Question

can any one describe what is Richie tensor and in other words what is tensor

Answer 1

There are a few different ways to understand tensors, and my opinion is that tensors don’t become truly intuitive until you see how they all fit together.

I will explain here the “geometric” point of view, which I think is easiest to absorb if you don’t know what a tensor is at all. The one-sentence version is that a tensor of order  results from associating a coordinate with every ordered -tuple of coordinate axes. (Therefore, a vector is a tensor of order 1: it has one coordinate per coordinate axis.)

As I assume you know, in three dimensions, a vector has three components, and the vector itself can be thought of as a combination of the basis vectors along the coordinate axes, weighted by the components: that is, when we write , this is a shorthand for . Each coordinate is associated with a corresponding unit basis vector parallel to a coordinate axis.

To generalize to higher-order tensors, we can start by pairing up the unit basis vectors and associating components with ordered pairs. There are nine such pairs: xx, xy, xz, yx, yy, yz, zx, zy, zz, so a tensor of order 2 would have nine components. These components would represent coefficients for a linear combination of nine unit basis tensors. That is, a second order tensor would have the form

You might wonder what I mean by  and similar constructions. Don’t try to visualize it yet. Just think of it as representing the ordered pairing of the unit basis vector  with itself. Likewise, represents the ordered pairing of  with .

Based on this concept, you can construct tensors of any order. A tensor of order 3 would have one component for every ordered triple of unit basis vectors, so it would have 27 components in all, each associated with one particular ordered triple, and the tensor itself would represent the linear combination of the 27 unit basis tensors, weighted by the corresponding components. A tensor of order 4 would have 81 components. And so on.

An example that motivates this concept of tensors is the Cauchy stress tensor, which describes stresses within a material. If you have a single particle, you only care about the net force on it, which is a vector, but if you have a continuous body such as a steel beam, then a point within the beam can be under considerable stress even though there is no net force on it. The way to describe these stresses is by assigning a coordinate to each pair of directions! Imagine a tiny cube inside the material, with its faces aligned with the coordinate planes. Now consider the face with normal vector . That face experiences some force due to the material on the other side pushing on it. That force is a vector, so it has three components. The faces with normal vectors of and  likewise have forces on them. All in all, you need nine component