What is a tensor?

Answer 1

The term ‘tensor’ has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function. In physics and engineering, the same term usually means what a mathematician would call a tensor field: an association of a different (mathematical) tensor with each point of a geometric space, varying continuously with position.
The rank of a particular tensor is the number of array indices required to describe such a quantity. For example, mass, temperature, and other scalar quantities are tensors of rank 0; but force, momentum and other vector-like quantities are tensors of rank 1. The novel aspects of tensor theory are seen from rank 2 onwards. A linear transformation such as an anisotropic relationship (relativistic mass) between force and acceleration vectors is a tensor of rank 2.
Tensors are important in physics and engineering. In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain; in this technique tensors are in effect made visible. Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear material by a fourth rank elasticity tensor.

Specifically, a 2nd rank tensor quantifying stress in a 3-dimensional/solid object has components which can be conveniently represented as 3x3 array. The three Cartesian faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number (being in three-space). Thus, 3x3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment (which may now be treated as a point). Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, the need for a 2nd order tensor is produced.

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond specifying that it requires a number of indexed components. In particular, tensors behave in specific ways under coordinate transformations. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra.

There are two ways of approaching the definition of tensors:

The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.
The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors.
Physicists and engineers are among the first to recognise that vectors and tensors have a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated. Similarly, mathematicians find there are some tensor relations which are more conveniently derived in a co-ordinate notation.

Answer 2

u must be knowing that there are 2 kinds of values : scalar and vector.

tensor is considered to be a third type .. used for multiplying with vectors

tensors are denoted by matrices ... an example is the following 2-dimension rotation tensor :

[ sin theta -cos theta ]
[ cos theta sin theta ]

this matrix when multiplied with a 2d vector, rotates the vector by an angle theta ... for example, lets rotate the vector (1,0) by 90 degrees.

the tensor wud be
[0 -1]
[1 0]
multiply this with the vector in column matrix form:
[1]
[0]
so, u get the answer as
[0]
[1]
which is (0,1) which is obviously rotated by 90 degrees.

A 3-dimensional tensor will be expressed as a 3x3 matrix and so on.

Answer 3

Can't say that I actually have any idea, but I do know how to use a search engine. I did find this at Wikipedia:

"General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann,"

If Einstein had great difficulty, I wish you luck.

You can read more here: http://en.wikipedia.org/wiki/Tensor

Answer 4

guy I hate it while human beings do this. enable me arise with Scalar-Vector-Tensor gravity. that would desire to sparkling the full difficulty up. I have no adventure interior the sphere, yet on condition that i'm an atheist it is going to not take me greater advantageous than approximately 20 minutes to derive each thing from first principals and are available up with a sparkling concept. /sarcasm That seems the point the theists anticipate from us.

Answer 5

Very briefly: a tensor is dimensional or dimension converting matrix for another matrix.

Answer 6

one more than a nine, sir.