Physical meaning of tensor fields

[espen180]

In the past I have worked with scalar and vecor fields. I get vector fields. at each point the field has a magnitude and direction. With tensor fields, from what I have seen in diagrams, each point has three vectors assigned to it, and they appear to be orthogonal to each other.

What is the physical interpretation of a tensor field, and what distinguishes them from ordinary vector fields?

 

[Fredrik]

What you said about scalar and vector field sounds more like a partial definition than like a "physical interpretation". I'm not even sure what "physical interpretation" means here.

Tensors are much more difficult to understand than vectors, because to really understand them you need to understand manifolds. One thing you need to know is that there's a vector space associated with each point p in a manifold M. This vector space is called the tangent space at p. It's usually written as T_pM, but I'll just write V here.

If V is a vector space over the real numbers, you can define the (algebraic) dual space V* as the set of all linear functions from V into the real numbers. When V is the tangent space at p, V* is called the cotangent space at p. The members of V are called "tangent vectors" or just "vectors", and the members of V* are called "cotangent vectors" or just "covectors".

A tensor (at the point p) of type (n,m) is a multilinear (linear in all variables) function

$$T:\underbrace{V^*\times\cdots\times V^*}_{\mbox{n factors}}\times\underbrace{V\times\cdots\times V}_{\mbox{m factors}}\rightarrow\mathbb R$$

A tensor field can be defined as a function that assigns a tensor at p to each point p. Vector fields are a special case of this. They are tensor fields of type (1,0). Note that a tensor of type (1,0) is a member of the dual space of V*, which I'll write as V**. To see why this assigns a tangent vector to each point in M, you need to understand that V** is isomorphic to V.

Recall that members of V** take members of V* to real numbers. We want to associate a v** in V** with each v in V, and to do that we must specify how v** acts on an arbitrary u* in V*:

v**(u*)=u*(v)

The map v $\mapsto$ v** is an isomorphism from V into V**. So each tensor field of type (1,0) defines a tangent vector at each point in M, via this isomorphism.

 

[atyy]

With tensor fields, from what I have seen in diagrams, each point has three vectors assigned to it, and they appear to be orthogonal to each other.

Maybe the 3 vectors are just the set of basis vectors?

Go with Fredrik's definition.

 

[dx]

The best way, IMO, to get a feeling for the 'physical meaning' of tensors is simply to carefully consider a few examples of them from physics. Here are a few simple ones: the dot product in Euclidean geometry (which, as you can easily check, is a bilinear function $$V \times V \rightarrow \mathbb{R}$$), the tensor of elasticity, the tensor of conductivity, the tensor of polarization. Here's what I suggest. Take each of these physical examples, and think about what the tensors in each case are supposed to represent physically. See if you can understand why, in each case, the tensor must be a linear function. The main idea of tensors is linearity, so if you can understand this, you understand tensors.

(Old fashioned treatments of tensors usually talk about representations of them in some basis, i.e. see them as a set of numbers. This makes it necessary, to make the definitions precise, to talk about how these representations transform when you change the coordinate system. It is best to stay away from books written in this style, because they are now obsolete.)

 

[espen180]

Thanks for your help and suggestions. I will do some reading, then, if need be, come back with more questions.