stylistic:tone, structural:scholar_edition, information:classification
Covariant derivative

The covariant derivative is only defined if a connection is defined. For any tensor field, a semicolon (;) placed before an appended lower (covariant) index indicates covariant differentiation. Less common alternatives to the semicolon include a forward slash (/) or in three-dimensional curved space a single vertical bar (|).

The covariant derivative of a scalar function, a contravariant vector and a covariant vector are:

where Γ are the connection coefficients.

For an arbitrary tensor:

An alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol ∇. For the case of a vector field A:

The covariant formulation of the directional derivative of any tensor field along a vector v may be expressed as its contraction with the covariant derivative, e.g.:

The components of this derivative of a tensor field transform covariantly, and hence form another tensor field, despite subexpressions (the partial derivative and the connection coefficients) separately not transforming covariantly.

This derivative is characterized by the product rule: