/sci/ - Science & Math

retard here

what the hell is math research? do you fags just sit down and think of stuff all day like philosphers and call it research or do you actually do productive things like testing theories

>>9715573
>what the hell is math research?
You know all the theorems written down in all those textbooks? Someone had to prove those. Also, coming up with conjectures, gathering data to support conjectures, etc.

Some recent research: https://arxiv.org/list/math/recent

>>9715573

Depends. There are mathematicians that essentially sit around proving interesting stuff, there are some devising new models and algorithms.

What's a nice list of facts about self-adjoint differential operators

>>9715564
Why are your books so expensive? I don't wanna read some trashy russian pdf. Come back to life and reduce the price pls.

*plays fast and loose with vector and variational calculus*
[eqn]
I[u] = \int_\Omega F(\mathbf{x}, u, \nabla u) \\
\begin{align}
\delta I[u] = 0 &= \delta \int_\Omega F(\mathbf{x}, u, \nabla u) \\
&= \int_\Omega \delta F(\mathbf{x}, u, \nabla u) \\
&= \int_\Omega F_u \delta u + F_{\nabla u}\cdot \delta \nabla u \\
&= \int_\Omega F_u \delta u + \underbrace{\int_\Omega F_{\nabla u}\cdot \delta \nabla u }_{*}
\end{align}
[/eqn][eqn]
\begin{align}
* &= \int_\Omega F_{\nabla u}\cdot \nabla \delta u \\
&= \int_\Omega F_{\nabla u}\cdot \nabla \delta u + \left(\nabla \cdot F_{\nabla u}\right)\delta u - (\nabla \cdot F_{\nabla u})\delta u \\
&= \int_\Omega \nabla\cdot(F_{\nabla u} \delta u) - (\nabla \cdot F_{\nabla u})\delta u \\
&= \underbrace{\int_\Omega \nabla\cdot(F_{\nabla u} \delta u) }_{= \oint_{\partial \Omega} (F_{\nabla u} \delta u)\cdot \vec{\mathbf{n}}} - \int_\Omega (\nabla \cdot F_{\nabla u})\delta u
\end{align}
[/eqn][eqn]
\begin{align}
\therefore \delta I[u] = \oint_{\partial \Omega} (F_{\nabla u} \delta u)\cdot \vec{\mathbf{n}} + \int_\Omega (F_u - \nabla \cdot F_{\nabla u})\delta u = 0
\end{align}
[/eqn]
Dirichlet boundary conditions [math]u|_{\partial \Omega} = u_0\ (\to \delta u|_{\partial \Omega} = 0)[/math] and the fundamental lemma of variational calculus ([math]\delta u[/math] is arbitrary) [math]\implies F_u - \nabla \cdot F_{\nabla u} = 0[/math] aka the Euler-Lagrange equations.

I'm looking up numerical methods for DEs and am doing an example with the implicit trapezoidal method.
If I have something like:
[math]
y' = -y + x
y(0) = 0
[\math]
and applied the method, would rearranging instead of using a fixed point method resulting in
[eqn]
\left(1 + \frac{h}{2}\right)y_{i + 1} =
\left(1 - \frac{h}{2}\right)u_i + \frac{h}{2}\left(x_i + x_{i+1}\right)
[/eqn]
fuck up convergence, order, etc.?

>>9716108
Sonofabitch
[math]
y' = -y + x
y(0) = 0
[/math]

[math]
\left(1 + \frac{h}{2}\right)y_{i + 1} =
\left(1 - \frac{h}{2}\right)u_i + \frac{h}{2}\left(x_i + x_{i+1}\right)
[/math]
inb4 fucking shit up again

>>9716110
>>9716108
Kill me.

[math]
y' = -y + x \\
y(0) = 0
[/math]

[math]
\left(1 + \frac{h}{2}\right)y_{i + 1} =
\left(1 - \frac{h}{2}\right)y_i + \frac{h}{2}\left(x_i + x_{i+1}\right)
[/math]

Retard here
is this true?

>>9716121
Yes.

>>9716124
thanks i might be able to do my trig homework now

>>9716121
Yeah? Why wouldnt it be?

>>9715652
a mathematician condenses years of insights and research into a book that might take him years to write and that only a few people will buy, and you call that unreasonable?

>>9716121
Underage plz.

>>9716145
Pfft, some obscure russian text is in no was similar to shitty books that are mostly just books with a lot of problems for brainlet professors to leave as a set and for colleges to force students into buying a text because they have ties with the publisher. If I really want a book is because my field uses it intensively or becauae I like it as a good reference, so I'm willing to pay up for that, but ffs, some shitty undergrad text I will never use again is a waste.

>>9715564
I found this a while back. "The World of Mathematics", four volumes about maths. Seems very comprehensive and I think it's something worth having around if you're interested in math.

https://archive.org/search.php?query=creator%3A%22James+R.+Newman%22

>>9716248
Eh, I bought the set a couple years ago from a used bookstore for a couple of bucks, but I haven't read it yet. It seems to be interesting from a historical perspective.

How do I become intelligent in mathematics? The highest level math I've taken is Calc II and I still feel ignorant on the subject (which is not a surprise). I want to be able to understand and apply math profoundly - and I really don't expect to get a comprehensive education on the subject by just going to classes.

>>9715898
1. It only requires [math]u \in C^1[/math] and [math]L \in L^1[/math] to write down the variational problem, while the EL equations require [math]u\in C^2[/math] and [math]F \in C^1[/math]. Shrinking your space on which [math]u[/math] lives while keeping the boundary conditions will sometimes leave you with no solutions at all.
2. The variation [math]\delta I[/math] is only the form you wrote down only if [math]I[/math] is Frechet differentiable on the Banach space that [math]u[/math] lives on.
3. The EL equation is a necessary condition for the extremization of the variational problem. There may exist extremizers that do not satisfy the EL equations.
4. What does [math]u|_{\partial \Omega}[/math] mean? For functions in the Lebesgue class there is no unique way of assigning values to the boundary of a function. In general a Sobolev trace map [math]\operatorname{tr}:L^p(\Omega) \rightarrow L^{p^*}(\partial\Omega)[/math] exists, but it is surjective. Only under very stringent conditions (that may not be compatible with the regularity of [math]u[/math] used in defining [math]I[u][/math]) is this an isomorphism.

