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.
Read more about this topic: Differentiable Manifold
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