Examples
Not all relationships in nature are linear, but most are differentiable and so may be locally approximated with sums of multilinear maps. Thus most quantities in the physical sciences can be usefully expressed as tensors.
As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The acceleration will in general not be in the same direction as the force, because of the particular shape of the ship's body. However, it turns out that the relationship between force and acceleration is linear. Such a relationship is described by a tensor of type (1,1) (that is to say, it transforms a vector into another vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. Just as the numbers which represent
a vector will change if one changes the coordinate system, the numbers in the matrix that represents the tensor will also change when the coordinate system is changed.
In engineering, the stresses inside a rigid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i.e. causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.
Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor and the polarization tensor.
Geometric and physical quantities may be categorized by considering
the degrees of freedom inherent in their description. The scalar
quantities are those that can be represented by a single number ---
speed, mass, temperature, for example. There are also vector-like
quantities, such as force, that require a list of numbers for their
description. Finally, quantities such as quadratic forms
naturally require a multiply indexed array for their representation.
These latter quantities can only be conceived of as tensors.
Actually, the tensor notion is quite general, and applies to all of
the above examples; scalars and vectors are special kinds of
tensors. The feature that distinguishes a scalar from a vector, and
distinguishes both of those from a more general tensor quantity is
the number of indices in the representing array. This number is
called the rank of a tensor. Thus, scalars are rank zero tensors (no
indices at all), and vectors are rank one tensors.
Approaches, in detail
There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.
The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the indices of the array.
This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.
The tensor field theory can roughly be viewed, in this approach, as a further extension of the idea of the Jacobian.
The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of