>>9716750
>[math]L\in L^1[/math]
Meant [math]F\in L^1[/math]

>>9716750
1. That's why all the good methods for solving variational problems numerically rely on the integral (so-called "weak") form.
2. Nerd shit.
3. See (1.)
4. Nerd shit.

>>9716819
>numerically
Where do you think you are? /eg/ is that way.
>weak form
That still requires [math]W^{1,p}[/math]-regularity, which does not mediate any of the concerns that I presented at all.
Read a fucking book for once.

>>9716750
>>9716829
Where do you think you are? /taiwan/ is that way.

How do I evaluate the indeterminate form here

>>9716907
>How do I evaluate the indeterminate form here
What indeterminate form?

>>9716907
Ask in >>>/sci/engi/.

>>9716921
Meant to be the antiderivative [math][x^ne^{-x}]^{\infty}_{0}[/math]

>>9716939
[math][x^ne^{-x}]^{\infty}_{0}[/math]

>>9716939
no idea how latex broke here it worked in the preview

basically terms like
[x^n*e^(-x)], x from 0 to inf
that occurs in antiderivatives

>>9716829
>Read a fucking book for once.
I'm already reading a book, specifically this one:
https://www.amazon.com/dp/1107022584

Are you saying that I should put that book down (metaphorically, since it is an ebook) and read one on functional analysis instead?

>>9716967
>Are you saying that I should put that book down
Yes.
>read one on functional analysis instead?
No. Read Struwe.
https://www.springer.com/gp/book/9783540740124

>>9716973
Hmm thank you for the suggestion but I like the current book I'm reading better. Here is an excerpt illustrating why.

>>9717022
Suit yourself. At least you know that you're handicapped.

Is there something like a "quantum" or "dark" set? A set which may be empty and yet may have elements at some point in time. I'm thinking of something like the phantom cobordism phenomenon in TQFT.

What are the evolutionary benefits of classifying modules over a ring using their socles instead of actual isomorphisms? This is strictly weaker than isomorphism, as every isomorphism of modules restricts to an isomorphism of socles, but every module and its injective envelopes have isomorphic socles. It seems like a nice idea to use them, but does it yield anything nice?

>>9716121
>Retard here
This wasn't necessary.

>>9716121
[math] (\frac{a}{b})^2 = \frac{a^2}{b^2} [/math]

>>9716907
Ask yourself "What's the derivative of x e^-x ?" and do some rearrangements

Is axiom of multiple choice equivalent to axiom of choice? Does AMC imply AC? here http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf it says that AC can be deduced from AC, here https://ncatlab.org/nlab/show/axiom+of+multiple+choice it says it's not
What am I missing?

>>9716721
Going to class gets you connections with professors. The only way to get into serious math shit is by having good professors help you out.

>>9717892
This.

>>9715573
>sit down and think of stuff all day
yes
>testing theories
these are called conjectures. When you have no idea how to prove a conjecture but want to understand it better, you check examples.

Should I just not bother with abstract algebra and go straight into universal algebra?

>>9718110
It could make life easier if you encountered things like free abelian groups or tensor products of modules first, so you would be more familiar with universal properties. On the other hand, I haven't read any textbook on universal algebra, so I don't know if they come with an introductory chapter on such things.

>>9718110
>Should I just not bother with abstract algebra and go straight into universal algebra?
Why are you asking this in a maths thread?

>>9718114
why not

>>9718121
>why not
What does it have to do with maths?

>>9715564
Why is calculus so generally disliked/unpopular?

>>9718124
nothing

>>9718124
what

>>9716121
Ofc

>>9718134
>what
What what?

>>9718110
I recommend this book:
https://math.berkeley.edu/~gbergman/245/3.0.pdf

But even the author says this:

"A student who has seen the concept
of free group introduced, but isn’t sure he or she thoroughly understood it would
be in a fine position to begin"

So I'd say learn some abstract algebra first, Dummit and Foote is nice, and universal algebra is a pretty cool topic, best of luck-

>>9718110
No, you should have at least a basic knowledge of group/ring/field theory. You should study them anyway. Then try to get into model theory and/or category theory. Then if you are interested in some particular field such as module theory you will find adeguate books. 'Universal algebra' is an outdated field.

>>9715564
Babby (and brainlet) first year maths major here. What are some good “how to prove” books that go over proof methods? Also how do I network with profs? All my classes have several hundred people in them

>>9718182
>What are some good “how to prove” books that go over proof methods?
Your textbooks, as you just try to understand how the claims are proved in those and suddenly you are the proof master.

>>9718182
Polya's "How to Prove it", and professors should notice you and your greatness. If they don't, you'll never make it.

>>9718110
You can also study it from a category theory perspective, check the handbook of categorical algebra, 3 volumes I think, probably a better choice

>>9718110
Putting aside the fact that nobody does universal algebra any more (because it's been largely subsumed by category theory), the entire point of the field is to generalize a bunch of disparate examples into one construction. You can't see this unless you actually know what the examples are.

>>9718182
Not many people develop relations with their profs in freshman courses. You can't really find anything worth talking with them about in a calculus course, and to them you're just another calculus drone right now anyway.
You can meet people in extracurricular stuff that the math department does (I always recommend people attend talks at their department if they're interested in math, even if they may not understand much), and you'll find it easier to get noticed in upper-level courses that are small enough for the prof to care about each student in them.

>>9718176
thanks

I am looking for a topic in modern algebra that I could feasibly research and present on over the course of a semester-long undergraduate seminar. I have had two semesters of algebra, but I do not know of any suitable topics. Would any of you be willing to recommend a few topics or point me in the direction of where I could find some?

>>9717876
>What am I missing?
Neurons.

>>9718845
Was there a specific section of your algebra sequence you enjoyed more than the others?
It's difficult to find an idea in "algebra" because it's so broad a field. Narrowing down your search to something like ring ideals or symmetry groups would make it easier to find a topic.

Ok, sorry for the cringy question but where can i get a gf mathematicians? I'm a student and i would like a gf to share maths problems. Where can i meet one? I know that i can meet one at my uni but it's too late (year ended for me). Can i ask on tinder for mathematicians only? Am I just too picky to search for a girl that can do math?

>>9718957
Why do you need a girlfriend _right now_?
Jerk off for the summer and then hit on some girls in your fall classes

>>9718845
how much do you know ?

>>9718193
>Polya's "How to Prove it"
there's no such book written by Polya

>>9719097
I'm doing nofap so i don't think it's an option and I want a gf because I want some affection.

