Editing Covariant derivative (section)

== Notation ==
In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.

Often a notation is used in which the covariant derivative is given with a [[semicolon]], while a normal [[partial derivative]] is indicated by a [[comma]]. In this notation we write the same as:
<math display="block">
\nabla_{e_j} \mathbf{v} \ \stackrel{\mathrm{def}}{=}\ {v^s}_{;j}\mathbf{e}_s \;\;\;\;\;\;
{v^i}_{;j} =
{v^i}_{,j} + v^k {\Gamma^i}_{k j}
</math>
In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives:
<math display="block"> \nabla_{e_k} \left( \nabla_{e_j} \mathbf{v} \right) \ \stackrel{\mathrm{def}}{=}\ {v^s}_{;jk}\mathbf{e}_s </math>

In some older texts (notably Adler, Bazin & Schiffer, ''Introduction to General Relativity''), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe:
<math display="block">\nabla_{e_j} \mathbf{v} \ \stackrel{\mathrm{def}}{=}\ {v^i}_{||j} = {v^i}_{|j} + v^k {\Gamma^i}_{k j}</math>