Loosely speaking, temperature is a tensor of rank zero, velocity is a tensor of rank one, stress and strain in deformable bodies are tensors of rank two, and the idea can be generalized to higher ranks. I give below a detailed description rather than a definition.

We chose a positive integer *n* which specifies the dimensionality of our problem. We also establish *n* independent real variables, , each of which is chosen from an interval of real numbers, . Our space will be represented by a set of *n*-tuples of real numbers, and we suppose that there is a one-to-one correspondence between each "point" of the space and each element of the just mentioned *n*-tuple set. Typically, the *n*-tuple set will be the *n*-fold Cartesian product of the intervals. The specific correspondence just supposed constitutes a coordinate system covering the space, and the are the coordinate variables. I have followed Sokolnikoff here.

For the purpose of using multiple 'coordinations' of the points in our space with ordered *n*-tuples of real numbers, *n* functions, are used. The functions together with their first partial derivatives need to be continuous, and it must also be the case that the Jacobian determinant is not zero at any point of the space. The set of all such 'sets of *n* functions', constitutes what we may call the set of all admissible coordinate systems. See for example Sokolnikoff page 51, who applies the name admissible to the tranformations between coordinate systems, rather than to the coordinate systems themselves. For simplicity, I follow the old authors and use the same symbol both for the coordinate variables and for the functions used in coordinate transformations. There should be no confusion, since the context of any given expression should make things clear. For example, in the expressions and , *x* is obviously a function in the first expression and a variable in the second. Similarly, *s* is a variable in the first expression and a function in the second. Of course, if we implement the theory with a computational package such as Mathematica, we will be forced to distinguish between the functions and the variables. But Mathematica allows us to put an overbar on symbols, and consequently we could have

and

which distinguishes the different objects without destroying the simplicity.

The Mathematica 4.1 help browser says "In *Mathematica*, a tensor is represented as a set of lists, nested to a certain number of levels. The nesting level is the rank of the tensor." This is essentially what Levi-Civita (page 65) described as a system. The following image illustrates what we could call a rank four system in two dimensions. The subscripts on *a* are the values of four indices, *i*, *j*, *k*, and *l*, each of which varies from one to two.

Each component, that is each , is defined to be zero in the above example, but each component could be any real number. If we associate the above object with a point in a two dimensional space, we might call it a point system. If we associate the above object with every point in a two dimensional space, we might call it a system field. These names have been applied here as a way of looking forward to the same ideas associated with tensors. However, the system idea does not have any association with the admissible coordinate systems which cover the space.

The general description presented in the next equation has been made very large because it contains subscripted superscripts which are attached to variables that occur in the numerator and denominator of differentiations. Consequently clear visibility requires large size. I'll give the definition without motivation. Subsequent links will provide examples, which will motivate the definition. Classical tensor analysis is a subject in which the theory provides its own examples.

where

and where

Let * r* and