CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND c) Find the work done in moving an object in this field from (0, 1, -1) to  2 c) . Ans. b)  sin 4 2z 15 4  y x  xz y   const.  2 3 Q6. Evaluate for each of the following cases; F nˆ dS  S a) and S is the surface of the plane 2x+y=6 in the first octant 2xˆj F  yiˆ  zkˆ cut off by the plane z=4. x iˆ S 2x+y+2z=6 b) 2xˆj 2 and is the surface of the plane in the F   y2  yzkˆ first octant. Ans. a) 108 b) 81 xy- Q7. Evaluate over the region in the plane bounded by the plane  x y dxdy 2 2 R 36 .   x y 2 2 Ans. 144 2x2 Q8. If 3ziˆ 2xyˆj 4xkˆ F  Evaluate a) dV F b) F dV V x=0, y=0, z=0 where is the closed region bounded by the planes and 2x+2y+z=4   Ans. a) 8 3 b) 8 3 DIVERGENCE THEOREM (GAUSS’ THEOREM) Z S, V For a closed surface enclosing a region in a vector field F   S  FdV F  nˆdS F  d S V V S S Y In general, this means that the volume integral on the left- hand side can be expressed as a surface integral on the right-hand surface. X O 71
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS c) Find the work done in moving an object in this field from (0, 1, -1) to  2 Ans. b)   y 2 sin x  xz 3 4 y  2z  const. c) 15  4 . Q6. Evaluate F  nˆ dS for each of the following cases; S a) F  yiˆ  2xˆj zkˆ and S is the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4. b) F  x  y2 iˆ 2xˆj  2 yzkˆ and S is the surface of the plane 2x+y+2z=6 in the first octant. Ans. a) 108 b) 81 Q7. Evaluate x 2  y 2 dxdy over the region in the xy- plane bounded by the plane R x 2  y 2  36 . Ans. 144 Q8. If F  2x2 3ziˆ 2xyˆj 4xkˆ Evaluate a) F dV b) F dV where V is the closed region bounded by the planes x=0, y=0, z=0 and 2x+2y+z=4 Ans. a) 8 3 b) 8 3   DIVERGENCE THEOREM (GAUSS’ THEOREM) For a closed surface S, enclosing a region V in a vector Z field F  FdV  F  nˆdS  F  d S V S S V S In general, this means that the volume integral on the left- hand side can be expressed as a surface integral on the right-hand surface. 71 X O Y