>>9719191
>I'm doing nofap

If you call hyperbolic geometry Lobachevskian geometry you are a fucking commie.

>>9718957
You can't. If one of the extremely rare research-oriented female math students wants a math bf, she could easily go for a postdoc/prof. I know of at least 3 such cases, which represents about half of the females that planned to go into research.

>>9718845
classification of real division algebras

although one of the biggest results of the above comes from topological considerations, there's a lot of different classifications from Hurwitz, Zorn, Frobenius, and the one from Adams

>>9718957
I feel you anon. I for one have resigned myself to the fact that I'm going to die a kissless, hugless virgin. Doing mathematics is better than doing any woman anyway.

>>9719368
https://www.youtube.com/watch?v=gXlfXirQF3A

>Humans exhibit a diversity of strategy “choices” that are solutions to the allocation problem between mating and parenting. Males can devote most of their effort to mating effort, usually involving competition with other males. Male commitment to parenting effort is not common in mammals but there are familiar examples like beavers, coyotes, gibbons, and some humans. In the jargon the polar strategies of male mammals are called “cad” and “dad” strategies.

>Females have a more restricted set of strategy choices because of their engineered commitment to parenting. At one extreme a human female can seek a dadly male who provides resources like food and protection to their joint offspring. At the other extreme, a human female can pay little or no attention to her mate choice, instead letting the guys work things out. In the jargon these female alternatives are called “coy” and “fast”.

>You can find a more detailed account of this game between the human sexes works in a chapter of our book (that the editor discarded as “too academic”) on our website here*. Briefly we are likely to find dad males/coy females in ecological situations where male labor and resources are critical for successful reproduction. Think of labor-intensive agriculture, European peasants and Asian farmers, as examples. In the United States in the past, “working class” meant stable mated pairs who together provisioned and cared for children. An archetype of working class in American television was Archie Bunker.

(*) http://the10000yearexplosion.com/human-cultural-diversity/

>>9719768
>Notice that in each of the above descriptions there are two hands clapping: in cad/fast social systems neither a coy female nor a dad male does very well while in dad/coy systems neither a fast females nor a cad male does very well. The two polar social types are deeply rooted in contemporary politics. The zany feminism of the 1980s (“a woman needs a man like a fish needs a bicycle”) precisely advocated the cad/fast setup. Our religious right with its chatter about “the natural family” and “stable marriages” and the like pushes hard for a dad/coy world.

Reading between the lines: nerdy male types lean very strongly towards a "dad" reproductive strategy. But contemporary females can largely take care of themselves, opting for a "fast" reproductive strategy. Not all humans pursue a particular strategy with equal intensity. Problem is that nerdy (especially mathy) males are high outliers on the "dad" end of the spectrum. What complicates matters in countries like the US or Canada, is that their populations are very diverse, being made up of immigrants. And we're not talking here merely about racial diversity, but also ethnic diversity. Southern Europeans for example tend to go for cad+fast strategies more often than northern Europeans, reason being that living in southern Europe wasn't as harsh as living in northern Europe, so a high degree of parental investment wasn't as necessary for survival.
Thirdly, women's behaviour viz. reproduction is more plastic than men's, meaning, in any given population there are fewer exclusive "coy" females than there are "dad" males (likewise, usually there are fewer exclusive "fast" females than there are "cad" males). Most women are equally adept at adapting both strategies, changing their choice depending on the environment they are in.

tl;dr - we're fucked.

What prompted this non-math related dump is >>9719191
>I want a gf because I want some affection.
Most women in your cohort aren't looking for affection. They're looking for dick.

>>9719768
>>9719770
>>9719781
>>9719784
Fuck off to some other thread.

>>9719785
No. Our fellow mathematician needs help.

>>9719788
Make a separate thread and link it here to you can help him.

>>9719788
>>>/adv/

>>9719761
It's the anime.

>>9716750
>3. The EL equation is a necessary condition for the extremization of the variational problem. There may exist extremizers that do not satisfy the EL equations.
>it is a necessary condition for extremization but it is not a necessary condition for extremization

>>9716111
whats your problem with this?

>>9718180
>>9718199
So basically you guys have no idea what universal algebra is about, got it.

>>9718176
You might also check out Cliff Bergman's book on the subject. (No relation to George Bergman.) George says that he's not a universal algebraist, he just uses the stuff. Do with that what you will.

>>9720047
t. grad student that got memed into an irrelevant field by his advisor

>>9720066
Yeah it sucks following your own path instead of being a slave to trends. If only I could study something fashionable.

>>9719877
?

>>9715564
Is there some universal formula of optimizing sum formulae?
Or it's a job of computers calculating sums?

>>9720221
>sum formulae
You mean... any expression involving a sum? That's way too broad.

How exactly do you deduce this from excision? Note M is an n-manifold

>>9720268
X = M

A = M-{x}

B = open ball around x


Then by excision H(B,A∩ B) = H(X,A)

Obviously B = R^n and so A∩ B = (R^n)\x

>>9720306
makes sense, thanks

>>9719964
Are you retarded? Actual honest question.

What trend line should I draw here?

y = a * e^b(x - c) with total least squares/deming/perpendicular distance regression?

>>9720778
Not enough data. Don't be that retard.

>>9719768
>>9719770
>>9719781

>>9720798
Here is the full data series. I am not sure how to visualize it better, but there is a clear trend to me.

>>9720811
Also, how to perform a deming regression in excel?

>>9720778
>>9720811
>>9720813
Why are you asking this in /mg/? Use >>>/sci/sqt/ or >>>/sci/engi/.

>>9720814
Didn't find any statistics thread, thought this would be closest.

>>9720778
Here, I drew a rollercoaster for you.

>>9720821
see
>>9720811

What does [n] here stand for? I've never seen this notation before.

>>9715634
they are self-adjoint

maximum integer closest to n

>>9721249
[n] = {1,2,3,...,n}

>>9721261
>[math]I\subset [n][/math]
>[math]I[/math] is a subset of an integer
Fucking retard.

>>9721292
im sorry that was dumb sorry!

Doing some research on this pattern. Is this what is known as the Thue Morse Sequence? For someone without much mathematical knowledge could you explain the significance? Thanks

1221211221121221

>>9721419
>the significance
None whatsoever.

