Editing Directional derivative (section)

== In differential geometry ==
{{see also|Tangent space#Tangent vectors as directional derivatives}}

Let {{math|''M''}} be a [[differentiable manifold]] and {{math|'''p'''}} a point of {{math|''M''}}.  Suppose that {{math|''f''}} is a function defined in a neighborhood of {{math|'''p'''}}, and [[total derivative|differentiable]] at {{math|'''p'''}}.  If {{math|'''v'''}} is a [[tangent vector]] to {{math|''M''}} at {{math|'''p'''}}, then the '''directional derivative''' of {{math|''f''}} along {{math|'''v'''}}, denoted variously as {{math|''df''('''v''')}} (see [[Exterior derivative]]), <math>\nabla_{\mathbf{v}} f(\mathbf{p})</math> (see [[Covariant derivative]]), <math>L_{\mathbf{v}} f(\mathbf{p})</math> (see [[Lie derivative]]), or <math>{\mathbf{v}}_{\mathbf{p}}(f)</math> (see {{section link|Tangent space|Definition via derivations}}), can be defined as follows.  Let {{math|''γ'' : [−1, 1] → ''M''}} be a differentiable curve with {{math|1=''γ''(0) = '''p'''}} and {{math|1=''γ''′(0) = '''v'''}}.  Then the directional derivative is defined by
<math display="block">\nabla_{\mathbf{v}} f(\mathbf{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}.</math>
This definition can be proven independent of the choice of {{math|''γ''}}, provided {{math|''γ''}} is selected in the prescribed manner so that {{math|1=''γ''(0) = '''p'''}} and {{math|1=''γ''′(0) = '''v'''}}.

===The Lie derivative===
The [[Lie derivative]] of a vector field <math> W^\mu(x)</math> along a vector field <math> V^\mu(x)</math> is given by the difference of two directional derivatives (with vanishing torsion):
<math display="block">\mathcal{L}_V W^\mu=(V\cdot\nabla) W^\mu-(W\cdot\nabla) V^\mu.</math>
In particular, for a scalar field <math> \phi(x)</math>, the Lie derivative reduces to the standard directional derivative:
<math display="block">\mathcal{L}_V \phi=(V\cdot\nabla) \phi.</math>

===The Riemann tensor===
Directional derivatives are often used in introductory derivations of the [[Riemann curvature tensor]]. Consider a curved rectangle with an infinitesimal vector <math>\delta</math> along one edge and <math>\delta'</math> along the other. We translate a covector <math>S</math> along <math>\delta</math> then <math>\delta'</math> and then subtract the translation along <math>\delta'</math> and then <math>\delta</math>. Instead of building the directional derivative using partial derivatives, we use the [[covariant derivative]]. The translation operator for <math>\delta</math> is thus
<math display="block">1+\sum_\nu \delta^\nu D_\nu=1+\delta\cdot D,</math>
and for <math>\delta'</math>,
<math display="block">1+\sum_\mu \delta'^\mu D_\mu=1+\delta'\cdot D.</math>
The difference between the two paths is then
<math display="block">(1+\delta'\cdot D)(1+\delta\cdot D)S^\rho-(1+\delta\cdot D)(1+\delta'\cdot D)S^\rho=\sum_{\mu,\nu}\delta'^\mu \delta^\nu[D_\mu,D_\nu]S_\rho.</math>
It can be argued<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press|location=Princeton|isbn=9780691145587|page=341}}</ref> that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
<math display="block">[D_\mu,D_\nu]S_\rho=\pm \sum_\sigma R^\sigma{}_{\rho\mu\nu}S_\sigma,</math>
where <math>R</math> is the Riemann curvature tensor and the sign depends on the [[sign convention]] of the author.