The Jacobian of the inverse of a transformation is the inverse of the Jacobian of that transformation

.

In this post, we would like to illustrate the meaning of

the Jacobian of the inverse of a transformation = the inverse of the Jacobian of that transformation

by proving a special case.

.

Consider a transformation , which is an one-to-one mapping from unbarred ‘s to barred coordinates, where .

By definition, the Jacobian matrix J of is

.

Now we consider the the inverse of the transformation :

By definition, the Jacobian matrix of this inverse transformation, , is

.

On the other hand, the inverse of Jacobian of the original transformation is

.

If , their (1, 1)-elementd should be equation:

Let’s try to prove that.

.

Consider equations

Differentiate both sides of each equation with respect to , we have:

.

.

,

results

…

— Me@2018-08-09 09:49:51 PM

.

.

2018.08.09 Thursday (c) All rights reserved by ACHK