Editing Covariant transformation (section)

==Contravariant transformation==

The ''components'' of a (tangent) vector transform in a different way, called contravariant transformation.  Consider a tangent vector '''v''' and call its components <math>v^i</math> on a basis <math>\mathbf{e}_i</math>.  On another basis <math>\mathbf{e}'_i</math> we call the components <math>{v'}^i </math>, so
:<math>\mathbf{v} =  v^i \mathbf{e}_i  = {v'}^i  \mathbf{e}'_i</math>
in which 
:<math>
     v^i = \frac{dx^i}{d\lambda} \;\mbox{ and }\;
  {v'}^i = \frac{d{x'}^i}{d\lambda}    
</math>

If we express the new components in terms of the old ones, then
:<math> 
  {v'}^i = \frac{d{x'}^i}{d\lambda\;\;} =
             \frac{\partial {x'}^i}{\partial x^j}
           \frac{dx^j}{d\lambda} =
             \frac{\partial {x'}^i}{\partial x^j} {v}^j
</math>
This is the explicit form of a transformation called the '''contravariant transformation''' and we note that it is different and just the inverse of the covariant rule. In order to distinguish them from the covariant (tangent) vectors, the index is placed on top.

===Basis differential forms transform contravariantly===

An example of a contravariant transformation is given by a [[differential form]] ''df''.  For ''f'' as a function of coordinates <math>x^i</math>, ''df'' can be expressed in terms of the basis <math> dx^i</math>.  The differentials ''dx''  transform according to the contravariant rule since
:<math>d{x'}^i = \frac{\partial {x'}^i}{\partial {x}^j} {dx}^j</math>