In short, once we have defined the covariant derivative on scalar fields and vector fields, we can work out the action on more general tensor fields

by induction using the identity:

, where

is either an arbitrary vector field or an arbitrary covector field, the choice being such that the convariant derivative of the contracted quantity

is known. Since

is arbirtrary we can then recover

.

Because it will be useful for later questions, I will work this out explicitly.

For each coordinate vector field

, define the coefficients:

(Note that

is tensorial in the

and

indices by construction, but in general it will not be tensorial in the

index).

Now for an arbitrary vector field

we use the Leibniz rule and the rule for the action of a covariant derivative on a scalar field:

We now calculate the covariant derivative of a covector field

. The Leibniz rule gives:

In coordinates this becomes:

(n.b. I changed dummy indices on the

term in the last step)

Since this is true for any vector field

we have:

Using induction, it is then straightforward to show that for an arbitrary tensor field

we get the following:

I'll prove the inductive step in the case of an arbitrary covariant tensor. The general case is similar (except that you need to contract contravariant indices with a covector field instead of a vector field).

Suppose we have proved the identity for covariant tensors with up to

indices. The case for a covariant tensor with

indices is handled by contracting with a vector field

(where we have renamed dummy indices in the final step). This is true for any vector field and so the induction step is proven.