I don't understand the difference between tensor and tensor field.

I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions:

If $A:(V^*)^r \times V^s\to K$ transformation is $K$-multilinear then $A$ is a tensor on $V$.

$M$ is a manifold, $\mathfrak{X}(M)$ is the vector fields' set that is an $F(M)$-module.

If $A$ is a tensor on $\mathfrak{X}(M)$ then we say $A$ is a tensor field on $M$.

I did not understand the last sentence. What is the difference between a tensor and tensor field?

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    $\begingroup$ Usually, a tensor field of a manifold $M$ is an assignment of a tensor to each point of $M$. Just like a vector field of $M$ gives you a vector at a particular point of $M$. $\endgroup$ – Lemon Jan 4 '13 at 14:23
  • $\begingroup$ I got it.I have another question.We said A is a tensor on V but how did we say A is a tensor on M.Shouldn't it be a tensor on X(M)? $\endgroup$ – Serkan Yaray Jan 4 '13 at 14:29
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    $\begingroup$ Where is it said that $A$ is a tensor on $M$? O'Neill's book says precisely that $A$ is a tensor on $\mathfrak{X}(M)$ and equivalently $A$ is a tensor field on $M$. $\endgroup$ – Willie Wong Jan 4 '13 at 14:31
  • $\begingroup$ Oh pardon you are right.I wrote it wrong. $\endgroup$ – Serkan Yaray Jan 4 '13 at 14:37

The difference is all in your head. Literally.

The difference in calling the same object $A$ a "tensor over $\mathfrak{X}(M)$" as opposed to "a tensor field over $M$" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there.

Calling something a tensor field instead of a tensor forces you to remember that $\mathfrak{X}(M)$ is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.

  • $\begingroup$ Oh it finally has been much helpful.Thanks for your help. $\endgroup$ – Serkan Yaray Jan 4 '13 at 15:27

Let's start with some simple explanations.

A tensor is "a mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space."

With each point in space , one can associate a set of scalars called a scalar field , and a bunch of vectors called a vector field , and one can also associate a set of tensors which would be called a tensor field .

A tensor field has to do with the notion of a tensor varying from point to point . A scalar is a tensor of order or rank zero , and a scalar field is a tensor field of order zero . A vector is a tensor of order or rank one , and a vector field is a tensor field of order one .

Some additional mathematical details.

$\mathbb {R}^n$ is a vector space representing the n-tuples of reals under component-wise addition and scalar multiplication .

A manifold is the natural extension of a surface to higher dimensions , and to spaces more general than $\mathbb {R}^n$ .

A tensor field on a given manifold M assigns to each point of M a tensor which is defined on the tangent space at the given point.

More precisely, a tensor field of type $\left(\begin{array}{c}r \\s \\\end{array}\right)$ on a manifold M is a mapping $T$ taking r differential fields and s vector fields on M to real-valued functions $f$ of class $C^k$ (having continuous partial derivatives of a certain order $k$ at each point ) on $\mathbb {R}^m$.

See also my related answer:



Once you know what a tensor is we can define a tensor field by varying it over a space in the same way as we think of a vector field.

There are two ways in which vectors are generally introduced in mathematics,

  • geometrically as orientated lines

  • and algebraically following the axioms of a vector space.

  • The physics of continuous materials and GR adds a third, its defined transformationally through its components. This relies on the local symmetry group of space.

Likewise, we can do the same for tensors - that is they have a geometric, algebraic and transformational definition. Here I'll concentrate on the geometric.

Vectors are usually introduced as a directed length.

A 2-tensor, geometrically speaking, is basically a directed area. They are defined by giving two vectors and these form the edges of the parallelogram, which we think of as exactly the 2-tensor. When generalised to higher spaces, Lang calls these blocks. So this parallelogram is a 2-block. We can see geometrically, that the 2-block depends linearly on the two edges. Thus we have 2-linearity.

Whereas a vector can scale in only one direction, a 2-tensor can scale in two independent directions; however, a 2-tensor can also internally scale. That is we can shrink one edge by a factor s, and expand the other edge by the same factor and this will give the same tensor. This phenomena is not seen in 1-tensors, aka vectors, since there is only one direction.

Now, although vectors are introduced as directed lengths, the length of a vector is actually additional information - this is the norm. Thus we ought to think of a vector as a directed scaling; and a 2-tensor as a 2-directed 2-scaling. Likewise for higher tensors.

In fact, there are two notions of length available for vectors. One is the unorientated notion of length, the norm, and this is the usual notion of length; and the second are the orientated notion of lengths. These are elements of the dual space and are usually known as covectors. Physically, they represent the potential energy of a particle moving along the vector in a uniform gravitational field.

For tensors, it isn't the unorientated length that generalises, that is the norm; but the orientated version. These are what are known as k-covectors (though really they ought to be called k-cotensors here since k-covectors are usually alternating). This geometrically speaking, gives the actual orientated area of an area element - it is a number, whilst the area element, like a vector is geometrical.

Categorically, a tensor is defined by a certain diagram and this simply says that any area function on the defining edges is the same as an area function on the tensor block.

A tensor field is then a field of tensors, that is a smoothly varying distribution of tensors over a manifold.

In terms of formalism, let M be a manifold

Then XM, the sheaf of vector fields, is the sheaf of sections of the tangent bundle TM over M, the base manifold. And it is a module over CM, the space of smooth functions on M. In fact, a sheaf of modules.

If we write $X^p_q$.M, for the sheaf of tensor fields of type$ (p,q)$, that is p-contravariant, and q-covariant, then this is the sheaf of sections of the $(p,q)$-tensor bundle $(x)^p_q.M := [(x)^p.TM] (x) [(x)^q.T*M].$ These are also module over CM.

It's probably useful to realise that the calculus of differential forms is basically the same as antisymmetric covariant tensor fields.


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