How do I evaluate the indeterminate forms in the antiderivative of this integral

>>9721620
>How do I evaluate the indeterminate forms in the antiderivative of this integral
What are the indeterminate forms?

>>9721622
Something like this

>>9721625
>Something like this
There's no indeterminate forms in your picture.

>>9721626
nevermind I just havn't done integration for a long time and forgot I can't just substitute infinity like definite integrals

>>9721620
just gonna guess but you can express the (x-1)^n term as a binomial coefficient series then pass the integral into the sum

>>9717028
I don't see a reason to go full autism with formalism for formalism's sake unless you're going to stick everything on a computer for algorithmic proof verification -- otherwise it just feels like a half-measure.

>>9715564

Am I the only one here that’s really good with ”real maths” like geometry and modelling stuff, but strugles with ”fictional maths”?

>>9722289
>fictional maths
Such as?

>>9722302
the ones he can't understand

>>9722302

Purely theoretical stuff like some integrals which aren’t related to the real world. Don’t get me wrong I get them, but I find applied stuff so much easier.

>>9722196
>formalism for formalism's sake
Where did she mention "formalism for formalism's sake"?

>>9722319
>I find applied stuff so much easier
That's a common issue among the less intellectually gifted ones, i.e. retards.

>>9722331

I find it hard to believe that I’m retarded. Also you know the mean iq of physicists is higher than that of mathematicians? Seems like applied stuff is higher iq.

>>9722336
>I find it hard to believe that I’m retarded.
That's a common issue among your kind. Feel free to discuss your experiences with physics, IQ and other non-mathematical stuff in suitable threads. I recommend >>>/sci/pg/ or >>>/sci/sqt/ or >>>/sci/engi/ or >>>/toy/.

>>9722340

Why does mentioning physics trigger you so hard? I just wanted to have a nice discussion but now I have to mention how physicists have higher iq, earn more money, are more employable, are more respected, etc.

>>9722345
Discussions about non-mathematical topics belong in other threads. Feel free to make a new thread discussing how empirical string theorists have higher IQs than mathematicians.

>>9722355

You’re the one who started talking about non mathematics retard. It’s funny how you backpedal now that you’ve been told.

>>9722358
>You’re the one who started talking about non mathematics retard.
How so? Your very first post begins with discussing non-mathematical garbage like "modelling stuff". Feel free to make another thread if you want to continue the discussion. I'm sure the people here would be glad to join you in having deep conversations about higher IQs of empirical string theorists.

>>9722366

How is geometry non mathematics you fart sniffer?

>>9722368
The kind of "geometry" your ilk usually studies can't really be called geometry. And "modelling stuff" is clearly non-mathematical and thus belongs in threads like >>>/sci/pg/ or >>>/sci/engi/ or >>>/sci/sqt/ and so on.

You worthless fucks drove out the people who actually discussed mathematical topics here. I hope you all and everyone you hold dear die painfully.

>>9722376

>Geometry isn’t maths
Now you’re just going full retard. I hope someone intelligent actually responds to me instead.

>>9722385
Are you even capable of reading and comprehending basic English sentences? My post clearly states that what you call "geometry" isn't really geometry in any mathematical sense of the word. Maybe you could try discussing it at >>>/sci/engi/?

>>9722388

>Geometry isn’t real geometry
Please stop.

>>9722394
Confirmed for literal retard. Come back to my posts when (or rather if) you gain the ability to understand basic sentences.

>>9722404

I fully understood it you asswipe. Your entire argument is based on the false assumption that when I say feometry I don’t mean all geometry. In conclusion you’re a faggot.

>>9722409
>faggot
Why the homophobia?

>>9722321
It recommended a book that was overladen with formalism at the expense of an intuitive treatment of the subject matter. Probably just another reference manual they confused for something that you're actually supposed to read.

>>9722459
Quite sad that anything you don't understand you treat as "unintuitive formalism".

>>9722540
Okay whatever you say.

Sorry for the pedantic question. I'm learning differetiak geometry and I think I'm getting the theory pretty well, but I find it frustrating that there are few concrete examples to see if everything is fine. I'm trying to do the exercises, but all of them don't involve a lot of concrete ejamples with some geometric object. So I go to a general relativity book, and the questions are weird and make no sense formally but at keast some of them are concrete. The problem is that looking at solutions, most of them are from physicists and it becomes a lot of plug and chug without care, in particular, I'm trying to compute the elements of the rieman tensor. However, because I'm trying to do everything formally from the ground up (yea I know it's tedious but I need my sanity checks), and I find a lot of the solutions involve the standard parametrization of the sphere. Now, I know this can be used as an atlas because there is a whole halfmeridian were the continuity gets fucked, so my question is if I can then just say that I take four of these parametrizations (to include the poles) and then just work the same in each one? By this I mean is that if from that I can conclude that globaly, the curvature is constant and positive?

>>9722708
Can't be used as an atlas*

>>9722708
assuming you're speaking about the 2-sphere, you need at least 6 if youre using hemispheres, just saying

>>9723127
You nees two, consider the stereograpgic projection from south and north pole.

>>9723134
yeah, but >he said >he's using the standard parametrization, which usually is taking open hemispheres over each pole

>>9722708
also alternatively you could show that there is a diffeomorphism between the charts

>>9723137
The standar parametrization only has problems in half of a meridian, not a complete great circle.

>>9723143
yes, let's say you put have two hemispheres, one with top being north and bottom being south pole. You have problems in half meridian. Now you define one point to be east and its opposite to be west, and put two hemispheres there. Now there are still two points that are not covered by the chart, and those are precisely the two points orthogonal to both east/west and north/south poles

>>9723150
>>9723143
the cover of do carmo's book actually has what im saying

>>9723151
But that's not the standard parametrization. That's using the algebraic equation. The standard parametrizatilon is the spherical coordinates.

>>9723154
Oh right XD im used to algebra :D

>>9723172
Are you being ironic? I meant x^2+y2+z^2=1

>>9723192
half ironic half legit

but x^2+y^2+z^2 is not a valid parametrisation in local charts of the sphere, if anything it should be the spherical polars

>>9723251
I mean, taking the charts, as they are in Do carmos book. Z=+-sqrt(x^2+y^2) x=+-sqrt(...

>>9723292
then you DO need 6

??

>>9723300
With that parametrization. Im taljing about spherical polar.

>>9715564
Suppose f is a nonnegative function on [0,1] whose Riemann integral is 0, and let A be the set on which f is strictly positive.

