CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND  8 Z For this case we use cylindrical coordinates.   cos, sin, x  y  z  z said region  cos2   sin      x2 y z z 2 2 2 2      z or z 2 Z 2 4 z 0 z0 z V 4  Z  ddzd 2 2 4 Hence  3 2 3 0 z0 Y 0 2 z  0 2 256 2 5 15 4 2 EXERCISE  2 2 P X Q1. If , where is single-valued and has continuous partial derivatives,  F  x, show that the work done in moving a particle from one point in P1 y, z x2  this field to another point is independent of the path joining the , , P2 y2 z2 two points. y 3y dz Q2. Show is an exact differential z3 cos x 4x3 zdx  2z3 y sin xdy  z sin x x4 2 2 2 of a function and find .   Ans.  sin  y z3 x x4 z  const. 2 2 Q3. Evaluate around the closed curve of figure F d r  Y X x x . 2 adjacent if F  yiˆ   y ˆj  y x Ans. 2 3 O
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS For this case we use cylindrical coordinates. x   cos, y   sin, z  z x2  y 2  z  2 cos2    2 sin 2   z  2  z or    z Z Z  8 said region V 2 4 z 0 z0 z  2 ddzd Z  4 Hence  2 4 2 3 0 z0 3 2 0 z 5 4 2 0 256 15  Y 2 2 EXERCISE X P  2 2 Q1. If F  , where  is single-valued and has continuous partial derivatives, show that the work done in moving a particle from one point P1 x, y, z in this field to another point P2 x2 , y2 , z2  is independent of the path joining the two points. Q2. Show y 2 z3 cos x 4x3 zdx  2z3 y sin xdy  3y 2 z 2 sin x x4 dz is an exact differential of a function  and find  . Ans.   y 2 z3 sin x x4 z  const. Q3. Evaluate F  d r around the closed curve of figure Y 2 adjacent if F  x yiˆ  x  y ˆj . Ans. 2 3 O y  x 2 X