Given a smooth function , if the Jacobian is invertible at 0, then there is a neighborhood containing 0 such that is a diffeomorphism. That is, there is a smooth inverse .
Inverse Function Theorem
See also
Diffeomorphism, Implicit Function Theorem, JacobianThis entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. "Inverse Function Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseFunctionTheorem.html