Is it possible to show A has measure 0 without simple functions or measure theory directly?

>>9723302
retard leave me alone

>>9723304
The least amount of charts you need to make an atlas is two. Taking stereographic projections. that there are other possible atlases with more charts doesn't contradict this fact. Spherical polar cooridnates have trouble with half a great circle, so you only need four of these charts. What so fucking difficult about this?

>>9723303
legit no

>>9723303
>is it possible to show something about measures without using theory developed to study measures?
lmfao

>>9723313
>>9723317
We've defined measure 0, but not the lebesgue measure.

>>9723303
Hmm, you can start by showing that if there exists an interval contained in [0,1] were the function is strictly positivez then then the integral must be non 0. From that, you can take the theorem that a function is rieman integrable, then the Jordan measure of their discontinuous points must be 0. And the points where it's strictly positive must be discontinuous I believe or there can't be a set of positive measure of continuous points or the integral would be positive.

>>9723341
That was more or less my idea. What I couldn't figure out is how to get around the case where the set A is dense in [0,1] (e.g. Thomae's function). It doesn't seem possible to get an arbitrarily small cover of A in general.

Can someone give me an intuitive explanation of when it's the case that

[math] \mu^{*}(X) = \mu(X) [/math]

Where [math] /mu [/math] is the Lebesgue measure.

>>9723373
You can define outer and inner measure by approximating the set with those n-intervals, and then taking the infimum or supremum, respectively, of the sums of geometric measures, you know, the usual way. To have a reasonable notion of measure, you would then like to have its outer and inner measure coincide. Therefore, [math]m^*(X)=m_*(X) \Rightarrow m(X)=m*(X)[/math].

>>9723379
Oopsie, [math]m(X)=m^*(X)[/math], not [math]m*(X)[/math]. Silly me~<3

Why is linear algebra so boring

>>9723309
Lol scratch that you only need two charts using polar spherical. Around z and y axis.

>>9723730
It's too clean. You need some bullshit that doesn't work how you want to spice things up

>>9723841
>You need some bullshit that doesn't work how you want
So linear algebra?

>>9724179
>>>/r/eddit

>>9724185
Excuse me?

I'm trying to compute the Feingenbaum constants [math]\alpha[\math] and [math]\delta[\math]. I've found a paper by Molteni that gives some details into the calculation, however I am not versed enough into approximations to understand everything.
Could someone give me insights in the part after (2.4) in page 2 ? I dont understand how he gets those initial coefficients.
Link to the paper : https://arxiv.org/pdf/1602.02357.pdf

>>9723841
What is the spiciest field of algebra?

>>9724464
Universal magma theory. My advisor is currently working in that field and I'm thinking of writing my thesis about some related topics.

>>9715652
shitty publishers, Manfredo's books are literally like R$30~R$60 here in Brazil lol.
thats like 8 dollars lmao.
https://loja.sbm.org.br/index.php/geometria-riemanniana.html
https://loja.sbm.org.br/index.php/geometria-diferencial-de-curvas-e-superficies.html

>>9724179
Linear Algebra is really intuitive. You probably need to approach it with a different mindset.

Whats the best introductory measure theoretic probability theory book?

>>9724646
stewart calculus

>>9724605
When did I ever say anything about it being unintuitive?

>>9724646
Ash if you don't know any measure theory
McKean if you do

>>9724787
if you had intuition, you would be able to use it how you want

>>9724787
>>You need some bullshit that doesn't work how you want
>So linear algebra?

Intuition: the ability to understand something immediately, without the need for conscious reasoning.

>>9724605
oh really? what's a flat module, intuitively

>>9724852
what kind of linear algebra course teaches homological algebra

>>9724864
>what kind of linear algebra course teaches homological algebra
What kind doesn't?

>>9724852
>intuitively
What do you mean precisely?

>>9724864
Any kind which deserves the name "linear algebra".

>>9724787
Well, you implied it.

I don't know if this is the proper place to ask, sorry if it isn't.

hello, I've been reading GEB(Godel Escher Bach), and I'm having trouble with understanding one of his example systems, the tq-system. The author describes a system xtyqz where x,y,z are strings with dashes and x*t = z.

The axiom schema: xt-qx, whenever x is a hyphen-string;
Rule of Inference: If xtyqz is a theorem then xty-qzx is a theorem; E.g --t-q-- is an axiom so --t--q---- is a theorem.

The author then introduces a way to define composites with Rule:
If x-ty-qz is a theorem then Cz is a theorem.

My understanding of a composite is that it is a value that can be divided evenly by itself, 1, and one or more other numbers. However, based on the my understanding, C(nothing) is valid because -t-q- is a theorem. Am I misunderstanding something? I don't want to move on without fully understanding this sample system. My best guess if that zero is not defined in this system, since that is what the author implies.

Page 72(https://www.physixfan.com/wp-content/files/GEBen.pdf)) if you want the original description

>>9724852
Think geometrically.

Flat R-modules correspond to certain sheaves over SpecR.

Geometrically, they are to locally-free sheaves over SpecR what fibrations are to fiber bundles in algebraic topology.

What's the word for all possible arrangements of fixed sized ordered tuples where each element can be one of elements from a (finite) set called?

Not sure if it's called permutation or combination and what notations do I use for it? e.g. for the set of tuples of size 3 with elements from {0,1}

>>9726178
https://fr.wikipedia.org/wiki/Arrangement

Got 338 as my answer for this question

Just want to confirm if it's correct

>>9726389
>hint tells you to look at a part you havent shown in pic

dumbass

Guys, I'm getting stomped in calculus because my high school basic math is pretty weak. Can someone please point me to direction of light and link me some resources so I can learn shit like logarithm from the ground up? Thanks in advance.

>>9726596
Algebra stuff would be of great help too.

Anyone want to have a go at these problems? I'm too drunk and high to do them. I will send $50 in bitcoin to anyone who does it correctly.

