CHAPTER 6 CURVILINEAR COORDINATES Transformation of Coordinates In general coordinates, the transformation equations of the form  (U1 ,U ,U );  ,U ,U );  ,U ,U ) x f y g(U1 z h(U1 2 3 2 3 2 3 where the functions f, g, h are continuous and single valued and whose partial derivatives are continuous. Then   and coordinate curves can be formed by keeping two of the three variable constant. Now ...     2 3 dU dU 2 3   dU1 is a tangent vector to the U1 – coordinate curve at P is a tangent vector to the U2 – coordinate curve at P is a tangent vector to the U3 – coordinate curve at P eˆ1  eˆ1  hU1 eˆ2 2 2 2  hU2 eˆ3 3 3 3  eˆ3  hU3 Then Eqn. (*) above becomes  , , Where are the scale factors. hU1 hU2 hU3
 CHAPTER 6 CURVILINEAR COORDINATES Transformation of Coordinates In general coordinates, the transformation equations of the form x  f (U1 ,U 2 ,U 3 ); y  g(U1 ,U 2 ,U 3 ); z  h(U1 ,U 2 ,U 3 ) where the functions f, g, h are continuous and single valued and whose partial derivatives are continuous. Then   and coordinate curves can be formed by keeping two of the three variable constant. Now    dU1  2 dU 2  3 dU 3 ... is a tangent vector to the U1 – coordinate curve at P is a tangent vector to the U2 – coordinate curve at P is a tangent vector to the U3 – coordinate curve at P eˆ1   hU1 eˆ1 2 2 2  hU2 eˆ2 eˆ3  3 3 3  hU3 eˆ3 Then Eqn. (*) above becomes  Where hU1 , hU2 , hU3 are the scale factors.