Directional Derivative - in Differential Geometry

In Differential Geometry

Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as (see covariant derivative), (see Lie derivative), or (see Tangent space#Definition via derivations), can be defined as follows. Let γ : → M be a differentiable curve with γ(0) = p and γ′(0) = v. Then the directional derivative is defined by

This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ′(0) = v.