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Curvilinear Coordinates Transform matrix

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$begingroup$

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!

mathematical-physics curvilinear-coordinates

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asked Mar 15 at 6:32

user560239user560239

388

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add a comment | 

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!

mathematical-physics curvilinear-coordinates

share|cite|improve this question

asked Mar 15 at 6:32

user560239user560239

388

$endgroup$

add a comment | 

$begingroup$

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!

mathematical-physics curvilinear-coordinates

share|cite|improve this question

asked Mar 15 at 6:32

user560239user560239

388

$endgroup$

share|cite|improve this question

asked Mar 15 at 6:32

user560239user560239

388

share|cite|improve this question

asked Mar 15 at 6:32

user560239user560239

388

asked Mar 15 at 6:32

user560239user560239

388

asked Mar 15 at 6:32

user560239user560239

388