A tensor is an array of numbers or functions that can be used to describe properties of a body, e.g. the scalar x for temperature, the vector for velocity and the matrix for stress.
Tensors are categorised by their ranks (also known as order) where:

 A zerothorder tensor is an array representing a quantity that only has magnitude and no direction, i.e. a scalar.
 A firstorder tensor is an array representing a quantity that has magnitude and a direction, i.e. a vector.
 A secondorder tensor is an array representing a quantity that has magnitude and two directions.
The above implies that the number of elements of a tensor is 3^{n}, where n is the order of the tensor.
Next, we shall investigate how the elements of a tensor, particularly a 2^{nd}order tensor, transform between two sets of coordinates. From an earlier article, u_{j}, which is a set of three quantities, is known as a first order tensor (i.e. a vector). Consider the product of two first order tensors u_{j }and v_{l }, whose elements with respect to another set of axes are given by:
As each term on the RHS of eq12a is a scalar, we can rewrite eq12a as
Since i = 1,2,3 and k = 1,2,3, there are 9 possible products of . In other words, is a set of 9 elements, each being a sum of nine quantities , e.g.
We can therefore represent the elements of by a 3×3 matrix . Similarly, in eq12b is a multiplication of two first order tensors and can be represented by a 3×3 matrix . This results in:
Eq13 indicates that each element of in one reference frame is a result of the transformation of in a different reference frame by . In general, a set of nine quantities (as is the case of a secondorder tensor) with reference to an orthogonal set of axes is transformed to another set of nine quantities with reference to another orthogonal set of axes by the transformation matrix .
Using the same logic, a third order tensor transforms as follows:
and a fourth order tensor:
In general, a tensor of rank n is a quantity that transforms from one set of orthogonal axes to another set of orthogonal axes according to: