Differentiation of Functions
There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. The directional derivative therefore looks at curves in the manifold instead of vectors.
Read more about this topic: Differentiable Manifold, Differentiable Functions
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