Can the Jacobian of an Inverse Transformation Prove to be 1?

[jakey]

Hi guys, let's say I have a transformation $$T$$ from $$(p,q)$$ to $$(u,v)$$. The inverse transformation would be $$T^{-1}$$ from $$(u,v)$$ to $$(p,q)$$

Now, $$J(T) = u_{p}v_{q} - u_{q}v_{p}$$. On the other hand, $$J(T^{-1})= p_{u}q_v - p_{v}q_{u}$$. But $$|J(T)J(T^{-1})| = 0$$ and not equal to 1. I know it's supposed to be 1 but how do you show it using this way?

thanks!

 

[jakey]

Oh by the way, $$J$$ here refers to the jacobian.