Editing Covariant derivative (section)

===Functions===
Given a point <math>p \in M</math> of the manifold {{mvar|M}}, a real function <math>f : M \to \R</math> on the manifold and a tangent vector <math>\mathbf{v} \in T_pM</math>, the covariant derivative of {{mvar|f}} at {{mvar|p}} along {{math|'''v'''}} is the scalar at {{mvar|p}}, denoted <math>\left(\nabla_\mathbf{v} f\right)_p</math>, that represents the [[Principal part#Calculus|principal part]] of the change in the value of {{mvar|f}} when the argument of {{mvar|f}} is changed by the infinitesimal displacement vector {{math|'''v'''}}. (This is the [[differential of a function|differential]] of {{mvar|f}} evaluated against the vector {{math|'''v'''}}.) Formally, there is a differentiable curve <math>\phi:[-1, 1]\to M</math> such that <math>\phi(0) = p</math> and <math>\phi'(0) = \mathbf{v}</math>, and the covariant derivative of {{mvar|f}} at {{mvar|p}} is defined by
<math display="block">\left(\nabla_\mathbf{v} f\right)_p = \left(f \circ \phi\right)^\prime \left(0\right) = \lim_{t \to 0} \frac{ f(\phi\left(t\right)) - f(p) }{t}.</math>

When <math>\mathbf{v} : M \to T_pM</math> is a vector field on {{mvar|M}}, the covariant derivative <math>\nabla_\mathbf{v}f : M \to \R </math> is the function that associates with each point {{mvar|p}} in the common domain of {{mvar|f}} and {{math|'''v'''}} the scalar <math>\left(\nabla_\mathbf{v}f\right)_p</math>.

For a scalar function {{mvar|f}} and vector field {{math|'''v'''}}, the covariant derivative <math>\nabla_\mathbf{v} f</math> coincides with the [[Lie derivative]] <math>L_v(f)</math>, and with the [[exterior derivative]] <math>df(v)</math>.