Editing Covariant derivative (section)

===Vector fields===
Given a point {{mvar|p}} of the manifold {{mvar|M}}, a vector field <math>\mathbf{u} : M \to T_p M</math> defined in a neighborhood of {{mvar|p}} and a tangent vector <math>\mathbf{v} \in T_pM</math>, the covariant derivative of {{math|'''u'''}} at {{mvar|p}} along {{math|'''v'''}} is the tangent vector at {{mvar|p}}, denoted <math>(\nabla_\mathbf{v} \mathbf{u})_p</math>, such that the following properties hold (for any tangent vectors {{math|'''v'''}}, {{math|'''x'''}} and {{math|'''y'''}} at {{mvar|p}}, vector fields {{math|'''u'''}} and {{math|'''w'''}} defined in a neighborhood of {{mvar|p}}, scalar values {{mvar|g}} and {{mvar|h}} at {{mvar|p}}, and scalar function {{mvar|f}} defined in a neighborhood of {{mvar|p}}):
# <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> is linear in <math>\mathbf{v}</math> so <math display="block">\left(\nabla_{g\mathbf{x} + h\mathbf{y}} \mathbf{u}\right)_p =  g(p)  \left(\nabla_\mathbf{x} \mathbf{u}\right)_p + h(p) \left(\nabla_\mathbf{y} \mathbf{u}\right)_p</math>
# <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> is additive in <math>\mathbf{u}</math> so: <math display="block">\left(\nabla_\mathbf{v}\left[\mathbf{u} + \mathbf{w}\right]\right)_p = \left(\nabla_\mathbf{v} \mathbf{u}\right)_p + \left(\nabla_\mathbf{v} \mathbf{w}\right)_p</math>
# <math>(\nabla_\mathbf{v} \mathbf{u})_p</math> obeys the [[product rule]]; i.e., where <math>\nabla_\mathbf{v}f</math> is defined above, <math display="block">\left(\nabla_\mathbf{v} \left[f\mathbf{u}\right]\right)_p = f(p)\left(\nabla_\mathbf{v} \mathbf{u})_p + (\nabla_\mathbf{v}f\right)_p\mathbf{u}_p.</math>

Note that <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> depends not only on the value of {{math|'''u'''}} at {{mvar|p}} but also on values of {{math|'''u'''}} in a neighborhood of {{mvar|p}}, because the last property, the product rule, involves the directional derivative of {{mvar|f}} (by the vector {{math|'''v'''}}).

If {{math|'''u'''}} and {{math|'''v'''}} are both vector fields defined over a common domain, then <math>\nabla_\mathbf{v}\mathbf u</math> denotes the vector field whose value at each point {{mvar|p}} of the domain is the tangent vector <math>\left(\nabla_\mathbf{v}\mathbf u\right)_p</math>.