If U is an open set in Rn and we have a function f : U → Rm, we say that f is smooth (or C∞) if all higher partial derivatives of f exist and are continuous. The composition of smooth functions is smooth. In the case m = n, if f : U → Rn is one-to-one onto f(U), with f(U) open in Rn, and both f and f−1 are smooth then f is a diffeomorphism (from U to f(U)).
KeywordsTangent Vector Limit Point Open Neighborhood Differentiable Manifold Matrix Group
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