CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Example 5.19 Evaluate over the entire surface of the region bounded by the cylinder F  nˆdS S 2x x2+z2=9, x=0, y=0, z=0 and y=8, if . F  6   yˆj zˆi xkˆ Apply divergence theorem F FdV  nˆdS  S V 3 8  ˆj  3 8 3 8 8  dx sind ,  2 On letting  x 3 cos ,  0 0 2  9 9 cos2  sind  18 0 4  2 0 Example 5.20 Evaluate over the entire surface of the region above the xy- plane bounded F  nˆdS S z2=x2+y2 z=4, by the cone and the plane if . F  4  xyz2  xzˆi ˆj 3zkˆ Z   x 2 y 2 z 2 By divergence theorem; Y F nˆdS FdV   S V z  4 4z   3dydxdz 4 z 2 z0 X
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Example 5.19 Evaluate F  nˆdS over the entire surface of the region bounded by the cylinder S x2+z2=9, x=0, y=0, z=0 and y=8, if F  6 zˆi  2x  yˆj xkˆ . Apply divergence theorem F  nˆdS  FdV S V 3 8  ˆj  3 8 3 8 8 On letting x  3 cos , dx  sind ,  2  0 0 2 9 9 cos2   sind 2 0 4 0   18 Example 5.20 Evaluate F  nˆdS over the entire surface of the region above the xy- plane bounded S by the cone z2=x2+y2 and the plane z=4, if F  4 xzˆi  xyz2 ˆj  3zkˆ . Z By divergence theorem; F  nˆdS  FdV S V z  4 x 2  y 2  z 2 Y 4 z z0 4z  2  3dydxdz X