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7:00 PM
@TedShifrin Yes, but that doesn't affect my idea, as far as I can see. But, as insinuated, I think making the idea precise (if it can be made precise) would be unheimlich difficult.
 
@Chris'ssis Ah... without the earlier one, it is hard to follow the later one.
 
Heinrich Unheimlich?
 
@robjohn I see DonAntonio has been suspended for voting irregularities.
3
Hm....
 
@TedShifrin Heinrich MirGrautVorDir
 
I wonder who could have caused that, @Pedro ...
 
7:02 PM
@PedroTamaroff it would appear so...
 
@Pedro hm, go on
 
@Studentmath Well, now I need to show we actually have a $K_{n,n}$.
The point now is to argue again like before, using the "maximization".
 
Yes
 
Well, and that's that.
 
You have to be rigirious regarding the maximization, as it is intutive but not obvious
But as you explained it before, it seems without fault
But wait, if you get to $K_{l,k}$, and afterwards to $l=k=n$, you don't have to go further.
 
7:07 PM
mr @Pedro, you ignoring me today?
 
@r9m how do you derive $x^2+y^2+z^2 \ge xy+yz+zx$ without diff calculus ?
 
@TedShifrin Sorry, I was a bit caught up with this combinatorics stuff.
I need to pass the next midterm and make up for the first.
 
Oooh noo failure?
 
r9m
@G.T.R well $x^2+y^2+z^2- xy-yz-zx = \frac{1}{2}((x-y)^2+(y-z)^2+(z-x)^2) \ge 0$ :-)
 
No problem, @Pedro ... Just checking ...
 
7:10 PM
@DanielFischer Not pleasant for his customees...
 
@r9m ah
 
@Studentmath I know of it, but I don't know it
 
@Mike 6.3.9 here (page 329) holds a small part of the proof, most is made by computers though. I don't ge the proof there too well still.. toihoctap.wordpress.com/2013/02/13/…
 
Talk about chutzpah!!!
 
I own West's book. Let me check.
Someone commented on my answer from 2 weeks ago saying he wants help with complex analysis.
 
7:12 PM
@r9m I switched the answer check to your answer because you taught me something :P
 
@Mike: I figured in summer the amount of homework begging would decline, but it seems to be getting worser and worser.
Wow, someone taught @GTR something? :D
 
@TedShifrin Ah, no, this is not a student. It's just someone with a poor grasp on english and manners.
@Studentmath That must be a new edition. 6.3.9 for me is the definition of thickness.
 
You talking about yours or the one I just referred to above? @Mike
 
The worsest is yet to come...
 
@TedShifrin Mine. Yours is terrible.
 
7:14 PM
@Mike yes, it's the second edition
 
I'm sure I'll be accosted for being rude to him with my comment.
 
Quite new.
 
@Studentmath I don't want to look at copyrighted stuff online (bad!), so can you guide me towards the page?
@Studentmath Hm, I have the second edition.
 
LOL ... duelling second editions.
 
Oh it's illegal?!
The question itself is on page 270.
 
7:15 PM
covers eyes and ears yet again
 
Has an hint at 514.
 
@TedShifrin Ya, why are you being rude to him?
 
@Studentmath Oh, I thought you were having trouble reading one of West's proofs.
 
yes.but it is a topology homework.pls help me in writing this solution. — user115608 4 hours ago
 
7:15 PM
@Pedro I still don't get why you are not done when l=k=n, since you have shown it to be $K_{l,k}$.
 
@MikeMiller Holy shit. This will be good.
 
Hmm @Mike: Martin is pretty accomplished.
 
@Mike ah sadly not that simple, trying to practice up for the test so doing few 'hard' questions.
 
@Studentmath Did I show that?
=P
 
Yeah, I can't help with this. I don't know what a lobe is, for instance. I also haven't thought seriously about graphs for a long time.
I probably never thought seriously about graphs.
 
r9m
7:16 PM
@G.T.R irk .. there is something in math110's answer too :P .. do you know weighted am-gm inequality ? he used it and later balanced weights by solving a system of liner equations that are necessary to get to the final expression ... :P its definitely not out of thr blue (he just hid his machinations) :P
 
@Pedro - ah, the maximilization part comes at the end
My bad
 
@TedShifrin Hence the wow. That guy's condescension is definitely not deserved.
 
