Oh, I should send you my 2-page handout for the physics majors in my old multivariable course, @Semiclassic. One exercise was to figure out grad, div, curl in different coordinate systems using forms. The other was a dirac delta thing.
Where taking the curl of a magnetic field should yield the volume current density, so taking the curl of the field produced by a line current should give that current back.
The main idea is that the roots of one of them lies on a circle on the complex plane, and the roots of the other lie on a tangent circle on the complex plane
@ted Heh. Want to know what's funny? The physics book I'm lifting this from uses the notation $\int d(z,\overline{z})=\int_{-\infty}^\infty\int_{-\infty}^\infty dx\,dy$
Which I guess means I'm wondering if there's a way to do that kind of Gaussian integral without having to go back to $x,y$. Otherwise it just seems unnecessary.
Just like how the roots of $x^n-1$ (which would be the $n$-th roots of unity) all lie on the unit circle, I claim that the roots of $x^n-a^n$ all lie on a circle of radius $|a|$ centered at the origin. @Typhon
@Ted i feel like the definition of "$d$" of a form that you're thinking of as a current should be inducing the usual exterior derivative on forms, so like if you call $C_{\omega}$ the current associated to $\omega$ then we should have $dC_{\omega} = C_{d\omega}$ (i think a formula like this holds for distributions iirc, so it's the same thing) so maybe it "sees" the residue in the sense that this formula fails
Nope, Eric, $d$ is defined on currents by forcing Stokes's Theorem.
If you have a global smooth form, then you're right, of course.
Remember that submanifolds (chains) are also currents, because you can integrate over them. So you want $d$ to include $\partial$ in the appropriate way.
i do feel strongly that for a form like the $dz/z$ that's locally integrable we should have a formula like $dC_{\omega} - C_{d\omega} = Residue$ or something
There's lots of mathematics for you to learn there. They make occasional errors, but it's still by far my favorite source for complex algebraic geometry and related topics.
@AkivaWeinberger I wrote a large section on complex numbers in the context of geometry for about 5 pages of my 50 geometry paper I wrote last fall. I know the algebra rules. I just don't know the fancy terms or "theory x"s.
