Thus a one-form is an order 1 covariant tensor field.

In other words, a two-form is a skew-symmetric covariant tensor field of order 2.

More generally, any covariant tensor field - in particular any differential form - on N may be pulled back to M using φ.

To obtain the covariant tensor F, multiply by the metric tensor and contract:

The transformations in this form can be made more compact by introducing the electromagnetic tensor (defined below), which is a covariant tensor.

A particular important case of the pullback of covariant tensor fields is the pullback of differential forms.

He gave up looking for fully generally covariant tensor equations, and searched for equations that would be invariant under general linear transformations only.

The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:

It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight 1.

A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

(with k factors of TM in the product), where TM is the tangent space to M at p. Equivalently, β is a totally antisymmetric covariant tensor field of rank k.