The covariant derivative

A covariant derivative is a tensor which reduces to a partial derivative of a vector field in Cartesian coordinates. For a scalar, the covariant derivative is the same as the partial derivative, and is denoted by

 

The covariant derivative of a contravariant tensor of rank one is given by

 

where is the so-called Christoffel symbol of the second kind given by

 

It can be shown that

 

The covariant derivative of a covariant tensor of rank one is given by the expression:

 

It can be shown that

 

The covariant derivative of a contravariant tensor of rank two is defined as follows:

 

It can be shown that

 

and

 

The covariant derivative of a scalar density (i.e. a relative scalar of weight 1) is defined as

 

It can be shown that

 

Hence

 

Another important identity is

 

and is called the geometric identity. This can be derived as follows. Suppose is constant. Then with the aid of (2.20) we have

Since this holds for all constant , (2.27) follows.