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

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

The metric is a symmetric tensor and is an important mathematical tool.

A symmetric tensor always has a complete set of orthogonal eigenvectors and real eigenvalues.

For example, a 2x2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal.

For a covariant symmetric tensor field we have:

It is a symmetric tensor .

On a simply connected domain in Euclidean space implies that for some rank k-1 symmetric tensor field .

Another way, connected with invariant theory is via the polarization identity for a symmetric tensor .

The permanent arises naturally in the study of the symmetric tensor power of Hilbert spaces.

Unless the fluid is made up of spinning degrees of freedom like vortices, is a symmetric tensor.

Around the 1930s, Voigt notation would be developed for multilinear algebra as a way to represent a symmetric tensor by reducing its order.

In special and general relativity, T is a symmetric tensor, but in other contexts (e.g. quantum field theory), it may not be.

In analogy with the theory of symmetric matrices, a (real) symmetric tensor of order 2 can be "diagonalized".

The sources of any gravitational field (matter and energy) is represented in relativity by a type (0, 2) symmetric tensor called the energy-momentum tensor.

Since g is symmetric as a bilinear mapping, it follows that g is a symmetric tensor.

In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:

Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress, strain, and anisotropic conductivity.

One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation .

For functions on a higher-dimensional Euclidean space R, there are more measures of distortion because there are more than two principal axes of a symmetric tensor.

Saint-Venant's compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincaré's lemma for skew-symmetric tensor fields (differential forms).

A newer version of MSTG, in which the skew symmetric tensor field was replaced by a vector field, is scalar-tensor-vector gravity (STVG).

The first term on the right is the constant tensor, also known as the pressure, and the second term is the traceless symmetric tensor, also known as the shear tensor.

This led Moffat to propose Metric Skew Tensor Gravity (MSTG), in which a skew symmetric tensor field postulated as part of the gravitational action.

It is shown that the Minkowski tensor in connection with Eckart's non-symmetric tensor—if one requires that the resulting tensor be symmetric—is equal to the sum of Eckart's symmetric tensor with the Abraham tensor.