Differential is a term, I first learned, in Mathematics Analysis. Now I have some problems about this term:
(1) Can we generalize differential to norm space?
(2) Can we generalize it into topological linear space?
(3) Can we generalize it into more general objects, like topological group, which can refer to Algebraic Object in Topology
Can someone give an explanation? Or reference? Thank you.
Derivative of any quantity gives you the rate of change of that quantity w.r.t. other quantities which influence it. So for example when we are taking ordinary $\frac{du}{dt}$ then you are measuring the rate of change of $u$ wrt $t$. Now wherever you need such a concept, you basically encounter derivative. For example the whole field of differential geometry originated when people started doing calculus on surfaces (or manifolds ?).
One more way to view derivative is to see it as an approximation to a quantity near certain points. We have notions of $Frechet\ derivative$, while dealing with Banach spaces. The operation of derivative when is equipped on algebraic objects like rings, fields etc. is a different field in itself and is known as $differential \ algebra$...see this http://en.wikipedia.org/wiki/Differential_algebra
