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Are there any deep reasons, why we use the same symbol, $\partial$, when describing two (apparently fundamental) different mathematical objects, namely the boundary of a set (in topology), as well as the partial derivative (in analysis) ?

Can one of these maybe be viewed in (some very sophisticated manner) as the other ?

(According to wikipedia, this symbol occurs in even more places...)

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there are many symbols that have multiple applications. I think it may just be an instance of "running out" of symbols. Or sticking to the original appearance of the symbol. –  picakhu Apr 6 '11 at 15:39
    
The integral $\int_a^b f(x) dx$ of a map $f:[a,b]\to\mathbb{R}$ is the same thing as the difference $F(b)-F(a)$ where $F$ is the antiderivative of $f$. Notice that $a,b$ are the boundary points of $[a,b]$, and that $f=F'$. This connection between integration over a cell and its boundary gives us Stokes' theorem. When integration happens in several dimensions, we need the partial derivatives in there. I'm not sure about if this choice of notation was intentional. –  Eivind Dahl Apr 6 '11 at 15:43

2 Answers 2

up vote 3 down vote accepted

I was going to say "no" but a google search brought up this MathOverflow thread: http://mathoverflow.net/questions/46252/is-the-boundary-partial-s-analogous-to-a-derivative

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Thanks. Prof. Tao's answer to the MO thread, to which you gave me the link, pretty much clears the issue. –  Morres Apr 8 '11 at 9:55

Yes, Stokes theorem $$\int_\Omega d\omega = \int_{\partial \Omega} \omega$$ looks nice in this notation--you just need to shift the d from the integrand to the integration manifold.

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