Difference Between Vector and Tensor

[abrowaqas]

Can we call a vector as Tensor, discuss What is different between Vector and Tensor? What happens if we permute the subscript of tensors

 

[GoldPheonix]

All vectors are, technically, tensors. All tensors are not vectors.

This is to say, tensors are a more general object that a vector (strictly speaking though, mathematicians construct tensors through vectors).

Tensors are technically defined through two different objects:

1.) Vectors.
2.) One-forms. ("dual" vectors)

Vectors are just objects for which you know what it means to add any two of them (vector addition), and what it means to scale-change it (scalar multiplication). One-forms, likewise, have all the same notions, except that they can operate on vectors and return scalars.

Examples are in order: The most prototypical examples are Euclidean vectors --points of space. Examples of one-forms would be the magnetic potential "vector" (It's actually not a "true" vector) or the gradient operator.The most important property, when you add a few other suitable assumptions, is that one-forms and vectors transform in particular manners under a change of coordinates. These are the properties that physicists are most often concerned with when discussing things like the theory of general relativity.

Tensors, by extension, as mathematical objects are "multilinear" operators; this is to say, they take in sets of vectors (and/or one-forms) and return another tensor (as opposed to linear operators which take in vectors and return vectors). These have varying uses.If you want to understand the general theory of tensors, you should understand abstract algebra (especially linear algebra), and if you want to understand tensor calculus you should understand the theory of differentiable manifolds, also.

 

[HallsofIvy]

The fundamental property (at least for Physics) of tensors is that "If a tensor is identically equal to 0 in one coordinate system it is equal to 0 in any coordinate system."

The reason that is important is that it means that if a tensor equation, A= B where A and B are tensors, is true in one coordinate system (so that A- B= 0), then it is true in any coordinate system (A- B= 0 in any coordinate system so A= B in any coordinate system).

A vector is a special type of tensor- a "tensor of order 1". You can construct tensors of higher order from vectors.

Note that, just as you can look at "vectors" from a purely applications point of view, always thinking in terms of $R^n$ or from the more abstract "Linear Algebra" point of view, so you can work with tensors in terms of their applications or in terms of abstract algebra.

 

[Hurkyl]

All tensors are vectors too. There's an ambiguity in language here.


When we are interested in one specific vector space -- for example, the tangent vector space to a point -- then it is conventional to call its elements "vectors", elements of its dual space "covectors" or maybe "one-forms", elements of its tensor algebra "tensors", and so forth.

(And any vector space 'is' a subspace of its tensor algebra, so in this terminology, vectors are tensors, as the others have said)

 

[abrowaqas]

i got the idea about tensors... so that means tensors are the combination of both vectors and scalars ... tensor become scalar when n= 0 in 3^n otherwise a tensor is a vector .. am i right... but what if we permute the tensor... i read in one of the source it says that if you permute the subcript of tensor you will get another new tensor. but didn't explain the idea.. so if you have idea .. kindly share

 

[Hurkyl]

i got the idea about tensors... so that means tensors are the combination of both vectors and scalars ... tensor become scalar when n= 0 in 3^n otherwise a tensor is a scalar .. am i right... but what if we permute the tensor... i read in one of the source it says that if you permute the subcript of tensor you will get another new tensor. but didn't explain the idea.. so if you have idea .. kindly share

Bilinear forms are tensors. More precisely, the space of bilinear forms is a tensor product of the dual space with itself.

A bilinear form, remember, is a scalar-valued function of two vectors

B(ax+by, z) = aB(x,z) + bB(y,z)
B(x, ay+bz) = aB(x,y) + bB(x,z)​

where a,b are scalars and x,y,z are vectors.

If B is a bilinear form, I can create a new bilinear form C by swapping arguments:

C(x,y) := B(y,x)​

If we represented bilinear forms via coordinates with respect to a basis -- i.e. with index notation -- then the coordinate form of C can be computed by swapping the two subscripts in the coordinate form of B:

[C]_ij = [B]_ji​