Understanding Tensors

Question

I don't seem to be able to visualize tensors. I am reading The Morgan Kauffman Game Physics Engine Development and he uses tensors to represent aerodynamics but he doesn't explain them so I am not really able to visualize them. Please explain in very simple ways. I just want to understand the basics.

Answer 1

From your context, I will restrict attention to tensors in 3 dimensions and in rectangular coordinates, where there is no distinction between up and down indices, no distinction between covectors and vectors.

The tensor visualization methods pretty much crap out at above 4 indices. At this point, I don't know how to visualize. I assume you are interested in low rank tensors.

First, review this answer: What is a tensor?. A 2-tensor is a vector of vectors, that is a vector each of whose components are vectors. A 3-tensor is a vector of 2-tensors, etc.

The way to visualize a 2-tensor is in terms of its invariant parts, there's the antisymmetric part:

$$ {1\over 2} (T_{ab} - T_{ba})$$

and the symmetric part:

$$ {1\over 2} (T_{ab} + T_{ba})$$

The antisymmetric part is a vector, actually an axial vector, and you can rearrange it into a vector, and visualize it this way:

$$ V_i = \epsilon_{ijk} T_{jk}$$

So I will assume your 2-tensors are symmetric. The symmetric part is further decomposable into a trace part, and a traceless part:

$$ T = T_{aa} $$ $$ T_{ab} - {1\over 3} T $$

One way to visualize these tensors is in terms of three orthogonal vectors A,B,C which have the property that:

$$ A_i A_j + B_i B_j + C_i C_j = T_{ij}$$

The directions A,B,C are called the principal axes, and the tensor is entirely defined by their direction, and the magnitude of A,B,C. The trace part is the sum of the magnitude of A,B,C. The transformation to the principal axes is diagonalizing a 3 by 3 matrix by a rotation

Examples of two tensors are the stress tensor, the moment of inertia, and the coefficients of a quadratic potential. The stress tensor is not so great to visualize with principal axes, the other two are.

The way to visualize the stress tensor is just to look at it as three vectors side by side in a matrix, each of which is a separate current. These are the currents of the x,y,z momentum as it flows around.

For rank 3 tensors, there are decompositions, but they aren't as useful for visualization. If two of the indices are antisymmetric, you can turn them into a single index, but this wrecks parity. For these, you can use one example, which will guide you--- the connection.

A three tensor $T_{abc}$ has the property that when acting on a vector, it is a two tensor, or a map from vectors to vectors. So this associates a linear map with each direction. You can visualize a three tensor as giving you an infinitesimal linear transformation for each direction, in a way that depends linearly on the direction. This visualization is probably good for any 3-tensor you will encounter.

For rank 4 tensors, there are two examples to guide your intuition:

• The elastic tensor, which tells you the stress tensor in terms of the strain tensor.
• The curvature tensor, which tells you the linear map associated with an infinitesimal closed curve (pair of directions).

A rank 4 tensor takes rank 2 tensors to rank 2 tensors. The stress is a rank two tensor, and so is the strain ( the partial derivative of the new position of a solid $X_i$ with respect to the old position $x_j$). The way to view rank 4 tensors is as these type of linear relations between two-tensors, usually symmetric.

The best interpretation is always to find the right linear map to visualize. This is why mathematicians define tensors by the linear map properties, even though every tensor has a lot of different linear maps. You should take it on a case by case basis.

Answer 2

You can look at tensor as a data file. The most important thing is not to visualize it but to understand what each element of this file means, and how to transform this file from one system (system of coordinates) to another.

Visualization here can be done in two steps:

1. Visualizing data - vector fields, scalar fields, oriented volume fields and sooo on.

2. Visualizing transformation - in your case its mostly rotation and translation.

In this day and age, people tend to be less able to visualize things; so look back to older books on the subject which are full of good and easy-to-grasp analogies. The older the book, the better.

First check out, Landau Lifshitz Volume 2 "Theory of Fields". At the end of the book, in the chapter about "General relativity and tensors", it will have the information I am referring to.

I'm not quite certain, but you can also check out a very good general book about physics which is called "Motion mountain". It should have chapter on tensors. It's free and you can download it from the web.

Thank you for thinking about this message.

Answer 3

Tensors are abstract algebraic or differential geometric objects (in the latter case we're actually talking about tensor-valued fields) which are made from 'products' of vectors and covectors.

There's no tangible explanation for what a tensor is because the concept is so general that tensors can fill very different roles:

Any linear map is an element of a tensor space as well as volume and surface elements, the cross product of vectors generalizes to the wedge product of anti-symmetric tensors (or, depending on use-case, a Lie bracket, but that's unrelated), the laws of general relativity are expressed in terms of tensor calculus (energy-momentum-tensor, metric tensor, Ricci tensor, ...), compound system in quantum mechanics are modeled as tensor spaces (the bosonic and fermionic Fock spaces of second quantization are a special case of that), the electric and magnetic fields are components of the electromagnetic field tensor, ...

There's not much you can do except familiarize yourself with the abstract definitions and rules and look into the geometric interpretation of particular tensors.

Answer 4

You could probably do a lot worse than taking a look at the video "What's a Tensor?" by Daniel Fleische on YouTube: http://www.youtube.com/watch?v=f5liqUk0ZTw

He gives a nicely visual, but maths-lite answer to that question by using children's blocks, small arrows, a couple of pieces of cardboard and a pointed stick.

Answer 5

Following @peter4075, Fleische's book, A Student's Guide to Vectors and Tensors is a gold mine of clear, useful, practical explanation, by someone who clearly knows their topic. Highly recommended. I got it through my local library interlibrary loan program and am loathe to surrender it.

Good luck!