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let's suppose I have two functions [itex]\phi:U\rightarrow V[/itex], and [itex]T:V\rightarrow V[/itex] that are both diffeomorphisms having inverse.

Furthermore [itex]T[/itex] is

*linear*.

I consider the function [itex]f(u) = (\phi^{-1}\circ T \circ \phi)(u)[/itex], where [itex]\circ[/itex] is the composition of functions.

Since [itex]T[/itex] is

*linear*, we already know that the Jacobian determinant is constant: [itex]J_T(v)=\lambda[/itex].

What can we say about [itex]J_f(u)[/itex], the Jacobian of

*f*?