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If I have a transformation T: $\bar{x}^i = \bar{x}^i(x^1,x^2,...,x^n)$, (where $ 0 \leq i \leq n $) with Jacobian Matrix $J = \left[\frac{\partial \bar{x}^i}{\partial x^j}\right]_{nn}$, is it true that $J^{-1}=\left[\frac{\partial x^i}{\partial \bar{x}^j}\right]_{nn}$? (If the inverse transformation $T^{-1}$ is defined.)

Is this just a result of the fact that in the process of calculating $\left[\frac{\partial x^i}{\partial \bar{x}^j}\right]_{nn}$ I would have to invert the transformation and thus am really calculating $\bar{J}$ for the inverse transformation, which I know to be $J^{-1}$? If possible please could you elaborate on this.

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Yes. This follows from the chain rule.

Let $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a diffeomorphisme (i.e. $T$ and $T^{-1}$ are continuously differentiable). Denote by $df_p$ the jacobian of a function $f$ at a point $p$. By the chain rule, $$ I_n = d(T^{-1} \circ T)_p = d(T^{-1})_{T(p)} dT_{p}, $$ from which it follows that $(dT_p)^{-1} = d(T^{-1})_{T(p)}$.

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