The order of a tensor

Tensors can be classified by their order, sometimes also called rank . As a first example, we consider again a vector. As already explained, its components are 

In a scalar, which is coordinate-independent, the component matrix is a single number. Since a scalar thus has dimension zero, it is a tensor of zeroth order. Using an index is thus not necessary. 

If we now move on to a quantity with components written as a (3 x 3) matrix, we get a tensor of second order. In this case, we need two indices to specify a component of the tensor. One example for a second-order tensor is the stress tensor a. 

Tensors of second order can be represented by a matrix in a specified coordinate system. In general, a matrix is simply a rectangular arrangement of numbers. Arbitrary matrices do not necessarily share the important tensor property of invariance If a coordinate transformation is done, a tensor s components may change, but the tensor itself does not. Thus, many quantities that are usually called matrices should better be denoted as second-order tensors. 

A third-order tensor is represented by a coordinate cube with 3x3x3 = 27 components. This scheme can be extended to arbitrarily high orders. A fourth-order tensor, having 3 = 81 components, cannot be imagined geometrically. Nevertheless, it is of great importance in material science (see section 2.4.2). 

---

To summarise, it can be said that the order of a tensor specifies the dimension of the hypercube of edge length three that contains the components in a certain coordinate system. In three-dimensional space, a tensor of order m has S components. 

---