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) ).
KeywordsVector Field Tangent Vector Limit Point Open Neighborhood Differentiable Manifold
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