Analysis on Manifolds

Abstract

The intuitive picture of a smooth surface becomes analytic with the concept of a manifold. On the small scale a manifold looks like a Euclidean space, so that infinitesimal operations like differentiation may be defined on it.

A function f from an open subset U of ℝⁿ into ℝ^m is differentiable at a point x ε U if it may be approximated there with a linear mapping Df: ℝⁿ→ℝ^m We can make this notion more precise by requiring that for all ε > 0 there exists a neighborhood U of x such that

$$ \left\| {f(x') - f(x) - Df(x)(x' - x)|| < \varepsilon \left\| x \right. - \left. {x'} \right\|} \right.\forall x' \in U $$

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