Differentiable Manifold - Differentiable Functions

Differentiable Functions

A real valued function f on an n-dimensional differentiable manifold M is called differentiable at a point p ∈ M if it is differentiable in any coordinate chart defined around p. In more precise terms, if (U, φ) is a chart where U is an open set in M containing p and φ: U → Rn is the map defining the chart, then f is differentiable if and only if

is differentiable at φ(p). The definition of differentiability depends on the choice of chart at p; in general there will be many available charts. However, it follows from the chain rule applied to the transition functions between one chart and another that if f is differentiable in any particular chart at p, then it is differentiable in all charts at p. Analogous considerations apply to defining Ck functions, smooth functions, and analytic functions.