CHAPTER 6 CURVILINEAR COORDINATES (r ) A vector associated with P can therefore be expressed as F (r )    where , , are the components of in the directions of F Fr eˆ1 F eˆ2 F eˆ3 Fr F F F the base vectors , , . Both cylindrical and spherical polar coordinates are eˆ1 eˆ2 eˆ3 orthogonal. Scale Factors: 1. For cylindrical polar coordinates, the unit base vectors are  ;  h  eˆ1 h  ; h  1 eˆ2  h  ;  hz 1  ˆe3 h 1 2. For spherical polar coordinates, the unit base vectors are ;   h  hr ; h In each case, h is called the scale factor. If U1, U2, U3 represent the coordinates in any coordinate system, we may obtain the scale factors as 2 2 dU1 dU dU 2 3 1. Rectangular Coordinate System U1  x, ,U ,U In rectangular coordinate system;  y, z 2 3  1, 1, 1 Then hU1  hx  hU2  hy  hU3  hz  U1  ,, ,U ,U 2. Cylindrical Coordinate System  z 2 3  Z P(,, z) Y    cos,  sin,  where x y z z cos  r O x 
CHAPTER 6 CURVILINEAR COORDINATES A vector F (r ) associated with P can therefore be expressed as F (r )  Fr eˆ1  F eˆ2  F eˆ3 where Fr , F , F are the components of F in the directions of the base vectors eˆ1 , eˆ2 , eˆ3 . Both cylindrical and spherical polar coordinates are orthogonal. Scale Factors: 1. For cylindrical polar coordinates, the unit base vectors are eˆ1  eˆ2  ˆe3     1 h 1 h 1 h ; h  ; h  ; hz  2. For spherical polar coordinates, the unit base vectors are hr  ; h  ; h  In each case, h is called the scale factor. If U1, U2, U3 represent the coordinates in any coordinate system, we may obtain the scale factors as 2 2 dU1 dU 2 dU 3 1. Rectangular Coordinate System In rectangular coordinate system; U1 ,U 2 ,U 3   x, y, z  Then hU1  hx  1, hU2  hy  1, hU3  hz  1 2. Cylindrical Coordinate System U1 ,U 2 ,U 3   ,, z  where x   cos, y   sin, z  z  cos x Z O  r P(,, z) Y