CHAPTER 6 CURVILINEAR COORDINATES Thus   cos  sin  eˆ eˆ1 iˆ ˆj sin  cos ; Similarly eˆ  eˆ2  iˆ  eˆz  kˆ ˆj 3. Spherical Coordinate System U1,U2  r,  ,U3 ,  , ) P(r, Z sin  cos, sin  sin , cos    x r y r z r  and y 1, sin     hr h r, h r r r X Z x Y O  General Curvilinear Coordinate System (U1, U2, U3) Any system of coordinates can be treated in like manner to obtain expressions for , , the appropriate units vectors . eˆ1 eˆ2 eˆ3 ; ;  ;  eˆ2 eˆ3 eˆ1  2 2 3 3 These unit vectors are not always at right angles to each other. If they are mutually perpendicular, the coordinate system is orthogonal. The unit vectors are orthogonal if    0. eˆ1 eˆ2 eˆ1 eˆ3 eˆ2. eˆ3   
CHAPTER 6 CURVILINEAR COORDINATES Thus eˆ  eˆ1  cos iˆ  sin  ˆj Similarly eˆ  eˆ2  sin  iˆ  cos ˆj ; eˆz  kˆ 3. Spherical Coordinate System U1,U2 ,U3   r, ,  Z P(r, , ) x  r sin  cos, y  r sin  sin , z  r cos  and hr  1, h  r, h  r sin  X x O  y r r Z Y General Curvilinear Coordinate System (U1, U2, U3) Any system of coordinates can be treated in like manner to obtain expressions for the appropriate units vectors eˆ1 , eˆ2 , eˆ3 . eˆ1  ; eˆ2  2 2 ; eˆ3  3 3 ; These unit vectors are not always at right angles to each other. If they are mutually perpendicular, the coordinate system is orthogonal. The unit vectors are orthogonal if eˆ1  eˆ2  eˆ1  eˆ3  eˆ2.  eˆ3  0.