Let be a dimensional smooth manifold, and a smooth function.
The smooth covector field is called the differential of , and is defined as
Relation to the more general differential
Note if is a function, we may take it differential is the pullback sense which would give
Which is an element of , thus is a differential 1-form.
Taking into account the isomorphism between and , the map described above and the definition of the differential of a function between manifolds gives the same function.
Coordinate representation
Let be coordinate functions on .
In these coordinates, we have the differential as
Derivative of a function along a curve
Let be a smooth manifold and a smooth curve.
For a smooth function we have
This is a good way to calculate the differential at a point in the manifold.