Many linear physical and mathematical objects require more than a one dimensional (vector) description. A tensor is a geometric object that, like a vector, exists independent of any coordinate parameterization, although the actual value of its components depend on the parameterization, just like a vector. A tensor is a generalization of the vector concept, and a vector is a specific example of a tensor of rank 1. The rank of a tensor refers to the number of indices it has. The dimension refers to the range of the indices and is determined by the underlying vector space (2 dimensions, 3 dimensions, etc.). If we let n = the dimension and r = the rank, then the number of components in a tensor is equal to n r.

I will give as an example a simple minded physical model of a symmetric sailboat. By symmetric I mean its bow and stern are identical. This model assumes that, due to the viscous medium the boat travels through, its velocity is proportional (read "linear") to the applied force. Due to the design of the hull a sailboat goes forward and backward much more readily than it goes sideways, so the frictional forces are anisotropic. We can express the motion of the sailboat as V = T F where V i are the components of velocity, F j are the components of the applied force and Tij are the components of a tensor representing the hull of the boat in the water. In general the velocity and the applied force are NOT parallel. The relationship between V & F is a two dimensional second rank tensor, T. It embodies the relationship between the boat's resistance to forward and backward motion in relation to its resistance to sideways motion. It is purely a function of the hull design. The motion of the boat couldn't care less what coordinate system we use to calculate or describe its motion.

Another simple example of a second rank tensor is the polarization distribution of incident light. For unpolarized light the intensity will be the same in all directions or circular. As the light gets more and more (linearly) polarized the distribution takes on the shape of an ellipse, and approaches a line as the polarization approaches 100%. Note: do not confuse the above with circularly or elliptically polarized light, which is a different thing altogether.

A tensor can be thought of as a multi-linear machine. You input vectors, or even other tensors, in one side, turn the crank, and out pops a new vector, tensor, or even a scalar on the other side.

An object often associated with tensors is something called the Metric Tensor. The metric tensor specifies the non-orthonormality of the basis vectors. How it varies from point to point more fundamentally defines how the space is curved. For non-orthonormal basis components the issues of "contravariant" and "covariant" and "dual or reciprocal basis" come up.

The components of a tensor are usually specified with respect to the dual or reciprocal basis of the vectors it operates on, unless the metric tensor is included as part of the equation. A tensor can have both contravariant and covariant components. The position of the indices (vertically) denote whether a particular set of components is contravariant or covariant.

One should take note that not all linear machines are tensors. A defining characteristic of a tensor is the way its components transform with a change of basis. In general, when transforming to a new coordinate system (where A is the transformation matrix), one must multiply by A once for each covariant index, and by A-1 once for each contravariant index.

The difference between a rank 2 tensor and a matrix is the difference between a formula and an object! A tensor is a description of a real physical phenomena, object, etc. such as the strain in a rigid solid. Such objects exist independently of any coordinate system.

For an excellent set of examples of real world tensors in the physical sciences (restricted to an orthonormal basis), check out The Feynman Lectures on Physics, vol II, chapter 31. The whole chapter is virtually a list of various tensors, including the polarization tensor of a crystal, the conductivity tensor of a crystal, the Moment of Inertia tensor of an asymmetric solid, the stress tensor of a solid, the strain tensor of a solid, as well as others.