What is a Tensor?

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.

Coordinated Space

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.

Admissible Coordinates

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.

Systems

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.

Oriented Relative Tensors of Weight W

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 s be positive integers that are greater than or equal to one. In the sense of the previous section, the multi-sub and superscripted A at the end of the long equation is a system of rank r+s. In keeping with the nature of such a system, let it be the case that each of the sub and superscripts on A is an index with a range from 1 to n, where n is a positive integer greater than or equal to two. Suppose that we are considering also a space of n dimensions which is covered by a coordinate system X - so that each point in the space is represented by an n-tuple of real numbers . Let each component of the system A be a function of position in the space - as indicated by the x in parenthesis at the end of the equation. The x here is an abreviation for , which specifies the coordinates of a point P in the space in the coordinate system X. Similarly y is an abbreviation for , which gives the coordinates of P in any other admissible coordinate system Y. J is the Jacobian determinant. That is, we take the determinant of the matrix in which the element in the i'th row and j'th column is found by differentiating (expressed as a function of the y's) with respect to . W is any integer (positive or negative or zero). The indices i and j (subscripted from 1 to r and s respectively) each also range from 1 to n and are free, which means they can take on any value in their range. The indices α and β (also subscripted from 1 to r and s respectively) each also range from 1 to n and are summed. Thus the expression to the right of the equals sign involves r+s sums. The set of all such systems generated by this equation as we consider all of the admissible coordinate systems is an oriented relative tensor of weight W and of rank r+s. Furthermore, the tensor is said to be covariant of rank r and contravariant of rank s. The A's are the components of the tensor in the X coordinate system, while the B's are the components of the tensor in the Y coordinate system. The equation is the equation of transformation for the tensor components between admissible coordinate systems. If sign(J) does not appear in the transformation equation, then the tensor is not oriented. If W=0, then the tensor is said to be absolute rather than relative. If only contravariant or covariant indices are present, then the the tensor is said to be contravariant or covariant respectively. If both contravariant and covariant indices are present, then the tensor is said to be mixed.