A tensor of rank in an -dimensional space possesses components which are, in general, functions of position in that space. A tensor of rank zero has one component, , and is called a scalar. A tensor of rank one has components, , and is called a vector. A tensor of rank two has components, which can be exhibited in matrix format. Unfortunately, there is no convenient way of exhibiting a higher rank tensor. Consequently, tensors are usually represented by a typical component: for instance, the tensor (rank 3), or the tensor (rank 4), et cetera. The suffixes are always understood to range from 1 to .
For reasons that will become apparent later on, we shall represent tensor components using both superscripts and subscripts. Thus, a typical tensor might look like (rank 2), or (rank 2), et cetera. It is convenient to adopt the Einstein summation convention. Namely, if any suffix appears twice in a given term, once as a subscript and once as a superscript, a summation over that suffix (from 1 to ) is implied.
To distinguish between various different coordinate systems, we shall use primed and multiply primed suffixes. A first system of coordinates can then be denoted by , a second system by , et cetera. Similarly, the general components of a tensor in various coordinate systems are distinguished by their suffixes. Thus, the components of some third rank tensor are denoted in the system, by in the system, et cetera.
When making a coordinate transformation from one set of coordinates, , to another, , it is assumed that the transformation is non-singular. In other words, the equations that express the in terms of the can be inverted to express the in terms of the . It is also assumed that the functions specifying a transformation are differentiable. It is convenient to write
The formal definition of a tensor is as follows:
When an entity is described as a tensor it is generally understood that it behaves as a tensor under all non-singular differentiable transformations of the relevant coordinates. An entity that only behaves as a tensor under a certain subgroup of non-singular differentiable coordinate transformations is called a qualified tensor, because its name is conventionally qualified by an adjective recalling the subgroup in question. For instance, an entity that only exhibits tensor behavior under Lorentz transformations is called a Lorentz tensor, or, more commonly, a 4-tensor.
When applied to a tensor of rank zero (a scalar), the previous definitions imply that . Thus, a scalar is a function of position only, and is independent of the coordinate system. A scalar is often termed an invariant.
The main theorem of tensor calculus is as follows:
If two tensors of the same type are equal in one coordinate system then they are equal in all coordinate systems.
The simplest example of a contravariant vector (tensor of rank one) is provided by the differentials of the coordinates, , because
The simplest example of a covariant vector is provided by the gradient of a function of position , because if we write
An important example of a mixed second-rank tensor is provided by the Kronecker delta introduced previously, because
Tensors of the same type can be added or subtracted to form new tensors. Thus, if and are tensors, then is a tensor of the same type. Note that the sum of tensors at different points in space is not a tensor if the 's are position dependent. However, under linear coordinate transformations the 's are constant, so the sum of tensors at different points behaves as a tensor under this particular type of coordinate transformation.
If and are tensors, then is a tensor of the type indicated by the suffixes. The process illustrated by this example is called outer multiplication of tensors.
Tensors can also be combined by inner multiplication, which implies at least one dummy suffix link. Thus, and are tensors of the type indicated by the suffixes.
Finally, tensors can be formed by contraction from tensors of higher rank. Thus, if is a tensor then and are tensors of the type indicated by the suffixes. The most important type of contraction occurs when no free suffixes remain: the result is a scalar. Thus, is a scalar provided that is a tensor.
Although we cannot usefully divide tensors, one by another, an entity like in the equation , where and are tensors, can be formally regarded as the quotient of and . This gives the name to a particularly useful rule for recognizing tensors, the quotient rule. This rule states that if a set of components, when combined by a given type of multiplication with all tensors of a given type yields a tensor, then the set is itself a tensor. In other words, if the product transforms like a tensor for all tensors then it follows that is a tensor.
So far, the space to which the coordinates refer has been without structure. We can impose a structure on it by defining the distance between all pairs of neighboring points by means of a metric,
The elements of the inverse of the matrix are denoted by . These elements are uniquely defined by the equations
The tensors and allow us to introduce the important operations of raising and lowering suffixes. These operations consist of forming inner products of a given tensor with or . For example, given a contravariant vector , we define its covariant components by the equation
By analogy with Euclidian space, we define the squared magnitude of a vector with respect to the metric by the equation
Finally, let us consider differentiation with respect to an invariant distance, . The vector is a contravariant tensor, because