 to 6. From the given surface on the xy-plane, z=0 so that along the y-axis, we integrate from 0 to 12 2x 3 6 2 3 Example 5.12 A vector field F  yiˆ  2 ˆj  kˆ exists over a surface S defined by x 2  y 2  z 2  9 bounded by x  0, y  0, z  0 in the first octant. Evaluate F  d S over the indicated surface. Z nˆ S : x 2  y 2  z 2 9  0 ; X 3   p 3 Y x 2  y 2  9, z  0  2xiˆ  2 yˆj  2zkˆ