CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Example 5.1 C If evaluate along the curve having parametric equations    xy z d r 2 between P1 (0,0,0) and P2 (3,2,1).  3t,  2t ,  x y z t 2 3 t  3t 2t  12t    xy z 2 2 3 8     3tiˆ  2t  r xiˆ yˆj zkˆ t ˆj 2 3kˆ  dt d r  dxˆ  dyˆ  dzk  1 8 2 1 24  k ˆ 11 5 4iˆ   Example 5.2 If evaluate along the curve C defined by      ,  2t,   5 xy y z d r x t y z t 2 2 between P1 (0,0,5) and P2 (4,4,7). 2 3 2 8 15 Example 5.3 Find the total work done in moving a particle in a force field given by t=1 to t=2. along the curve from 3xyiˆ 5zˆj 10xkˆ 1, 2t , F   x  t  y  z  t 2 2 2      Total Work F d r  3xydx 5zdy 10xdz   2 Example 5.4 If , evaluate where C is the curve in the xy plane, , from 3xyiˆ 2x   F y ˆj F  d r y 2 2 (0,0) (1,2). to  0,  0  1,  2 For x y and x y Thus 2 57
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Example 5.1 If   xy 2 z evaluate  d r along the curve C having parametric equations x  3t, y  2t 2 , z  t 3 between P1 (0,0,0) and P2 (3,2,1).   xy 2 z  3t 2t 2 t 3   12t 8 r  xiˆ  yˆj  zkˆ  3tiˆ  2t 2 ˆj  t 3kˆ d r  dxˆ  dyˆ  dzk   dt 1 8 2 1  4iˆ  24 ˆ  5 11 k Example 5.2 If   xy  y 2 z evaluate  d r along the curve C defined by x  t 2 , y  2t, z  t  5 between P1 (0,0,5) and P2 (4,4,7). 2 3 2 8 15 Example 5.3 Find the total work done in moving a particle in a force field given by F  3xyiˆ 5zˆj  10xkˆ along the curve x  t 2  1, y  2t 2 , z  t 2 from t=1 to t=2. Total Work  F  d r      3xydx 5zdy  10xdz 2 Example 5.4 If F  3xyiˆ y 2 ˆj , evaluate F  d r where C is the curve in the xy plane, y  2x 2 , from (0,0) to (1,2). For x  0, y  0 and x  1, y  2 Thus 2 57