Covariant Derivative - Coordinate Description

Coordinate Description

This section uses the Einstein summation convention.

Given coordinate functions

,

any tangent vector can be described by its components in the basis

.

The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination . To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field along .

the coefficients are called Christoffel symbols. Then using the rules in the definition, we find that for general vector fields and we get

so

The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field u. In particular

In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.

The covariant derivative of a type (r,s) tensor field along is given by the expression:

Or, in words: take the partial derivative of the tensor and add: a for every upper index, and a for every lower index .

If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then you also add a term

If it is a tensor density of weight W, then multiply that term by W. For example, is a scalar density (of weight +1), so we get:

where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.