Understanding Tensors

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.

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:

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.