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: *e.g.*,
the tensor (rank 3), or the tensor (rank 4),
*etc.* The suffixes are always understood to range from
1 to .

For reasons which 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), *etc.* 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 , *etc.*
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, *etc.*

When making a coordinate transformation from one set of coordinates,
, to another, , it is assumed that the transformation in
non-singular. In other words, the equations which 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

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Note that

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The formal definition of a tensor is as follows:

- An entity having components
in the
system and
in the system is said to
behave as a
*covariant tensor*under the transformation if

- Similarly,
is said to behave as a
*contravariant tensor*under if

- Finally,
is said to behave as
a
*mixed tensor*(contravariant in and covariant in ) under if

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 which 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 which only exhibits tensor behaviour 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 above 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,
, since

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The simplest example of a covariant vector is provided by the gradient
of a function of position
, since if we
write

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An important example of a mixed second-rank tensor is provided by
the Kronecker delta introduced previously, since

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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.

Let

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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 neighbouring 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

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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

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By analogy with Euclidian space, we define the *squared magnitude*
of a vector with respect to the metric
by the equation

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Finally, let us consider differentiation with respect to an invariant distance,
. The vector is a contravariant tensor, since

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