@DanielF: This guy is acting a lot like B11b, whom I got suspended a while ago. He may be back with a new persona.
 
user105491
Does anyone in here participate in the homotopy theory chat room?
 
or personae :-)
 
7:18 PM
Well, @Mike, even we relative experts occasionally deserve to be condescended to :P
 
@r9m I prefer tortuous, enriching solutions to polished gems
 
Probably right, @skull ... I thought I saw something a few weeks ago.
 
This one's my guy. I didn't even properly answer his question, so I don't know why it's me he's asking.
 
You like turtles, @GTR?
 
@TedShifrin I don't think so, B11b scanned his/her beautiful handwriting and was generally polite enough to use complete words.
 
7:19 PM
@SanathDevalapurkar I don't think anyone in this room knows too much about homotopy.
 
r9m
@G.T.R LOL :P
 
I know some homotopy at the level of part of Hatcher's chapter (not the supplements!), but that's it. None of the categorical stuff.
 
@Mike: I've quit answering Rafik. I learned after one answer, if I remember correctly.
 
@MikeMiller Yeah, I know you know some.
 
user105491
@PedroTamaroff Oh. I had a question regarding infinitesimal thickenings of topoi, so I first decided to check if it could be answered here, and go there if necessary.
 
7:20 PM
lol
 
that's a whole lotta love (music national day in France, and there's a rock band performing right in front of my window (sleep? none!))
 
yeah, @Ted, I should have learned earlier - sometimes I don't pay attention to who the poster is
 
wow, @GTR, which arrondissement are you in?
 
@Ted Nation 11 ème. I'm gonna take some pictures to show you how crazy it is :P
 
ah, I was there several times last summer ... one of my friends lives near there
 
7:22 PM
@SanathDevalapurkar What is a topoi?
 
topoi is the plural of topos
topos is the romanization of the greek word τόπος, for place
 
What is a topos?
 
user105491
@PedroTamaroff It's the plural of topos. A topos is a category which looks like a category of spaces.
 
user105491
That's roughly speaking.
 
What does that even mean?
Isn't there a clear definition?
 
7:25 PM
if there was, we wouldn't be talking about it
 
@TedShifrin wat
 
user105491
@PedroTamaroff There is, if you want. I work with Grothendieck topoi; let me define that. In $1$-category theory, at least, a Grothendieck topos is a category which is cocomplete and has a small generating set.
 
user105491
Alternatively, it is the category of sheaves on a small site.
 
@blue: he said tortuous ...
 
@SanathDevalapurkar What is $1$-category theory?
This looks like super advanced stuff.
 
user105491
7:27 PM
@PedroTamaroff It's ordinary category theory .
 
@PedroTamaroff A 1-category is just a category.
 
OK, what is $n$-category theory.
 
Why do you assume $n$ comes after 1?
Maybe next is $\chi$-category theory.
 
Ugh ... I remember when it used to be fun to be in here ...
 
Me too
 
7:29 PM
It's always fun in here, @Ted.
 
@TedShifrin I have a joke for you.
What do you call a crocodile in a vest?
 
Maybe I should talk about injective $\mathbb Z$-modules instead? To lighten the mood.
 
user105491
@PedroTamaroff In essence, an $n$-category is a category enriched over $(n-1)\mathbf{Cat}$, where enrichment is defined as follows: the hom-set is an $(n-1)$-category (note - it's an inductive definition). But perhaps I'm getting too technical here!
 
user105491
@MikeMiller Ah, yes. That's be fun! :-)
 
@PedroTamaroff Well, if he's drunk, I call him a cab.
 
7:31 PM
Silly Mike. An investigator.
 
lol
 
I hate you.
 
The zoologist will remind us an alligator a crocodile is not.
4
 
@SanathDevalapurkar What is "the hom-set"?
@MikeMiller OK, this one is better.
 
@TedShifrin Luckily, we have assassins to take care of the zoologists.
 
7:32 PM
Yesterday I saw a kidnapping.
 
r9m
@PedroTamaroff That would be Alligator in a vest .. :P
 
@Pedro has been in a very punny (or puny?) mood of late. It must be that graph theory test.
 
I didn't wake him up.
 
@PedroTamaroff :O
@r9m True!
 
@TedShifrin It is not graph theory, it is combinatorics.
 
7:33 PM
@ted see your facebook messages
 
Then why are you doing all this graph theory, @Pedro?
 
Graph theory $\cap$ combinatorics $\neq \varnothing$
 
@TedShifrin It is part of the course.
 
