inverse function theorem
If a smooth map's best linear approximation at a point is reversible — it stretches space without collapsing any direction — then the map itself can be undone, at least nearby. This is the intuition: invertibility of the derivative, a purely linear-algebra condition, is enough to guarantee that the full nonlinear map is locally a clean two-way correspondence.
Precisely: let f: R^n -> R^n be continuously differentiable (C^1) on an open set, and suppose at a point a the Jacobian matrix Df(a) is invertible (equivalently, its determinant is nonzero). Then there are open neighborhoods U of a and V of f(a) such that f maps U bijectively onto V, the inverse f^{-1}: V -> U is also C^1, and D(f^{-1})(f(a)) = (Df(a))^{-1}.
Two honest caveats. The conclusion is strictly local: the global map can still be many-to-one even though every point has an invertible derivative — for instance, a map can wrap a region around and overlap itself. And the hypothesis is sufficient, not necessary: a map can be invertible at a point where its derivative is singular, but then the inverse need not be differentiable.
f(x, y) = (e^x cos y, e^x sin y) has Jacobian determinant e^{2x}, never zero, so f is locally invertible everywhere. Yet it is not globally injective: f(x, y) = f(x, y + 2pi). Invertible derivative everywhere, but the global map is many-to-one.
Local invertibility everywhere need not give global invertibility.