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
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:

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

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

Thanks!

**Addendum**. Although **Q2** has not been addressed, it seems appropriate to accept one
among the wealth of insightful responses to **Q1**. Thanks to all!

taken with orientationwhich ensures that the boundary of a boundary vanishes. And of course in this setting the boundary is really just the differential of the complex computing the homology of your space, which shows the connection to the classical "d", since you could also have computed the cohomology of the space using the de Rham complex where the differential is just the ordinary differential. $\endgroup$ – Dan Petersen Nov 16 '10 at 16:54Groups of homotopy spheres Iby Michel Kervaire and John Milnor (1962) where the notation is 'b'.) $\endgroup$ – Maxime Bourrigan Mar 30 '11 at 23:24