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Curvilinear Coordinates Transform matrix


Showing Jacobi identity for Poisson BracketThe Gaussian IntegralIs the derivative in the Euler-Lagrange equation defined as an implicit function?On the relationship between the volume element in curvilinear coordinates and the JacobianWhy are these two ways of evaluating the curl of a vector field not actually equivalent?Inverse of a (coordinate) transformation - which method is best?How to derive the transformation that maps general curvilinear coordinates to their cartesian image.How do I derive the volume element $ dV = sqrtg du^1 du^2 du^3 $ in a 3D curvilinear coordinate system?How to find inverse of general curvilinear coordinatesDivergence in Orthogonal Curvilinear Coordinates - Is there a mistake?













0












$begingroup$


In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_ijq_j$. Equation 4.100 lists $a$ as:
$$a_ij=frach'_ih_jfracpartial q_i'partial q_j tag4.100$$
Where $h$ is the scaling factor.



This makes sense when the unprimed system is (any) curvilinear system and the primed system is Cartesian. You would get $a_ij=frac1h_jfracpartial x_ipartial q_j$ .



This doesn't seem to work when the primed system is a curvilinear system and the unprimed system is Cartesian. You would get $a_ij=frach_i1fracpartial q_ix_j$. This, however is just an alternate definition of the Cartesian-->Curvilinear system



My question is why can't the most general form of the transformation matrix (4.100) produce a Curvilinear--> Cartesian matrix? I know I can just transpose the Cartesian-->Curvilinear matrix, but I am really confused as to why I can't get it straight out of 4.100 without need for transpose. Thanks in advance!










share|cite|improve this question









$endgroup$
















    0












    $begingroup$


    In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_ijq_j$. Equation 4.100 lists $a$ as:
    $$a_ij=frach'_ih_jfracpartial q_i'partial q_j tag4.100$$
    Where $h$ is the scaling factor.



    This makes sense when the unprimed system is (any) curvilinear system and the primed system is Cartesian. You would get $a_ij=frac1h_jfracpartial x_ipartial q_j$ .



    This doesn't seem to work when the primed system is a curvilinear system and the unprimed system is Cartesian. You would get $a_ij=frach_i1fracpartial q_ix_j$. This, however is just an alternate definition of the Cartesian-->Curvilinear system



    My question is why can't the most general form of the transformation matrix (4.100) produce a Curvilinear--> Cartesian matrix? I know I can just transpose the Cartesian-->Curvilinear matrix, but I am really confused as to why I can't get it straight out of 4.100 without need for transpose. Thanks in advance!










    share|cite|improve this question









    $endgroup$














      0












      0








      0





      $begingroup$


      In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_ijq_j$. Equation 4.100 lists $a$ as:
      $$a_ij=frach'_ih_jfracpartial q_i'partial q_j tag4.100$$
      Where $h$ is the scaling factor.



      This makes sense when the unprimed system is (any) curvilinear system and the primed system is Cartesian. You would get $a_ij=frac1h_jfracpartial x_ipartial q_j$ .



      This doesn't seem to work when the primed system is a curvilinear system and the unprimed system is Cartesian. You would get $a_ij=frach_i1fracpartial q_ix_j$. This, however is just an alternate definition of the Cartesian-->Curvilinear system



      My question is why can't the most general form of the transformation matrix (4.100) produce a Curvilinear--> Cartesian matrix? I know I can just transpose the Cartesian-->Curvilinear matrix, but I am really confused as to why I can't get it straight out of 4.100 without need for transpose. Thanks in advance!










      share|cite|improve this question









      $endgroup$




      In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_ijq_j$. Equation 4.100 lists $a$ as:
      $$a_ij=frach'_ih_jfracpartial q_i'partial q_j tag4.100$$
      Where $h$ is the scaling factor.



      This makes sense when the unprimed system is (any) curvilinear system and the primed system is Cartesian. You would get $a_ij=frac1h_jfracpartial x_ipartial q_j$ .



      This doesn't seem to work when the primed system is a curvilinear system and the unprimed system is Cartesian. You would get $a_ij=frach_i1fracpartial q_ix_j$. This, however is just an alternate definition of the Cartesian-->Curvilinear system



      My question is why can't the most general form of the transformation matrix (4.100) produce a Curvilinear--> Cartesian matrix? I know I can just transpose the Cartesian-->Curvilinear matrix, but I am really confused as to why I can't get it straight out of 4.100 without need for transpose. Thanks in advance!







      mathematical-physics curvilinear-coordinates






      share|cite|improve this question













      share|cite|improve this question











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