The following is from J.S. Milne's notes on modular forms. He gives a definition of compatible coordinate neighborhoods.
What is the significance of requiring that the derivative be nowhere vanishing?
2026-03-26 06:21:27.1774506087
Significance of requiring a nowhere vanishing derivative
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No significance.
If you assume only that they are holomorphic, interchanging the role of $i, j$ gives
$$z_j \circ z_i^{-1} :z_i (U_i\cap U_j) \to z_j(U_i\cap U_j)$$
is holomorphic. But this is the inverse of $z_i \circ z_j^{-1}$. Thus both of them has non-vanishing derivatives, which can be checked using chain rule.