Vectors, Forms, and Tensors

- To an unaided eye vectors come in two varieties: column
and row vectors. Column vectors correspond to vectors proper,
row vectors are often thought to correspond to
*forms*. - Forms and vectors are isomorphic, so you can always view the correspondence between forms, vectors, columns, and rows the other way round.
- A vector is more than a row or a column of numbers:

These objects, , are called*basis vectors*if*every*vector of a vector space can be decomposed uniquely into a sum of those vectors with some coefficients. We call such a sum a*linear combination*of basis vectors. - Forms are linear maps, usually denoted by bold Greek letters,
that assign a number to every vector.
This is how they go about it:

- We can always find such basis
in the
*form*space that

where is the Kronecker delta. In that case:

- Operations such as
are
*always*between a form and a vector, and, consequently, the contracted indexes,*i*here, should always be on different levels, i.e., an upper with a lower index. - In sloppy algebra the distinction between forms and vectors is often ignored, but we will try our best not to.
- If a given space is only a vector space, you cannot do with it anything other than contracting a vector with a form.
- If a space is equipped in a
*metric*then you can evaluate scalar products of two vectors. - A metric is a tensor of rank 2, i.e., a form with
2 slots, into which you can put 2 vectors:

- A metric can be used to take a
*scalar product*of two vectors:

- In Cartesian coordinates, i.e., when the basis vectors are
orthonormal, and in a flat (
*Euclidean*) space*g*_{ii}= 1 and*g*_{ij}= 0 for . - It is only in that context that an expression such as

makes a geometric sense. In curvilinear coordinates or in a curved space the coefficients*g*_{ij}will be functions of coordinates in general, although sometimes they can be made cartesian in a small neighbourhood of any given point. If that is the case, a space like that is called*Riemannian*(locally*Euclidean*). - Vectors and forms are geometric objects, they have life
of their own independent of systems of coordinates. Their
coordinates, which is what you get in a
*column*or a*row*, will vary depending on a choice of a system of coordinates, whereas the vector itself remains unchanged. - As vectors and forms are represented by their coordinates,
e.g.,
*v*^{i}or , tensors are also represented by their coordinates, e.g.,*g*_{kl}. If vectors were to be mapped on*columns*and forms on*rows*, tensors of rank 2, such as the metric tensor, would be mapped on matrices of rank 2. - Tensors, like vectors and forms, are geometric objects.
Matrices of their coordinates,
*g*_{kl}, are not: they represent a given tensor in a particular system of coordinates only, and may vary as the coordinates vary. The tensor itself, like a vector and a form, remains undisturbed by the change in a system of coordinates. - Operations such as a multiplication of a vector by a matrix usually correspond to a situation in which a tensor gobbles up a vector and produces some other vector, or a form on output. Sometimes even another tensor. Tensors can do things like that to each other, and even to themselves! An operation that converts a Riemann tensor into a Ricci tensor is one of those: here the Riemann tensor eats itself and what's left is the Ricci tensor.
- Maxima, Maple, and Mathematica have special packages for
operations on tensors, vectors, and forms. But programs
such as Matlab, Octave, and Calc operate on matrix
*representations*of geometric objects only. You have to supply the geometric thinking and understanding yourself. - Matrices, columns, and rows also have some life of their
own, especially in context of systems of algebraic equations,
which
it may not always be convenient to consider in geometric
terms - although the latter usually
*does*pay off handsomely. - Because expressions such as
*always*imply the sum, , people often drop the symbol of the sum itself and just write . This is understood as follows: if two indexes are the same, e.g.,*i*, and one is at the bottom, e.g., , whereas the other one is at the top, e.g.,*v*^{i}a summation over is implied, where*n*is the dimension of the vector space. In summary:

This is called the Einstein summation convention. The indexes*must*be on different levels, which implies the interaction of forms with vectors. Only in a Euclidean space with an orthonormal system of coordinates, can you sum over indexes that are on the same level, as in*v*^{i}*w*^{i}.