Ah, ok.
 
@Ted he is feeling masochist today
 
7:33 PM
Oui, monsieur @GTR
 
@MikeMiller Luckily, we have law enforcement guys to take care of you.
 
user105491
@PedroTamaroff Given objects $X,Y$ of the category $\mathcal{C}$, the class of morphisms in $\mathcal{C}$ from $X\to Y$ is called the hom-set. If this is an $(n-1)$-category, then $\mathcal{C}$ is an $n$-category. Inductively, you can define $\infty$-categories, and then define $(\infty,n)$-categories.
 
@SanathDevalapurkar Ah, but do we always require ${\rm Hom}(X,Y)$ to be a set? Why not a class?
 
user105491
@PedroTamaroff Are you asking why it's called a hom-set?
 
user105491
@PedroTamaroff Found my mistake, and edited it. Thanks for the clarification.
 
user105491
7:36 PM
@PedroTamaroff Are you a grad student?
 
@SanathDevalapurkar I am not sure.
 
user105491
@PedroTamaroff What does that mean?
 
The I-do-your-homework squad really gets on my nerves lately.
 
I am not sure.
 
@DanielF: You and I need to lead a revolt.
 
7:37 PM
@DanielFischer take a break pal
 
But most of our colleagues are on their side, not ours.
 
I think it's 50/50.
 
yep^
 
@TedShifrin "Do you hear the people sing...."
 
Pedro, you doing planar graph theory?
 
7:39 PM
@Ted The same user posted the same question before, and I guess reposted it when it wasn't written as something they could copy paste.
 
@PedroTamaroff nice comment :D
 
@Studentmath Why planar?
 
user105491
@PedroTamaroff +1 for the comment!
 
@TedShifrin revolts are for revolutionaries
 
@TedShifrin Yes.
 
7:40 PM
Trying to prove something, if it might be on your test you might want to try it too
Otherwise not :P
 
@Alyosha: I have no recollection of what this refers to :P
 
Analysis and Abstract Algebra limits.
 
Yes to analysis and abstract algebra limits?
 
ohhh ... right
they are analogous, @Alyosha, but different concepts.
 
'Can analysis limits be done using abstract algebra limits, or is it just a coincidence of terminology?'
That's what I thought.
 
user105491
7:42 PM
@Alyosha I think there was an M.SE/MO question regarding that somewhere.
 
@SanathDevalapurkar Do you recall the name of it?
 
@SanathDevalapurkar Thank you. I think the MO one is what I wanted.
 
user105491
@Alyosha The accepted answer in MO is the same as the accepted answer in M.SE
 
I meant the question.
But yes.
 
user105491
7:45 PM
@Alyosha Yeah. So, what's your motivation?
 
@SanathDevalapurkar I'm learning about limits in category theory, and am not fond of analysis, so wondered whether it was applicable there.
 
user105491
@Alyosha As the answers say, it could be applicable, but it'd be very hard to do actual computations using CT in Analysis.
 
r9m
@G.T.R okay ,, I'm following it :-)
 
@SanathDevalapurkar I feared as much.
 
user105491
7:51 PM
@Alyosha Anyhow, have you finished learning about limits or have you just begun?
 
@SanathDevalapurkar Just begun. Are there any bits of intuition you can offer?
 
user105491
 
Sigh. Another one with attitude. What is going on?
 
I don't get this step. We assume the graph is planar and has a ring of length 4, $R=\{v_1,v_2,v_3,v_4\}$. We also assume (for contradiction) it is minimal 5-chromatic graph. Now if we observe the outer-lobe+$v_1v_3$ and the inside-lobe+$v_1v_3$, each is four colourable, so far so good. But now they claim that we can assign the colours so that in each the colour given to $v_1,v_2,v_3$ will be the same. But what if in one colouring $v_1,v_3$ have the same colour and in the other not?
 
@TedShifrin I think everyone hates you Ted.
It is settled.
[jk]
 
8:01 PM
I'm inclined to stop trying to help.
 
@Ted people are usually ungrateful
 
@SanathDevalapurkar @PedroTamaroff We should delete our comments, so as not to clog up the question.
Keep the bit asking about the coefficients, though.
 
user105491
@MikeMiller OK.
 
user105491
Done.
 
It didn't used to feel that way on this site, and I only started about 15 months ago ...
 
8:03 PM
@TedShifrin I mainly contribute for my own learning. Answering stuff solidifies my knowledge.
 
The longer something is/the longer you do anything, the probability to encounter such behaviour increases. On the other hadn the probability to encounter positive behaviour also increases, so..
 
Luckily I don't feel a need to solidify my knowledge in some jerk's homework.
 
No, @Mike, you just want me to help you with yours :D
And pretty much I can't any more ...
 
@Ted Hey, it's not homework.
Well, I guess it is. I'm at home and I'm doing these problems.
 
Are you doing summer school, @Mike? You've already graduated ... So you're just hanging out until you move?
 
8:06 PM
@Ted I'm studying and I'm working for a professor on a problem.
I thought I had the problem solved, but I didn't, so now it's interesting again.
 
Sorta like life ...
 
:)
Right now I'm being frustrated by a literature search coming up short, but c'est la vie.
 
What a twit. Says he knows how to do a volume with a triple integral but cannot do it with a double integral. Do you really know how to do it? :D
 
Don't be mean!
I bet you were a twit at some point in your life, too.
 
That guy downvoted me a lot in the past -> DonAntonio
 
8:13 PM
glares
Oh, really, @Chris'ssis? How do you know?
 
@TedShifrin Well, I had some quarrel with him in the past and he used to downvote me, but I think he used a different account to do that.
 
Wow ... Well, he apparently went and downvoted me a bunch after I chewed him out for doing someone's takehome exam problem.
 
Hmm. Is gambling allowed on MSE?
 
yes
 
@TedShifrin I'd like to see his account totally erased. (serial downvoting ...)
 
8:16 PM
Isn't it against the site rules @Ted?
 
@GTR: With regard to the problem you posted, do you know Wirtinger's inequality?
 
@MikeMiller gambling rep ?
 
It isn't quite there, but it smacks of the right flavor.
 
@G.T.R No, I bet someone a quarter.
 
Isn't what against site rules, @Studentmath?
 
8:17 PM
Seriel downvoting
 
@Studentmath Yes. Hence his suspension.
 
poker.SE
 
@Ted I had the occasion to discuss Wirtinger inequality with @r9m
 
But, truth be told, I'd like people to be suspended for doing students' exam problems.
Of course, the student should be suspended too for posting them.
But I give up ...
 
Who gives take-home exams?
 
8:17 PM
@skullpatrol Well, it's a gamble about math.
 
Stupid professors who don't give a **** about what their students are actually learning?
 
@Studentmath I had a take-home exam in complex analysis. It allowed the professor to give us better, longer tests. But it also had an in-class portion.
 
Or maybe they're naive/over-trust the students
 
@GTR: So? It smacks of that. But I don't see the trick yet.
 
True.
 
8:18 PM
Tests aren't better if students are tempted to cheat. And everyone is.
 
It wasn't a big class, though... I think he had good reason to trust the lot of us.
 
Well, in a small class he could always check the net to see if someone uploaded the question.
 
Hell, when I took point-set topology from Munkres, he gave problem sets which counted as takehome exams (and in-class midterm and final). This was way before internet or anything else. But we were allowed to use the book. One of his questions was solved many chapters later in the book, and a number of us found that and used it. The problem was then thrown out (legitimately).
 
@Chris'ssis blabhabhabhasb $$\int^1_0\frac{\arcsin(x)}x\;\mathrm dx$$
 
I was given a few tests to do home in school, was naive and never cheated
 
8:20 PM
maybe you can expand on this family?
 
Well, @Studentmath, you're all the better for your honesty and your true learning.
 
I doubt these few tests have advanced me in life, but yeah I don't have any guilty conscience about cheating
 
let them cheat themselves out of learning
 
I remember cheating on a memorization-based history test in 8th grade. I never got caught, but I've always felt horrible about it. I so hated that sort of testing. Always have.
 
@Alizter What about that (elementary) integral?
 
8:22 PM
I cheated on a math-test in 3rd grade or so, I still feel bad about it. It wasn't even intentional
 
@Chris'ssis It looks obvious. Sub then parts but then it diverges. Maybe creating integrals in such a manner can give more evil ones :)
 
@TedShifrin Well, at a certain point, there's nothing you can do. One problem on the test had us proving the FTA step-by-step. Certainly someone could google it, but they could do that too if it was a homework... and it's a good experience to actually do the details of the proof.
 
@Chris'ssis Not just for trig..
 
@Alizter I see.
 
You could do that on an in-class exam, quite fairly, @Mike.
@GTR: Someone's posted a solution for you.
 