>>9726608
Note that a representation of gl(1) on a vector space V is characterized by the image of a, say A. Then, a submodule of V is exactly the same as a subspace of V that is invariant under A.
(i) trivial
(ii) If A is diagonalisable, then either A is scalar, in which case any basis gives a decomposition of C^2 as a direct sum of two submodules of dimension 1 (hence irreducible), or A has distinct eigenvalues, in which case C^2 is the sum of the two eigenspaces (which are submodules as A-invariant subspaces).
(iii) If A is not diagonalisable, then there is a scalar t such that A-t*I is nonzero nilpotent. Then, Ker(A-t*I) is a nontrivial proper submodule. It's actually the only one (indeed, a nontrivial proper submodule of C^2 is necessarily an eigenspace), hence this module is indecomposable.
(iv) Consider a representation of gl(1) on V := C^d. It is completely characterized by the image of a, which we still denote by A.
Then, a submodule of V is precisely the same as an A-invariant subspace of V.
Assume that V is indecomposable. Then, by the characterization of submodules as A-invariant subspaces, the Jordan normal form of A has exactly one block with eigenvalue t, say J_d(t). Hence, V is equivalent to the representation obtained by setting A = J_d(t).
Conversely, given any t, the representation of gl(1) obtained by setting A = J_d(t) is indecomposable. To do this, note that in that case the invariant subspaces of V are precisely the (A-t*I)^k, for k = 0..d, hence we cannot have a nontrivial decomposition of V as a direct sum of submodules.
Finally, note that these representations are pairwise inequivalent.

Do you ever get discouraged at how advanced most of the research being done now is? It feels like you need to spend a decade studying one very specific area to be able to contribute some tiny result that inches your sub-sub-field a tiny bit forward. I don't know if I'd be ok spending my life doing that. I'd rather study much more broadly and try to understand all of mathematics as best I can. Should I just give up and go applied in grad school? Or should I skip grad school and just go for the data science meme or something.

t. dumb undergrad

>>9726534
hint uses answer from earlier question and I won't bother to take a photo of it

Pretty sure you can do it using other ways

>>9727577
No because most of it is bullshit. Compare our current technological capacity with the supposedly advanced level of mathematics we've attained. Still burning dinosaur corpses, still can't do fusion, still can't predict the path of a hurricane more than a week ahead of time. If any of the "research" mathematicians were doing was really that important, we'd have solved some of these problems by now. Instead we've got armies of supposed geniuses trying to understand shit like IUT. What a waste!

>>9727759
I don't care about "importance", since some dumb businessmen would inevitably co-opt any useful discovery anyway. It's the sense of meagerness or futility that worries me.

>>9727577
>>>/adv/

>>9727577
Yes, but not really because of how difficult the material is. I would say lots of published stuff isn't even beyond the reach of a good graduate student to read except for translating all the super-niche jargon used.
The part of research I dislike is that most results do not push the field forward at all, not even a tiny amount, they just make it bigger. Everybody but the leaders of their fields in academia spends their lives building a colossal pile of pointless results that do nothing for the field except keep the author his job.
It's always been like this, you can go to the basement of your university library and flip through thousands of pages of stupid crap from the 50s that nobody has ever read too.

>>9720049
>(((no relation)))

>>9719781
Just go to church and marry a girl from there. That's what I did.>>9719781

>>9716108
Numerical methods are pathetic. Try using the symmetries of the ODE to integrate it at least once. HINT: Solvable algebras mean you can get the general solution.

>>9723151
The cover has quite a covering.

>>9723292
You're missing the 1's when you're writing the parameterization

$Z= \pm \sqrt{1+x^2+y^2}$

>>9727577
Have you taken any applied courses? They do things very differently in applied math. Mostly numerics.

>>9722381
We should make Upper Mathematics General /UMG/ for graduate mathematics and above only. I am tired of all the calculus posts.

>>9723730
Because row reduction is boring and everything comes down to that.

>>9715564
I've replaced my recreational programming time with math. I browse /sci/ now instead of /g/. I feel myself becoming infinitely more autistic. It's for the better, right?

>>9726178
>What's the word for all possible arrangements of fixed sized ordered tuples where each element can be one of elements from a (finite) set called?
That's called a permutation. Permutations can be defined on any set X. In general, the permutation group

[eqn] P(X) = \{ f:X\to X : f^{-1}\; is\; bijective \} [\eqn]

>>9726178
These are called Permutations. You can define permutations on any set.

[math] P(X)=\{ f:X \to X : \exists f^{-1} \} [/math]

>>9723292
You're missing the 1's when you're writing the parameterization

[math] Z= \pm \sqrt{1+x^2+y^2} [/math]

>>9727994
>Numerical methods are pathetic
Your contributions to the field of mathematics are pathetic.

>>9728032
1-x^2-y^2

>>9728045
My mistake.

>>9728030
>These are called Permutations. You can define permutations on any set.
His/her description doesn't seem to require uniqueness of each element, sounds more like the elements of some n-fold Cartesian product of the set

>>9721620
Just use induction. Show that this holds for n=0,1, then assume that there is a natural number [math] k \geq 0 [/math] such that you proved this for all numbers up to this [math] k [/math]. Then use the cases [math] n=1,k [/math] to show that the formula holds when [math] n=k+1 [/math]

>>9715898
I don't know... Maybe you should use the Einstein convention to derive this in general.

>>9716121>>9716134
Good thing this is an anonymous board. Otherwise I would have given you an F since I talked about this in class as an example two months ago and you should've been taking notes.

>>9715621
<.01% of all mathematicians are in any kind of textbook.
So I'll ask again what the hell is math research?

https://arxiv.org/pdf/1805.01755
Two theorems about the P versus NP problem
Tianheng Tsui
Two theorems about the P versus NP problem be proved in this article (1) There exists a language L, that the statement L∈P is independent of ZFC. (2) There exists a language L∈NP, for any polynomial time deterministic Turing machine M, we cannot prove L is decidable on M.

>>9728106
I don't know enough CS to tell if this is a meme or not.

>>9716750
1. The boundary conditions are not a problem. Usually functions are assumed to have compact support in Calculus of Variations in non-compact regions, and satisfy fixed boundary conditions in order to derive the EL equations. If one has moving boundary conditions, then there are formula for this too. The EL equations are of Kovalevskaya form which means that local solutions necessarily exist by Cauchy-Kovelevskaya's existence theorem. Differential/Integral constraints which are not listed are the only things that can ruin this. One only needs to assume [math] u\in C^1 [/math] to derive the Euler Lagrange equations. Please read Gelfand-Fomin, first chapter, fourth section, Theorem 3.
2. The first variation does not exist if the functional is not Frechet Differentiable.
3. The EL equation is a necessary condition for all differentiable functionals defined on [math] C^1 [/math].
4. He's just using poorly thought shorthand. No one assumes any isomorphism properties. They aren't even necessary.

