Covariant Derivative - Notation

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:

\nabla_{e_j} {\mathbf v} \ \stackrel{\mathrm{def}}{=}\ v^s {}_{;j}e_s \;\;\;\;\;\; v^i {}_{;j} = v^i {}_{,j} + v^k\Gamma^i {}_{k j}

Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates, but also depends on the vector v itself through .

In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe:

\nabla_j {\mathbf v} \ \stackrel{\mathrm{def}}{=}\ v^i {}_{||j} \;\;\;\;\;\;

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