Editing Covariant transformation (section)

==Co- and contravariant tensor components==
===Without coordinates===
A [[tensor]] of [[type of a tensor|type (''r'', ''s'')]] may be defined as a real-valued multilinear function of ''r'' dual vectors and ''s'' vectors. Since vectors and dual vectors may be defined without dependence on a coordinate system, a tensor defined in this way is independent of the choice of a coordinate system.

The notation of a tensor is
:<math>\begin{align}
            &T\left(\sigma, \ldots ,\rho, \mathbf{u}, \ldots, \mathbf{v}\right) \\
  \equiv {} &{T^{\sigma \ldots \rho}}_{\mathbf{u} \ldots \mathbf{v}}
\end{align}</math>
for dual vectors (differential forms) ''ρ'', ''σ'' and tangent vectors <math>\mathbf{u}, \mathbf{v}</math>. In the second notation the distinction between vectors and  differential forms is more obvious.

===With coordinates===

Because a tensor depends linearly on its arguments, it is completely determined if one knows the values on a basis <math>\omega^i \ldots \omega^j</math> and <math>\mathbf{e}_k \ldots \mathbf{e}_l</math>
:<math>
  T(\omega^i,\ldots,\omega^j, \mathbf{e}_k \ldots \mathbf{e}_l) =
    {T^{i\ldots j}}_{k\ldots l}
</math>
The numbers <math>{T^{i\ldots j}}_{k\ldots l}</math> are called the '''components of the tensor on the chosen basis'''.

If we choose another basis (which are a linear combination of the original basis), we can use the linear properties of the tensor and we will find that the tensor components in the upper indices transform as dual vectors (so contravariant), whereas the lower indices will transform as the basis of tangent vectors and are thus covariant. For a tensor of rank 2, we can verify that
:<math> 
     {A'}_{i j} = \frac{\partial x^l}{\partial {x'}^i}
                       \frac{\partial x^m}{\partial {x'}^j} A_{l m}
</math> '''covariant tensor'''
:<math>  
     {A'\,}^{i j} = \frac{\partial {x'}^i}{\partial x^l}
                 \frac{\partial {x'}^j}{\partial x^m} A^{l m}
</math> '''contravariant tensor'''

For a mixed co- and contravariant tensor of rank 2
:<math>
     {A'\,}^i{}_j= \frac {\partial {x'}^i} {\partial x^l}
                 \frac {\partial x^m}      {\partial {x'}^j} A^l{}_m
</math> '''mixed co- and contravariant tensor'''