In Calculus and PDE, people study rather complicated functions, equations on
relatively simple spaces (real line or n-dim Euclidean spaces). On the other hand,
in topology, people study complicated spaces with relatively simple function theory
on them. We are going to introduce a kind of calculus that takes the underlying
topological space into account. Thus we can see how topology interacts with calculus
naturally. The kind of new Calculus is called differential geometry. From this point
of view, Calculus and topology are finally unitfied into differential geometry.