Fdtxjr

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Is the inverse of the Jacobian equivalent to the Jacobian of the inverse?

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0

$begingroup$

$ widetilde rho = left [
beginmatrix
rho & theta & phi \
endmatrix
right ]^top ; $ and $ widetilde x = left [
beginmatrix
x & y & z \
endmatrix
right ]^top . ; $ Let $ widetilde x = rho left [
beginmatrix
sin theta cos phi & sin theta sin phi & cos phi \
endmatrix
right ]^top . ; $

$$
underline J = left [ beginmatrix
fracpartial xpartial rho & fracpartial xpartial theta & fracpartial xpartial phi \
fracpartial ypartial rho & fracpartial ypartial theta & fracpartial ypartial phi \
fracpartial zpartial rho & fracpartial zpartial theta & fracpartial zpartial phi \
endmatrix right ] = left [ beginmatrix
fracpartial widetilde xpartial rho & fracpartial widetilde xpartial theta & fracpartial widetilde xpartial phi
endmatrix right ]
$$

$$
underline J ^top underline J =
left [ beginmatrix
h_rho^2 & 0 & 0 \
0 & h_theta^2 & 0 \
0 & 0 & h_phi^2 \
endmatrix right ]
$$

$$
hat widetilde h_q = fracpartial widetilde xpartial q div h_q
$$

$$
underline J = left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]
$$

$$
underline J^-1 =
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]^-1
left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
^top
$$

Proposition: As $ underline J = fracmathbf d widetilde xmathbf d widetilde rho, ; $ now $ underline J^-1 = fracmathbf d widetilde rhomathbf d widetilde x. ; $

Is this proposition correct? If false, please indicate a case where the proposition fails. Thank you.

edit: The small error has been fixed.

inverse jacobian

share|cite|improve this question

edited Mar 14 at 18:00

EricVonB

asked Mar 14 at 17:16

EricVonBEricVonB

12710

$endgroup$

add a comment | 

0

$begingroup$

$ widetilde rho = left [
beginmatrix
rho & theta & phi \
endmatrix
right ]^top ; $ and $ widetilde x = left [
beginmatrix
x & y & z \
endmatrix
right ]^top . ; $ Let $ widetilde x = rho left [
beginmatrix
sin theta cos phi & sin theta sin phi & cos phi \
endmatrix
right ]^top . ; $

$$
underline J = left [ beginmatrix
fracpartial xpartial rho & fracpartial xpartial theta & fracpartial xpartial phi \
fracpartial ypartial rho & fracpartial ypartial theta & fracpartial ypartial phi \
fracpartial zpartial rho & fracpartial zpartial theta & fracpartial zpartial phi \
endmatrix right ] = left [ beginmatrix
fracpartial widetilde xpartial rho & fracpartial widetilde xpartial theta & fracpartial widetilde xpartial phi
endmatrix right ]
$$

$$
underline J ^top underline J =
left [ beginmatrix
h_rho^2 & 0 & 0 \
0 & h_theta^2 & 0 \
0 & 0 & h_phi^2 \
endmatrix right ]
$$

$$
hat widetilde h_q = fracpartial widetilde xpartial q div h_q
$$

$$
underline J = left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]
$$

$$
underline J^-1 =
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]^-1
left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
^top
$$

Proposition: As $ underline J = fracmathbf d widetilde xmathbf d widetilde rho, ; $ now $ underline J^-1 = fracmathbf d widetilde rhomathbf d widetilde x. ; $

Is this proposition correct? If false, please indicate a case where the proposition fails. Thank you.

edit: The small error has been fixed.

inverse jacobian

share|cite|improve this question

edited Mar 14 at 18:00

EricVonB

asked Mar 14 at 17:16

EricVonBEricVonB

12710

$endgroup$

add a comment | 

$begingroup$

$ widetilde rho = left [
beginmatrix
rho & theta & phi \
endmatrix
right ]^top ; $ and $ widetilde x = left [
beginmatrix
x & y & z \
endmatrix
right ]^top . ; $ Let $ widetilde x = rho left [
beginmatrix
sin theta cos phi & sin theta sin phi & cos phi \
endmatrix
right ]^top . ; $

$$
underline J = left [ beginmatrix
fracpartial xpartial rho & fracpartial xpartial theta & fracpartial xpartial phi \
fracpartial ypartial rho & fracpartial ypartial theta & fracpartial ypartial phi \
fracpartial zpartial rho & fracpartial zpartial theta & fracpartial zpartial phi \
endmatrix right ] = left [ beginmatrix
fracpartial widetilde xpartial rho & fracpartial widetilde xpartial theta & fracpartial widetilde xpartial phi
endmatrix right ]
$$

$$
underline J ^top underline J =
left [ beginmatrix
h_rho^2 & 0 & 0 \
0 & h_theta^2 & 0 \
0 & 0 & h_phi^2 \
endmatrix right ]
$$

$$
hat widetilde h_q = fracpartial widetilde xpartial q div h_q
$$

$$
underline J = left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]
$$

$$
underline J^-1 =
left [ beginmatrix
h_rho & 0 & 0 \
0 & h_theta & 0 \
0 & 0 & h_phi \
endmatrix right ]^-1
left [ beginmatrix
hat widetilde h_rho & hat widetilde h_theta & hat widetilde h_phi
endmatrix right ]
^top
$$

Proposition: As $ underline J = fracmathbf d widetilde xmathbf d widetilde rho, ; $ now $ underline J^-1 = fracmathbf d widetilde rhomathbf d widetilde x. ; $

Is this proposition correct? If false, please indicate a case where the proposition fails. Thank you.

edit: The small error has been fixed.

inverse jacobian

share|cite|improve this question

edited Mar 14 at 18:00

EricVonB

asked Mar 14 at 17:16

EricVonBEricVonB

12710

$endgroup$

share|cite|improve this question

edited Mar 14 at 18:00

EricVonB

asked Mar 14 at 17:16

EricVonBEricVonB

12710

share|cite|improve this question

edited Mar 14 at 18:00

EricVonB

asked Mar 14 at 17:16

EricVonBEricVonB

12710

asked Mar 14 at 17:16

EricVonBEricVonB

12710

asked Mar 14 at 17:16

EricVonBEricVonB

12710