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 formula_3 (such a collection is called a manifold). If we adopt a new coordinates system formula_4 then for each "i", the original coordinate formula_5 can be expressed as a function of the new coordinates, so formula_6 One can express the derivative of "f" in old coordinates in terms of the new coordinates, using the chain rule of the derivative, as