Differential Topology
In differential topology, a vector field may be defined as a derivation on the ring of smooth functions on a manifold, and a tangent vector may be defined as a derivation at a point. This allows the abstraction of the notion of a directional derivative of a scalar function to general manifolds. For manifolds that are subsets of Rn, this tangent vector will agree with the directional derivative defined above.
The differential or pushforward of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the Jacobian matrix.
On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.
The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
Together with the interior product (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a Lie superalgebra.
Read more about this topic: Generalizations Of The Derivative, Derivatives in Geometry
Famous quotes containing the word differential:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)