@XanderHenderson I wouldn't say I do it. It was my interest for a while, sure, but now I focus on other things. What just strikes me as odd is how people are not accepting of using dimension theory for separable metrizable spaces in their arguments
I always wonder: if $1+1=2$ is an axiom, does that mean someone could create a whole new 'system' of mathematics if they made $1+1=3$ (whatever that might mean)? Or are there some axioms that cannot form a 'system'?
If that is true is there a 'math field' where mathematicians just change the rules and try to build a new 'system'?
@BenSteffan ngl that is a terrible name for a field, as an outsider like me can't just know such thing exist because if I hear about it I will say they mean the general logic
if you encounter "surgery theory" or "deformation theory" or something like this in a mathematical text, you wouldn't assume that refers to medicine, e.g.
@BenSteffan Yeah because medicine don't use math, but engineers and programmers do use math (or abuse it) so I might suspect that "surgery theory" is a pure math thing, but model theory seems like a name for a course in engineering
@Jakobian I am in engineering school right now, I hate the way they never show a proof for anything and just "take my word for it, it works", so I generally assume applied math don't care about proofs but how to use the theorems, maybe I am wrong?
I feel like this is precisely the kind of cliche that makes people turn away from applied mathematics (however vague what this means is; and thus from a lot of mathematics)
@Jakobian It made me hate any form of applied math like physics (although I liked it in high school), in the first year we took thermodynamics and Schrodinger equation without taking linear algebra or ode or multi variable calculus or pde,
I remember struggling to memorise the solution of second degree PDE without knowing what the hell was going in any step
@pie "Applied math" is a spectrum. At my phd institution, there was an applied mathematics research group in the mathematics department. They were a rigorous bunch who also had industry ties. They cared about proof, or at least about having a strong understanding of the theoretical underpinnings of what they were doing.
@XanderHenderson I'm not going to argue with that, but definitely, philosophy is not a subfield of math
@pie again, the dichotomy between "applied" and "pure" math is really more of an issue with the philosophy with which you do math with. This only shows that you dislike that particular exposition to physics. Concluding more would be a stretch.
But you can always study PDE theory and see that, it really isn't that bad after all if you learn it properly (or just hate PDEs)
@BenSteffan even the numerical analysts, they are rigorous. If you write dog as dog=cat +(dog-cat), they make sure (dog-cat) is small for large enough values of cat
Yeah, and they need to be, or else you'll encounter failed rocket launches and stuff
@pie i wouldn't classify physics as applied math, really, though there is a lot of math applied in it. Applied mathematicians still must be good at theory, physicists needn't even touch analysis textbooks
@BenSteffan what's your opinion on taking theoretische Physik 1, 2, 3 and be done with the Nebenfach with that? Can't get more mathematical than that, huh haha
Experimentalphysik is a recommended pre-req though
@Jakobian I mean, math is a subfield of philosophy. If philosophy were a subfield of math, then the two would be equal, and that doesn't seem to be the case...
the sequence $f_k(x) = e^{-|x-k|}, k \in \mathbb{N}, x \in \mathbb{R}$ can be uniformly bounded by $M = 1$ a.e. in $\mathbb{R}$ and $$\lim_{k \to \infty}f_k(x) = 0:=f(x), \forall x \in \mathbb{R}$$ but $$\lim_{k \to \infty} \int_{\mathbb{R}} |f_k(x)-f(x)|dx = \int_{\mathbb{R}} e^{-|x-k|}dx = 2 \nrightarrow 0$$
Generally speaking, introductory texts like Lin Alg Done Right are meant to give you a baseline, which is supposed to help you solve problems later on.
The tools taught in lin alg are often of practical use in, for example, 3d graphics. They also show up in many numerical schemes for solving problems (e.g. linear optimization problems (in the form of "linear programming"), root finding algorithms, etc). Lin alg shows up all over the place in computer and data science.
