Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after reflection and some research,
I find little support for my unpremeditated claim.
Just sticking to the topological boundary (as opposed to the boundary
of a manifold or of a simplicial chain),
$\partial^3 S = \partial^2 S$
for any set $S$.
So there seems to be no possible analogy to
Taylor series.  Nor can I see an analogy with the
fundamental theorem of calculus.
The only tenuous sense in which I can see the boundary
as a derivative is that $\partial S$ is a transition between $S$
and the "background" complement $\overline{S}$.

I've looked for the origin of the use of the symbol $\partial$ in topology without luck. 
I have only found [references][1]
for its use in calculus.
I've searched through _History of Topology_ (Ioan Mackenzie James)
online without success (but this may be my poor searching).
Just visually scanning the 1935 _Topologie_ von Alexandroff und Hopf, I do not
see $\partial$ employed.

I have two questions:

> <b>Q1</b>. Is there a sense in which the boundary operator $\partial$ is
analogous to a derivative?

> <b>Q2</b>. What is the historical origin for the use of the symbol
$\partial$ in topology?

Thanks!


  [1]: http://en.wikipedia.org/wiki/%E2%88%82