Differentiable manifold
Directional differentiation

Given a real valued function f on an n dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows. Suppose that γ(t) is a curve in M with γ (0) = p, which is differentiable in the sense that its composition with any chart is a differentiable curve in R. Then the directional derivative of f at p along γ is

d d t f (γ (t)) | t = 0. {\displaystyle \left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.}

If γ and γ are two curves such that γ (0) = γ (0) = p, and in any coordinate chart ϕ {\displaystyle \phi },

d d t ϕ ∘ γ 1 (t) | t = 0 = d d t ϕ ∘ γ 2 (t) | t = 0 {\displaystyle \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}}

then, by the chain rule, f has the same directional derivative at p along γ as along γ. This means that the directional derivative depends only on the tangent vector of the curve at p. Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.
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