What is a tensor and can any examples of their use be given?

Asked by: Matthew Allen

Answer

Tensors are most easily understood by discussing the progression of tensor 'ranks'. Generally when one talks about tensors, though, one is referring to tensors of rank two or higher.

A scalar quantity is simply a number -- it has only magnitude. A scalar can be designated a tensor of rank zero.

A vector quantity has magnitude and direction. In two dimensional space, for example, it was x- and y-components, and in three dimensional space, it has 3 components. Vectors can have any number of dimensions. These components are commonly shown in a one dimensional column matrix.

           a
           b
    v  =   c 
           .
           .
           n

A vector can be designated a tensor of rank one.

A tensor of rank two is represented by a matrix:

          aa  ab  ac ... an
    T2 =  ba  bb  bc ... bn 
          ca  cb  cc ... cn
          .   .   .    .
          .   .   .    .
          ma  mb  mc ... mn

A rank-three tensor is represented with a cubic matrix, with components coming out of your computer screen.

(Tensors with rank higher than three are harder to represent; the most common notation is known as Einsteinian Notation, which makes use of indices. Note that a rank-four tensor is represented by a hyper-rectangular matrix. )

Visualizing tensors is very difficult, akin to visualizing hyperdimensional objects. One way to think of tensors is in terms of fields.

A scalar field is created by simply assigning scalar quantities (numbers) to each point in space. Think of temperature -- each point in the room has a different temperature.

A vector field is created by assigning vectors to each point. An electric field is an example -- a test charge placed at a point in space will move at a certain speed and direction as represented by the vector at that point.

A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge.

Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.
Answered by: Aman Ahuja, Physics Student, WPI, Massachussets