CHAPTER 6 CURVILINEAR COORDINATES Thus we have 2 2 2 2    0 Z 2 0 0 v  a 2 2 2 2 2 r0 0 Y  5  .r r 2  0 5  . a 2 dV  dxdydz X 2  Grad, div and Curl in Orthogonal Curvilinear Coordinates ) (a) Grad V Let a scalar field  exist in space and let be the d change in  from a point p to Q say. If the position vector Q of p is then that of Q is r r  dr u3 u2 2 3   Let grad dU P u1 2 O  iˆ dU 3  )U  )U2  )U3 eˆ1 eˆ2 eˆ3 1 Where )U1, are the components of grad  in the U1, U2, U3 directions. U2, U3 Also   dU1 dU dU 2 3 2 3    hU1 dU1eˆ1 hU2 dU hU3 dU 2eˆ2 3eˆ3 Now since dU  dU  dU3 2 2     )U1 )U2 )U3    e1 e2 e3 U1 U2 U3    hU1 dU1 hU2 dU hU3 dU3 2
CHAPTER 6 Thus we have CURVILINEAR COORDINATES v   a . 5 2 2 2 5   0 0  0  0  2  0 a r0 r 2 .r 2 X Z 2 2 2 2 2 2 2  dV  dxdydz Y Grad, div and Curl in Orthogonal Curvilinear Coordinates (a) Grad V ) Let a scalar field  exist in space and let d be the change in  from a point p to Q say. If the position vector of p is r then that of Q is r  dr u3 Q Let grad   2 dU 2  3 dU 3 O iˆ u2 P u1  )U 1 eˆ1  )U2 eˆ2  )U3 eˆ3 Where )U1, U2, U3 are the components of grad  in the U1, U2, U3 directions. Also dU1  2 dU 2  3 dU 3  hU1 dU1eˆ1  hU2 dU 2eˆ2  hU3 dU 3eˆ3 Now since dU  dU 2  dU3 2    )U1 e1  )U2 e2  )U3 e3    U1 hU1 dU1  U2 hU2 dU 2  U3 hU3 dU3