CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Example 5.7 2x iˆ 3y If , evaluate around the triangle (i) in the indicated 4x F   y  F d r ˆj  2 direction. 2 Y Y (2,1) 2 x  y (II) O (2,0) O (I) X X  F d r  Where C1 is from (0,0) to (2,0) where y is constant. C2 (2,0) (2,1) x Where is from to where is constant. Where C3 is from (2,1) to (0,0) where x=2y.  dx 3y 4xdy   Hence F  d r 2 1 0 2 2 0 3 2 0 2  4  8  32 32 23 143 14 3 Example 5.8 x x Evaluate around the closed curve C (fig (ii) above), if . F d r F  yiˆ   y  ˆj x x     F d r ydx ydy  C1 y=x2 C2 y2=x. Where is curve and is 1 2 2 Thus 4 3  1 1 2 2 where y  x 1 2 1 0 3 3 2 3 4 2
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Example 5.7 If F  2x  y 2 iˆ  3y 4x ˆj , evaluate F  d r around the triangle (i) in the indicated direction. Y (2,1) Y 2 x  y 2 O (2,0) (I) X O (II) X F  d r  Where C1 is from (0,0) to (2,0) where y is constant. Where C2 is from (2,0) to (2,1) where x is constant. Where C3 is from (2,1) to (0,0) where x=2y. Hence F  d r   dx  3y 4xdy 2 1 0 2 2 0 3 2 0 2  4  32 8 32 23  143 14 3 Example 5.8 Evaluate F  d r around the closed curve C (fig (ii) above), if F  x yiˆ  x  y ˆj . F  d r x ydx  x  ydy   Where C1 is curve y=x2 and C2 is y2=x. Thus  1 2 2 1 2 2 1 0 1 1 4 3 where y  x 2 4 2 3 3 2 3