>>9716967
You should read Gelfand-Fomin. The best book on the topic. Functional analysis is mostly used in numerics. If you're interested in tha

>>9728083
It depends on what field someone is in. If someone is in Applied mathematics, they are looking at real world systems and trying to create a mathematical model to describe it based on experiments or some assumptions. Then they try to make predictions from their mathematical model that could be measured in a laboratory. In algebra, people try to classify groups, semi-groups, algebras, integral domains, etc and often apply these things to computer science or trying to solve discrete systems. In Geometry, people try to classify manifolds and sometimes use them to answer other questions, like solving DE's, or physics problems, etc. In analysis, people look at sequences and things defined through sequences and often use these things to solve DE's or develop estimation methods, etc. There are also people who don't quite fit into any of these fields strictly who kind of walk the line between all of this too. For example, there are a lot of people who do symmetry analysis of DE's who don't do any geometry and do maybe a little bit of analysis and applied math.

Does anyone have that 1 image of useful math formulas and properties from pre-calculus all the way to calc 3? It has a black background, white text, and at the bottom left has the "you should be able to solve this" anime girl.

>>9728083
>>9728132
As a follow up. Here is an example of a few big guys in mathematics and their research.

http://www-users.math.umn.edu/~olver/paper.html
http://www.crm.umontreal.ca/~wintern/publications/Articles.pdf
https://www.math.ucla.edu/~tao/

>>9728166
None of those links have anything to do with mathematics.

>>9728180
>None of those links have anything to do with mathematics.
define "mathematics"

>>9728187
This seems like one of those deeper philosophical non-mathematical questions which are best discussed at >>>/lit/.

>>9728180
Now I know you're just trolling. Nice try.

>>9728197
>you're just trolling
In what way? None of the garbage you listed is even remotely close to being mathematical. Maybe you should try discussing physics and engineering in appropriate threads? Look around the catalog, there are special places for you and your kind to gather.

>>9728199
Maybe you should read the papers.

>>9728205
I don't usually engage in reading CS and engineering "papers". Wouldn't want to accidentally get infected, you see. Have you tried sharing the deep insights of your fields at >>>/g/ and the engineering threads on this board? I'm sure they would welcome such discussion since those threads are the appropriate gathering place for you and your type of people to discuss engineering and CS, not a thread called "math general".

>>9728219
READ THE PAPERS YOU DIMWIT.

>>9728231
As I have said, I wouldn't want to get infected by CS garbage. You should really considering advertising the works of your colleagues over at places where people actually care about that sort of stuff. I would suggest looking into >>>/g/ and >>/sci/engi/ and maybe even >>>/sci/slg/ or >>>/sci/pg/. There are a lot of places for your kind to gather so you should make good use of them instead of trying to shit up these threads with off-topic links about "Image Processing and Computer Vision Papers" and other such trash.

>>9728219
>I don't usually engage in reading CS and engineering "papers".
Mathematicians use "we", not "I".

What are some terms begging to be well-defined, other than "number", "mathematics", and "point"?

This brainlet
>>9728392
Thinks that calculus is a liberal conspiracy to poison academia, can someone go tell him why he should kill himself or something? He's insulting pure math.

>>9728418
>can someone go tell him why he should kill himself or something?
Not your personal army, do it yourself.

>>9728427
I'm trying, but he's insulting math and misinforming others. If you don't feel like it fine, but this isn't a "personal army" request.

>>9728054
problem is that I don't know what's on the left side and I intended to actually calculate it using the formula here

>>9728489
Pick n=0,1. The LHS is 1! or 0!=1. Pick n=2. The LHS is 2. Pick n=3. The LHS is 6. I don't think you can actually derive the formula. No one ever derived the formula. What they did is take cases, n=1, 2, 3, 4, 5, notice a pattern, then write down what they think the formula would be. After that, they used induction to prove it actually worked. Many formulas involving natural numbers like the one you're looking at are found this way.

>>9728526
I need to clarify that the left side is the number of derangements in a permutation with n elements, NOT FACTORIAL

>>9728537
Use the formula for derangements and induction.

>>9728543
I don't need to prove anything at all using induction or not
I'm only looking for ways to manually evaluate the integral here

>Sudden eureka moment last night on a proof im working on
>couldn't sleep due to excitement until 3am
>only to find out today it's invalid due to an elementry arithmetics error early in the proof

How do I deal with this feel

>>9728106
That's some very nice Engrish.

>>9728106
>The statement is highly confusing, and what does this mean?!
I recommend everyone read this.

>>9728695
>elementry arithmetics error early in the proof
Stop doing things related to arithmetics.

I hate this general.

>>9728695
I don't know. It just goes away after a while. I have suffered through such episodes too many times, so I can attest to that.

>>9728994
>I hate this general.
Why?

>>9727869
Would anons there actually know enough to offer advice?

>>9728106
>The statement 3 is a φ ∨ ¬φ form, so it is ture

>tfw studying mathematical statistics
>tfw too bad at real analysis to get simple shit
Let [math]X_1,X_2, \dots ,X_n[/math] be a sequence of random variables such that [math]P(X_n = \frac{k}{n})=\frac{1}{n}[/math] for [math]k=1,2, \dots , n[/math]

Determine the limit distribution of [math]X_n[/math] as [math]n \to \infty[/math]

So let [eqn]F_{X_n}(x) = P(X_n \leq x) = \sum_{\{ k:\frac{k}{n} \leq x \}}\frac{1}{n} = \sum_{\{ k:k \leq \lfloor nx \rfloor \}}\frac{1}{n}[/eqn]
Then this is the riemann integral of f(x) = 1 right and thus as [math]n \to \infty[/math] then [math]F_{X_n}(x) = \int_0^x 1 dx[/math] right, but isnt it quite weird that it becomes a mapping from real to real when in the definition [math]F_{X_n}(x) = P(X_n \leq x)[/math] its a mapping from the reals to a subset of the rationals?

I promise to read baby rudin over the summer so I am less shit

>>9729338
>but isnt it quite weird that it becomes a mapping from real to real
Why would it be weird? The reason the reals even need to exist is that the rationals are not closed under taking limits.

