@Meow Right, so the tangent line to $(0, 0)$ on the curve is a vertical line. $\theta$ = 90 degrees, $\tan(\theta)$ is undefined. Function's not diff at that point.
I think it's even possible to have a smooth graph which is nowhere differentiable but I am not sure.
I can certainly make it non-differentiable on the rationals, say, by plugging in a small $y = \sqrt[3]{x}$ at each rational point.
No, I take it back. I don't believe the first thing is possible.
If that had to happen every point would have to have a vertical tangent. Certainly can never happen.
But you can probably have a smooth graph with function nowhere differentiable on the rationals, or on a Cantor set and stuff like that using this idea.
There's a stronger version of the implicit function theorem (which I learned about on MSE, actually) wherein you don't even need to assume $C^1$ as long as you have partial derivative nonzero.
I want to be like the guy in those movies where someone asks "Who is he?" and a guy with a deep voice says "He's only the best God damn algebraic geometer this country has"
You need graduate algebra, commutative algebra, and some course in the complex (smooth) version of stuff helps with intuition (at least Riemann surfaces, hopefully a bit of complex geometry).
But they strongly recommend point-set, complex, and the third quarter algebra. For some reason, though, commutative algebra is a class in the winter quarter while algeo is fall
Whatever the case, though, I'll see how to partition classes at some point. The high priority will be those classes that you kinda need to know lest you're completely lost in grad school, followed by some number of classes that will be for "need to make sure I won't be broke if academia doesn't blow over well."
Basically, the hope is grad, and if not, I'll take corresponding topics at the undergrad level, like commutative algebra, algebraic geometry, rep theory, all that
because I think that dichotomy is horseshit. (and realized this after already specializing more towards algebra because I thought I fit more on that side.)
@Eric In that if you enjoyed your undergraduate analysis sequence more than your undergraduate algebra sequence, you should not base very much of your career on that fact.
That's at least what I did (or the opposite of what I did), and I hugely regret it.
yeah, I think ultimately I'll be in good shape, thinking about the future and the uncertainty of where ill be going for grad school just makes all these things more scary
What would you guys say are those topics that you just kind of are supposed to know lest you get lost in grad school? From what I've heard, that applies to real analysis, complex, point-set, and algebra for sure, but are there other subjects of the sort?
@Daminark I don't think that's a useful question to answer. Look up the prelim / basic quals at various schools. At the minimum you want to pass those.
In any case most of the point is less for you to learn specific things and more to get better at understanding what you're thinking about
Hi @robjohn. The question I pinged you and DanielF on was raised by Balarka: If you have a graph of a continuous function $f\colon\Bbb R\to\Bbb R$ and the graph is a smooth (say $C^1$?) curve, on what set $S$ can $f$ fail to be differentiable?
Oh, this actually follows (in part) from what @Alessandro is now pondering. The derivative is a pointwise limit of continuous functions, so can only be discontinuous on a set of first category.
Looking at it from the x-axis, valleys and peaks are clearly smooth, and any point of non-smoothness can be clarified by looking st it from the y-axis.
You only get the vanishing of the Christoffel symbols if it's torsion-free. Otherwise, you get $\Gamma^i_{jk}+\Gamma^i_{kj} = 0$ at the center of the normal coordinates.
If you want to interpret torsion as obstruction to integrability, you want to know what the "canonical charts" are in the Riemannian setting. I guess that's geodesic normal coordinates.
Robert Bryant and Robbie Gardner ran a weekend seminar years ago where we went through one of Cartan's papers in which torsion was made understandable.
I've forgotten the details, but integrating torsion of an affine connection around a little loop gives you the translation the base point undergoes when you use the affine connection to parallel translate around the loop.
