@Perturb: This is why I recommend studying curves and surfaces before jumping into more abstract stuff. Of course, you should know all this stuff from Guillemin & Pollack. The only difference is that they used parametrizations rather than charts (so $\phi^{-1}$ in place of $\phi$).
So just going back to what I asked, say I have a smooth manifold $M$ and a smooth function $f : M \to \mathbb{R}$, along with a chart $(U, \phi)$ containing a point $p \in M$. Then $\frac{\partial}{\partial x^i}\bigg|_p(f)$ is just the usual partial derivative $\frac{\partial f}{\partial x^i}(\phi(p))$ taking place in $\mathbb{R}^n$ (which is really the directional derivative in the direction of the basis vector $e_i$)
Can you verify if the statement is true or false please? Let $a,b\in \mathbb{R}_{+}$ with $a<b$, then \begin{equation*} \frac{b}{a}\leq \frac{1+b^{2}}{1+a^{2}}\text{.} \end{equation*}
I just learned that the jacobson radical and the frattini subgroup are both intersections of maximal substructures (or normal ones). Are there other examples of that?
@Takashi I think the inequality is true if and only if $ab<1$. Take $a=1/4$ and $b=1/2$ for example. Then $b/a=2$, but the right side is equal to $20/17<34/17=2$
another approach: the proposed inequality can be rearranged to $a+1/a\leq b+1/b$. Note that the form of this inequality remains the same if we replace a or b with their reciprocals. But these operations do not generically preserve the inequality a<b, so the proposed inequality cannot be true for all 0<a<b.
if ab<1, though, then a<1/b and a<b. so in that case, replacing b->1/b leaves all inequalities unchanged. so in that scenario it is indeed possible for the inequality to be valid.
Ah, I see. Let think about this a little bit longer.
Let $r : \Bbb{R} \to (0,1)$ be a retraction. Since $\frac{1}{n} \in (0,1)$ for every $n$, and since $r$ is continuous, $r(0) = \lim_{n \to \infty} r(\frac{1}{n}) = \lim_{n \to \infty} \frac{1}{n} = 0 \notin (0,1)$, a contradiction.
Ah I didn't take ODEs so idk for sure but to my understanding, even if it's a computational course, they probably don't give things that are way too fancy that you can't just figure it out yourself, no?
I wouldn't allow a table of integrals, either, in a low-level differential equations course. And for upper-level, there wouldn't be much in the way of explicit solving ...
Your friend should have been told before the exam what the situation was.
Typically, introductory DE courses are intended explicitly to make sure students learn some of their integral calculus skills (or basic linear algebra, for systems of ODE).
That will be a ton of graduate real analysis, if it's really graduate.
Plus knowing Stokes's and Divergence Theorem arguments.
Graduate courses are not typically plug-and-chug, unless it's an engineering course. Undergraduate PDE is ... Separation of variables and Fourier series.
I mean actually actually idk if I'd have enjoyed it, one of my friends who used to be gung ho about analysis took it and was like meh maybe not this kinda analysis, ever since he's started shifting more toward geometry/topology
My days of anxiety were the typewriter/snail mail days. We all suffered in silence (or to our immediate friends), but I don't remember doing that much.