@XanderHenderson ive seen some parts of that book lol. I intended on reading its pdf every night as a bedtime reading thing, at the start of the semester
some problems ofc need pen and paper, like for eg: Proving that for any 3 vector spaces V1, V2, V3 over a field with more than 2 members, the union of the 3 is a vector space if and only if one of them contains the other two. This was a notorious exercise for me, cant imagine solving without pen n paper. but say for eg, proving that for finite dim vector spaces, the only two sided ideals of $L(V,V)$ are (0) and the space itself, argument can be constructed without pen n paper ofc
for me, its not that I can't tell if a proof in my head is wrong, but rather, when looking at a proof, I can't tell if the steps are correct or not without writing them down explicitly sometimes
because those type of fields are computation heavy
similarly, abstract algebra has either computation heavy, or just very conceptual approach, and I would need paper to either get through computations or orient myself among the definitions
in general topology there is much less of this in comparison, and proofs are relatively tame
I'm not even talking about "problems" because constructing an argument in your head is another thing entirely
I mean it of course has elements of "verifying a proof" in itself, since you do so with your own proof, but it also has the additional step of having to make an argument for why something is true
But I do agree that a lot of non-computational type exercises are picked to be easy enough so that they can often be "done in your head"
this conceptual kind of thinking, if its not super long, can often be done without the aid of paper
@Jakobian u mean like actually learning a field? that I havent obviously. I did linear algebra throughout the day / week, with pen n paper ofcourse, and atop of that, I did a bit of bedtime reading of LADR. Reasons are similar to the points you raised, memorization of definitions, getting into the context of things, making notes, etc.
the semicircular arc in the upper half-plane, defined by $z = Re^{i\theta}$ for $\theta \in [0, \pi]$
By the residue theorem, since $f(z)$ is holomorphic on $\partial D$ except at isolated points:
$\int_{\partial D} f(z) \, dz = 2 \pi i \sum \text{Residues of } f(z) \text{inside} \partial D$
I write the integral along the contour as: $\int_{\partial D} f(z) \, dz = \int_{-R}^{R} f(z) \, dz + \int_{\Gamma_R} f(z) \, dz$
When $R \to \infty$, the integral along $\Gamma_R$ tends to zero, under the assumption that the function $f(z)$ decreases quite rapidly on the semicircular arc
if $f(z) = g(z) e^{iaz}$ and $g(z)$ is analytic and bounded on $\Gamma_R$, with $a > 0$, then: $\int_{\Gamma_R} f(z) \, dz \to 0 \quad \text{as } R \to \infty$
@SineoftheTime I have to add one more thing : (if S is a set and * : SxS --> S is a binary operation, then for all s there exists x such that x * s = s * x = s)
@SineoftheTime ↓
For the first part i'm trying to prove that associativity holds using z1, z2, z3, each written in trigonometric form but i get stuck in the calculations and im not sure it's the right approach. For the second part, S1 is a subset of Z, the identity element in (Z, *) is 1 and for every z in Z 1 is the only element such that 1 * z = z * 1 = 1. 1 is not in S1 since abs(1) is not greater than 1, therefore, S1 cannot be a monoid
I definitely need associativity of R, and a stronger statement that rz = zr for complex z and real r.
Not sure where I would need associativity of complex multiplication. Point taken, though, that this approach is not the best way to show complex multiplication is associative.
@Jakobian This, more or less. But also: you need to know it in your soul, and the only way to get there is to write it down yourself. Probably five or six times, definitely at least twice without constantly looking at the book.
The point, @nickbros123, is that the goal is to really internalize the mathematics, and that doesn't happen from reading alone.
> 1-8 Theorem. If $A \subset \mathbb{R}^n$, a function $f: A \to \mathbb{R}^m$ is continuous if and only if for every open set $U\subset \mathbb{R}^m$ there is some open set $V\subset \mathbb{R}^n$ such that $f^{-1}(U) = V\cap A$.
I have been thinking some time about the "if" direction and I feel like I'm going in circles.
Attempt. Let $\epsilon > 0$ and $x_0\in A$. Now $U=\{y: |y-f(x_0)|<\epsilon\}$ is open in $\mathbb{R}^m$. By hypothesis, there is a $V$ open in $\mathbb {R}^n$ such that $f^{-1}(U)=V\cap A$. Notice $x_0 \in V$, therefore there exists $\delta > 0$ such that $B(x_0,\delta) \subset V$. Then $$\forall x \in A, |x-x_0| < \delta \implies x \in V\cap A \implies |f(x)-f(x_0)| < \epsilon.$$
