Who Was Pavarotti's Understudy? What Are Second Order Tensors?

[aditya ver.2.0]

What are tensors?

 

[kishlaysingh04]

Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor.

 

[Chestermiller]

If you are interested exclusively in second order tenors, then they have little physical meaning on their own, but their real value is in their ability to linearly map vectors into new vectors. For example, the stress tensor maps a unit normal to a surface into the stress vector on that surface.

Chet

 

[WWGD]

mathematically, they are just multi-linear maps, and they form what one calls an algebra (somewhat like a ring in abstract algebra) under tensor product in general and wedge product for the alternating ones, which are the so-cold differential forms. All these tensors help lay out the infrastructure for doing Calculus on manifolds . The Riemannian metric itself is a 2-tensor--maybe the best-known of all tensors--which is equivalent to saying it is a bilinear map (non-degenerate ).

 

[HallsofIvy]

In differential geometry, tensors are objects (like scalars and vectors which are, as Kishlaysingh said, tensors of order 0 and 1, respectively) that transform homogeneously as we change coordinate systems. That means that the coordinates of a tensor in the new coordinates system are sums of products of some number depending on the relationship between the coordinate systems and the components of the tensor in the original coordinate system.

Specifically, that means that if a tensor has all "0" components in one coordinate system, it has all "0" coordinates in any other (the "0" tensor is the same in all coordinate systems). And that means, very importantly, the if an equation involving tensors, say A= B, where A and B are tensors, is true in one coordinate system, it is true in any coordinate system. That follows from the fact that if A= B is true in one coordinate system them A- B= 0 in that coordinate system and so in any coordinate system.

 

[Fredrik]

Let V be a finite-dimensional vector space over $\mathbb R$. The dual space of V is the vector space V* defined as follows. First we define the set V* as the set of linear functions from V into $\mathbb R$. Then we define addition and scalar multiplication on V* in the following way: For all f,g in V*, and all $a\in\mathbb R$, we define af and f+g as the elements of V* such that
\begin{align}
&(af)(v)=a(f(v))\\
&(f+g)(v)=f(v)+g(v)
\end{align} for all v in V. A tensor of type (n,m) over V is multilinear map
$$T:\underbrace{V^*\times\cdots\times V^*}_{n\text{ factors}}\times \underbrace{V\times\cdots\times V}_{m\text{ factors}}\to\mathbb R.$$ "Multilinear" means "linear in each variable. For example, if T is a tensor of type (1,2), then we can write a typical element of its range as $T(\omega,u,v)$. In this case, "multilinear" means that all of the maps
\begin{align}
&\omega\mapsto T(\omega,u,v)\\
&u\mapsto T(\omega,u,v)\\
&v\mapsto T(\omega,u,v)
\end{align} are linear. The best place I know to read about tensors as defined above is "A first course in general relativity" by Schutz. Schutz defines components of tensors, and explains how they "transform" under a change of ordered basis for V.

Many people who say "tensor" actually mean "tensor field". Tensor fields are harder to define properly. I will skip most of the technical details. An n-dimensional smooth manifold is (roughly) a set M together with a bunch of functions $x:U\mapsto\mathbb R^n$ called coordinate systems or charts. Each U is a subset of M, and their union is equal to M. There's an n-dimensional vector space $T_pM$ associated with each point p in M. This space is called the tangent space at p. If p is a point in M, then every coordinate system $x:U\to\mathbb R^n$ such that U contains p, defines an ordered basis for $T_pM$. So a change of coordinate system induces a change of ordered basis. A tensor field of type (n,m) is a function T, defined on a subset of M, such that for each p in that subset, T(p) is a tensor of type (n,m) over $T_pM$. If you want to know this stuff in detail, you will have to study a book on differential geometry, such as "Introduction to smooth manifolds", by John M. Lee.

 

[Blazejr]

Tensor fields can also be defined on affine spaces. That's probably a bit easier to grasp at the begining, but once you do it generalizing to manifolds is trivial when you learn about them.

 

[Incnis Mrsi]

If you are interested exclusively in second order tenors, then they have little physical meaning on their own…

Maxwell’s field possibly conveys more “physical meaning” than all your stress and strain tensors combined. 2-forms are important in gauge theories in general. Also, metric of the spacetime is a (0,2) or (2,0) tensor, this time a symmetric one.

 

[Chestermiller]

Maxwell’s field possibly conveys more “physical meaning” than all your stress and strain tensors combined. 2-forms are important in gauge theories in general. Also, metric of the spacetime is a (0,2) or (2,0) tensor, this time a symmetric one.

Hi, Incnis Mrsi.

Uh oh. It looks like I hit a nerve with what I said. If so, I apologize. Are you saying that the Maxwell field and the metric of spacetime have more real world practical applications than the stress and strain tensors of mechanics, or are you just saying that they are easier to interpret physically?

Chet

 

[Incnis Mrsi]

Hello.

Are you saying that the Maxwell field and the metric of spacetime have more real world practical applications than the stress and strain tensors of mechanics, or are you just saying that they are easier to interpret physically?

We argued about “physical meaning”. Although stress tensor is an example of such (as well as its relativistic generalization, stress–energy tensor), and not only as a linear operator, Maxwell field is more fundamental. Of course, it has countless practical applications (from electric generators to microwave ovens), although probably less than continuum mechanics. In one context (a force acting on a charge of given 4-velocity) it, as a relativistic 2-form, really looks like a linear operator. But the d ⋆F = J equation demonstrates it has “physical meaning” namely as a tensor field: Hodge dual and differentiation are operation on tensors, not on linear operators.

 

[NumericalFEA]

It is all about physics. There are fields that can be described at any particular point in space by a single value (parameter), such as temperature; they are called "scalar fields". There are fields that require three parameters, such as velocity; they are called "vector fields". However, there are fields that cannot be, at any particular point in space, described by one or by three values; they require six values (taking into account the symmetry of corresponding 3x3 matrix). Examples include strains and stresses in solid bodies. Such fields are neither scalar nor vector, they are 2nd order tensor fields.

 

[mal4mac]

If you are interested exclusively in second order tenors...

I am, who was Pavarotti's understudy?