It is also possible for a tensor field to have a "density".

A tensor field is an element of this set.

It is therefore a tensor field of rank three.

Note that the tensor field only needs to be defined on the curve for this definition to make sense.

The same is true in general relativity, of tensor fields describing a physical property.

The solution is a metric tensor field, rather than a wavefunction.

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

This identity hold for tensor fields of all orders.

The result can be generalized to higher rank symmetric tensor fields.

The theory seems to contain an asymmetric tensor field and a source current vector.

Functions, tensor fields and forms can be differentiated with respect to a vector field.

Other physically important tensor fields in relativity include the following:

This notation allows the most efficient expressions of such tensor fields and operations.

Tensor fields, which associate a tensor to every point in space.

On a manifold, a tensor field will typically have multiple indices, of two sorts.

Vector fields are one kind of tensor field.

Problem: Let be a positive definite symmetric tensor field defined on the reference configuration.

The term tensor is in fact sometimes used as a shorthand for tensor field.

Has rich feature set for scalar, vector, and tensor field visualization.

Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.

In modern mathematical terminology such an object is called a tensor field, but they are often simply referred to as tensors themselves.

For a tensor field, , the laplacian is generally written as:

In general, one can define various divergence operations on higher-rank tensor fields, as follows.

Many mathematical structures informally called 'tensors' are actually 'tensor fields'.

Some care is required, because it is common to see a tensor field called simply a tensor.