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:
  
  $$\boldsymbol{v} = \sum_{i=1}^n v^i \boldsymbol{e}_i$$
  
  
  These objects, $\boldsymbol{e}_i, \, i=1\ldots n$, 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:
  
  $$\begin{eqnarray*}\langle \boldsymbol{\eta}, \boldsymbol{v} \rangle &=& \langle ... ...a_i v^j \langle \boldsymbol{\omega}^i, \boldsymbol{e}_j \rangle \end{eqnarray*}$$
  
  
• We can always find   such basis $\boldsymbol{\omega}^i\, i=1\ldots n$ in the form space that
  
  $$\langle \boldsymbol{\omega}^i, \boldsymbol{e}_j \rangle = \delta^i{}_j$$
  
  
  where $\delta^i{}_j$ is the Kronecker   delta. In that case:
  
  $$\begin{eqnarray*}\langle \boldsymbol{\eta}, \boldsymbol{v} \rangle &=& \langle ... ...& \sum_{ij} \eta_i v^j \delta^i{}_j \\ &=& \sum_i \eta_i v^i \end{eqnarray*}$$
  
  
• Operations such as $\sum_i \eta_i v^i$ 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:
  
  $$\boldsymbol{g} = \sum_{ij} g_{ij} \boldsymbol{\omega}^i \otimes \boldsymbol{\omega}^j$$
  
  

• A metric can be used to take a scalar product of two vectors:
  
  $$\begin{eqnarray*}\boldsymbol{g}(\boldsymbol{v}, \boldsymbol{w}) &=& \langle \s... ...w^l \delta^i{}_k \delta^j{}_l \\ &=& \sum_{ij} g_{ij} v^i w^j \end{eqnarray*}$$
  
  
• 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 $i \neq j$.
• It is only in that context that an expression such as
  
  $$\sum_i v^i w^i$$
  
  
  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., vⁱ or $\eta_j$, 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 $\sum_i \eta_i v^i$ always imply the sum, $\sum_i$, people often drop the symbol of the sum itself and just write $\eta_i v^i$. This is understood as follows: if two   indexes are the same, e.g., i, and one is at the bottom, e.g., $\eta_i$, whereas the other one is at the top, e.g., vⁱ a summation over $\sum_{i=1}^n$ is implied, where n is the dimension of the vector space. In summary:
  
  $$\eta_i v^i \equiv \sum_{i=1}^n \eta_i v^i$$
  
  
  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 vⁱ wⁱ.