Editing Covariant transformation (section)

===The derivative of a function transforms covariantly===
The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function.  Consider a scalar function ''f'' (like the temperature at a location in a space) defined on a set of points ''p'', identifiable in a given coordinate system <math>x^i,\; i=0,1,\dots</math> (such a collection is called a [[manifold]]).  If we adopt a new coordinates system <math>{x'}^j, j=0,1,\dots</math> then for each ''i'', the original coordinate <math>{x}^i</math> can be expressed as a function of the new coordinates, so <math>x^i \left({x'}^j\right), j=0,1,\dots</math> One can express the derivative of ''f'' in old coordinates in terms of the new coordinates, using the [[chain rule]] of the derivative, as
:<math>
  \frac{\partial f}{\partial {x}^i} =
    \frac{\partial f}{\partial {x'}^j} \;
    \frac{\partial {x'}^j}{\partial {x}^i}
</math>

This is the explicit form of the '''covariant transformation''' rule. The notation of a normal derivative with respect to the coordinates sometimes uses a comma, as follows
:<math>f_{,i} \ \stackrel{\mathrm{def}}{=}\  \frac{\partial f}{\partial x^i}</math>
where the index ''i'' is placed as a lower index, because of the covariant transformation.