>>9729356
So the act of taking a limit of a mapping to the rationals becomes a mapping to the reals? Where can I read more about it since a cursory glance through baby rudin didnt make it clear to me

>>9729359
It obviously won't turn _all_ of them into real-valued functions, but it certainly can. Lots of integrals end up doing precisely that; a riemann sum of only rational numbers ends up with a limit that becomes something irrational (for example integrating 1/x makes a logarithm out of rational values)
You might also check out the chapter on sequences and series of functions.

It may be a very idiotic question but i don't want to pester my math teacher over it (too much).
Anyway, i just learned a bit about (not sure what the correct name is) "base change matrices" (you have two bases in vector spaces and this matrix allows you to go from one base [A] to another [B], plus it's inverse allows you to do the opposite (so from B to A))

What i wonder is: i know (well, more of a gut feeling) that this can be used to change the reference frame in physics but i don't exactly understand how and how it is adapted to cases where:
1) Relative position is constant
2) Relative position is not constant but relative velocities are, directions don't change too (example: two or more points moving away/toward each other at fixed velocities)
3) Relative accelerations change, directions don't change (for example someone looking at a falling object in absence of other forces [ie the object goes straight down])
4) Acceleration isn't constant in module and direction but it follows a known pattern (planetary motion and how it translates if you go from heliocentric to geocentric to jupitercentric etc)

Pic related for the 1st case, how do i make all my bases, matrices and transitions when analyzing the position of A from O1 and then O2? Assume that i use the unit vectors and the "direction" of the reference frame doesn't change (ie: i translate it from O1 to O2)

Please explain it like you would with a brainlet with down syndrome (cuz i'm one). I assume the base would look like {e1;e2;e3} where ei is the relevant column of the identity matrix nxn (in this case 3x3)

Thanks

I have a few questions about an expression of the form [math]\displaystyle \lim_{y\to\infty} \partial{y} \frac{\infty}{dx}[/math]. Is it always defined for all values of dx? If so, why? Is it because dx is so infinitesimally small that we can use it to divide infinity without any problems? And what is meant by the subexpression [math]\partial{y}[/math]?

>>9729436
Oh, i forgot, the coordinates in the images are calculated with O1 as the origin, more for not getting bogged down with letters during explanation

>>9729443
>And what is meant by the subexpression ∂y
https://en.wikipedia.org/wiki/Partial_derivative

>>9729450
I don't get what this formalism is saying. Is it like dx but made into a partial function which may have undefined error values at certain points? Like when you take the derivative of the function dividing say the speed of light by infinity?

>>9729436
>>9729443
>>9729456
>>>/sci/sqt/

>>9729456
From what i understand, is the derivative in x of a function with y taken as a constant

What can you do with Reeb graphs?

>>9729443
>>9729456
dude what

>>9729492
morse theory

>>9729492
put my penis inside

>>9728994
I hate myself more than you hate this general.

https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news/

Place your bets, will Mochizuki be BTFO?

>>9730335
>literally tagged "shameless rumor-mongering"
k-keep me posted

>>9730335
>They are preparing a detailed writeup explaining the issue, which should be available publicly in the next month(s).
>month(s)

>If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope.

why can't textbooks just say that?

>>9730638
Because Stokes's theorem is a nontrivial statement.

bump for math in Brasil

>>9715564
If:
3.1415926535
3.1415926536
3.1415926537
3.1415926538
3.1415926539
3.141592654
3.141592653
3.141592652
3.141592651
3.14159265
3.14159266
3.14159267
3.14159268
3.14159269
3.1415927
3.1415928
3.1415929
3.141593
3.141592
3.141591
3.14159
3.1416
3.1417
3.1418
3.1419
3.142
3.141
3.14
3.13
3.12
3.11
3.1
3
2
1
0
Then:
π = 0
Q.E.D

>>9730807

1 ≠ √2
1.4 ≠ √2
1.41 ≠ √2
1.414 ≠ √2
1.4142 ≠ √2
1.41421 ≠ √2
1.414214 ≠ √2
1.4142136 ≠ √2
1.41421356 ≠ √2
1.414213562 ≠ √2

√2 does not exist

>>9715564
>Do Carmo died on 30 April 2018 at the age of 89.[7]

;_;

>>9730810
Y u do dis 2 me? :(
This was going to win me my Fields Medal.

>>9730731
How is a tautology a non-trivial statement?

>>9727577
Endoplasmic reticulum looking muhfugga

How do I find someone willing to help me through a textbook (for money)? Do I just go to the math department of my local university and start asking grad students?

>>9731568
What good would this do you? You can get any question you ever have about a topic answered online for free

>>9731568
You could actually take a class.

>>9731588
Takes too long/Stack Exchange is full of gay nerds.

>>9731602
I prefer to learn on my own terms, not someone else's.

>>9731667
>I prefer to learn on my own terms, not someone else's.
So you think you can be one of the cool autodidact kids? Welcome to the loser club for failed self-studiers. We are waiting for you.

>>9731682
I'm not sure what constitutes a "failed self-studier". I've learned plenty of things on my own, and had a fine time doing it.

>>9731695
Fuck off. Don't lie to me.

>>9730807
true in engineering logic which using the rounding axiom

Trying to calculate the intersecting content of some high dimensional convex polytopes. I figure the best way to go about this is by monte carlo raycasting, right? Anyone have any tips?

>be joe shmoe undergrad with no experience doing anything
>receive offer from prof to do a research project on shit I know nothing about
Is he just trying to blow leftover grant money? I can't see any other purpose given my gross lack of foundations in the field he works in

>>9732597
What is his field then, fuckface?

What's a good book for a rigorous introduction to Complex Analysis?

>>9731682
>self-studiers
There is no other kind.

Does this imply that in zf every set is a set of sets except the empty set?

>>9732645
Visual complex analysis

>>9733083
No the axiom of union does not imply that, in ZFC everything is a set.

>>9733087
>No the axiom of union does not imply that, yes the axiom of union implies that.

>>9733083
>set of sets
What do you mean?

>>9733133
>Implying you can have a union of non-sets.
What do you mean?

>>9733155
>>Implying you can have a union of non-sets.
Who are you quoting?

>>9733157
>Who are you quoting?
You are whom I'm quoting.

>>9733155
>Implying you can have a union of non-sets.
You can in many non-standard models of ZF (of which there are uncountably many).

>>9732645
Depends on what your background/aim is.
Gamelin's book is a good one, but goes beyond an introductory course.
You can try reading some notes such as Robert Ash's
https://faculty.math.illinois.edu/~r-ash/