I am reading through Renteln: Manifolds, Tensors and Forms to hopefully one day tackle relativity (the hyperlink goes directly to the page in question). My question is probably a fairly simple one, the transition function between two overlapping coordinate charts on some region of a smooth differentiable manifold is defined as:
$$\psi_j\circ\psi^{-1}_i:\psi_i(U_i\cap U_j)\rightarrow\psi_j(U_j\cap U_j)$$
I'm pretty sure I understand the implication of this, that the two regions locally resemble $\Bbb R^n$ and that if two coordinate charts overlap they are effectively interchangeable at their intersection and to related each other by the transition function. My question is can this order be reversed, i.e:
$$\psi_{j}^{-1}\circ\psi_i:\psi_j(U_i\cap U_j)\rightarrow\psi_i(U_j\cap U_j)$$