Guys, say we have $f\colon\mathbb C\to\mathbb C$. My book says we can write $f(z)=u(x,y)+iv(x,y)$. I can see why this could be true, but why aren't we worried that something goes wrong, because $\mathbb C\neq\mathbb R^2$? For instance, my book used the chain rule on such a function, without ever bothering if this is correct. But the multiplication is exactly what makes $\mathbb C$ different, so what guarantees that everything goes well?
There's no issue in thinking of it as $\Bbb R^2$. There is more structure coming from multiplication of complex numbers, but that doesn't mean that forgetting about that makes it wrong!
If you have $f(z(t))$ for a curve $z(t)=x(t)+iy(t)$, then you can use your complex multiplication structure to write the chain rule as $$\dfrac d{dt} f(z(t)) = f'(z(t))z'(t)$$ when $f$ is analytic/holomorphic.
You can check that the "usual" chain rule gives that result.
No, this is explicitly for $f\colon\Bbb C\to\Bbb C$, although you know that it corresponds to $(u,v)\colon\Bbb R^2\to\Bbb R^2$ and we could do chain rule with that.
If the codomain is $\Bbb C$, you can again take $f(z) = u(z)+iv(z)$, so you're getting $f(z)\leftrightarrow (u(x,y),v(x,y))$ and that's where Cauchy-Riemann comes from.
Sure, @philmcole. An interesting thing to realize is that in even dimensions you can have rotations in orthogonal $2$-dimension planes and have no net rotation at all.
btw Ted, did I already tell you a friend of mine watched your lectures and recommended it to others, when we were struggling with Spivak's "Calculus on Manifolds"?
@Balarka: You should see me trying to write on a computer monitor (as I have to for my AoPS classes). Handwriting is absolutely horrendous. And I hate whiteboards ... the pens are always dry.
I made a colleague at UGA very angry when I first arrived and made that claim. (His handwriting and boardwork were execrable. His teaching wasn't much better.)
@MatheinBoulomenos I had a very bad ex super visor who is making my life miserable. Although, I am doing pretty good now in graduate school, but I am want to do something that has application in real world.
There is something quite mysterious here. Drawings are inherently 2-dimensional but the pictures we create in our mind’s eye are indeed inherently 3-dimensional.
If $\{f_n\}$ is a sequence of nonnegative measurable functions converging pointwise to $0$ a.e. on $E$, does this imply $\lim_{n \to \infty} \int_E f_n = 0$?
My favorite is to cup my hands together and draw a disc, so that my fingers form the boundary circle. I then pinch together my fingerpoints, which gives a figure 8. Now forget the disc, and do a twist to move one circle to the other.
It's clear that I've never been a topologist ... although when I applied for jobs UWisconsin made me check one from a list of fields, and topology was the closest they had for me.
the craziest homological algebra lesson was in the TA session for étale cohomology. The prof just assumed that spectral sequences are common knowledge, but it was pointed out to him that they weren't covered in any of the prerequisited courses.
So the TA defined spectral sequences, derived elementary properties like 5-term exact sequences and then went one to construct 4 "examples" of spectral sequences which were still really general, like the one we ended up in the end was the Grothendieck spectral sequence which has basically every spectral sequence in algebra or algebraic geometry as special cases
