Editing Directional derivative (section)

===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.