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
Explore with Wolfram|Alpha
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