CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND  2x  4xdx Since y dy 2 and hence we have 1 2 1 3 5 Example 5.5 Find the work done in moving a particle once around a circle C in the xy plane, if the circle has center at the origin and radius 3 and if the force field is given by  ˆ 2x 3x 2 4zk      F y zˆ y 0 3cos , 3sin  0  2      In the plane z So that d r dxˆ dyˆ where x y where     Then F d r   2x x    ydx ydy 2 2 2 0 9 4 1 1 4 4 18  Theorem: If everywhere in a region R of space, where is single-  x, F y, z valued and has continuous derivatives in R, then: 1) is independent of the path C in R joining P1 and P2. d r  C R. 2) 0 around any closed curve in In such a case is called a  F d r F  conservative vector field and  is its scalar potential. A vector field is F conservative if and only if 0 so that    F or F , an exact differential. F d r  F1dx  F2dy  F3dz  d 
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Since y  2x 2 dy  4xdx and hence we have 1 2 1 3 5 Example 5.5 Find the work done in moving a particle once around a circle C in the xy plane, if the circle has center at the origin and radius 3 and if the force field is given by F  2x y  zˆ   ˆ  3x 2 y  4zk In the plane z  0 So that d r  dxˆ  dyˆ where x  3cos , y  3sin  where 0  2 Then F  d r       2x ydx  x  ydy 2 2 4 9 2 0 1 4 1 4  18 Theorem: If F  everywhere in a region R of space, where x, y, z is single- valued and has continuous derivatives in R, then: 1)  d r is independent of the path C in R joining P1 and P2. 2) F  d r  0 around any closed curve C in R. In such a case F is called a conservative vector field and  is its scalar potential. A vector field F is conservative if and only if  F  0 or F  so that F  d r  F1dx  F2dy  F3dz  d , an exact differential.