 The differential of "f" at a point "x" is the map
where "X""x" is a tangent vector at "x", thought of as a derivation.That is formula_35 is the Lie derivative of "f" in the direction "X", and one has d"f"("X")="X"("f").Equivalently, we can think of tangent vectors as tangents to curves, and write
In either case, d"f""x" is a linear map on "T""x""M" and hence it is a tangent covector at "x".