What are tensors?

Question

I am having problem to admit why in a laminar flow of a fluid we assume a small area and take a direction normal to it.Similarly in stress tensor analysis introductory diagram area of each face of the cube is treated as a plane with direction.Please explain me why area turns from a directionless quantity to a quantity with direction and also in that context explain me what is a tensor in a simple language.

Answer 1

Tensors are a broad class which includes the scalars and vectors. The transformation of a quantity under coordinate transformation defines whether the quantity is a scalar etc. For instance we know that the magnitude of a vector is a scalar since it does not change under a transformation but the componenet of a vector does. As regards area the direction is usually associated with a direction perpendicular to the infinitisimal area. It is useful, for instance in obtaining various quantities like electric flux, flow of current etc.

Answer 2

Since I cannot delete an accepted answer, I will edit, because I think the comments are correct, that I was misrepresenting what actually a tensor is.

We live in a three dimensional space and it is easy to understand that three numbers in an (x,y,z) coordinate system on each of the axis define a point in space and the line joining with the origin is a vector, i.e. it has a direction. Vectors are useful because they model forces. There are zero dimensional vectors, i.e. points, there are two dimensional vectors defined in a plane also.

Continue with this link for a correct formulation.

Notice that the effect of multiplying the unit vector by the scalar is to change the magnitude from unity to something else, but to leave the direction unchanged. Suppose we wished to alter both the magnitude and the direction of a given vector. Multiplication by a scalar is no longer sufficient. Forming the cross product with another vector is also not sufficient, unless we wish to limit the change in direction to right angles. We must find and use another kind of mathematical ‘entity.’

For detailed exposition also go to the duplicate indicated above.

To model physical systems one has to consider the independent variables defining the system, whether a Cartesian coordinate system of n dimensions can be defined with the proper coordinate transformations. Tensors are used in modeling physics problems wherever there are variations in quantities that depend on two vectors and one can fit the mathematical demands of tensor algebra.