8:23 PM
@Chris'ssis I am very happy and over my head at the moment. So excuse my if I am being quite random
 
@TedShifrin I suppose. But that wasn't the takehome exam, it was one of the six questions, most of which were similarly enlightening.
I've forgotten what they enlightened me of, but I guarantee they did.
 
Uh huh ...
 
@Alizter :-)
 
I don't like Sal Khan's voice, it's too business like. (like a used car salesman)
 
8:27 PM
@Ted this is the solution I already knew. Wait to see how elegant r9m 's one is
 
Well, get him to post it. I was trying to think about it. Grr.
 
@r9m Ted mandates that you post your solution :D
 
de-mandates
 
Oui, je l'ordonne! @GTR @r9m
 
r9m
@TedShifrin I won't post it now .. (I've sent the outline to GTR on fb) ... I want to learn how others approach it too :-) .. @G.T.R will that be a problem ? :-)
 
8:32 PM
@r9m well post it here since Ted is very impatient :P I also hope others will come up with something different
 
I need an idea how to say Morera without actually saying Morera.
 
Integrate around closed path? @Daniel
 
@TedShifrin This question. I wonder how to tell the OP to use Morera without naming it.
 
r9m
Well my idea is $F(x) = \int_0^x f(t) \,dt$, then $F(0)=F(\pi)=0$, and from applying integration by parts to obtain $\pi f(x) = \int_0^x tf'(t)\,ft + \int_x^{\pi} (\pi - t)f'(t)\,dt $ and Cauchy Schwarz inequality after squaring to the RHS :)
 
Hello everyone :)
 
8:36 PM
Yeah, @r9m, that's the sort of trick that shows up in these sorts of problems. I like this better than Fourier series, by far. Plus, the proof that's posted bothers me. How does he know there are no $\sin$ terms?
 
Hi Prof. @Ted
 
Heya @Sab !! Did you pass?
 
I guess.
No results yet just got on holidays.
But I know I will pass. Although I will just pass
 
@DanielF: The OP is trying to differentiate under the integral sign, which is perfectly acceptable. Really, a technique that's needed later, anyhow. But Morera is the slick proof, yes.
 
I didn't do well in my test and I'm sure about it.
 
8:38 PM
Well, @Sab, I don't want to be mean, but the adviser in me (with 35 years' experience) suggests that maybe a career in math isn't necessarily the right path for you ... You should consider that if the exam in first calculus was not good.
 
@Ted I'm starting to doubt myself. As you said I'm not sure anymore
 
P.S. @r9m: I like that argument.
OK, @Sab, as I said, I don't want to be mean, but you should explore some options ... would be my advice ... while you take the next math.
 
Thing is I will pass, but that's not what I want. I want to get above 80% minimum
 
@TedShifrin That part was added later. And for the one with only analyticity in $z$ given for rational $t$, I don't see how differentiating under the integral would be doable without violating the Geneva convention.
 
Right. If you really want to be serious about studying mathematics, the first year courses should be relatively easy for you. @Sab
 
r9m
8:39 PM
@TedShifrin Thank you very much !! ^_^ :D (skips in joy)
 
Exactly. I realize maybe that my study habits were not good
Or maybe I didn't do enough
or maybe I slacked off somewhere along the way
 
too much chat room
 
Well, @Sab, studying the week before the final doesn't make up for regular hard work. Math is not a subject learned by last-minute cramming.
 
That's 1 point
 
@DanielF: Where am I supposed to see just rational $t$? [goes to look again]
 
8:41 PM
@Ted yes there's something odd... he should make the symetric of $f$ over $[-\pi,0] and then replicate it by periodicity to make it an even function. But he got the gist
 
Thinking about it, I just did all my homeworks and tuts during my first semester
I didn't actually learn/revise before one week
 
Oh, I see it now, @DanielF. Then we absolutely need Morera.
 
So probably this was the case
 
I find that many of my students are in bludgeon mode to get homework done, but they don't take the time to reflect and absorb what they've learned doing it. @Sab ...
 
The funny thing is I can do integrals and derivatives easily
I mostly lose marks on fundamental things
I realized that I focus too much on the hard stuff, master them and don't put time on the very little details(which make me lose marks)
 
8:43 PM
Wait @DanielF ... If I only know for rational, how do I use Morera? Is it even true?
 
we got 5 integration questions worth 20 marks, I did them all in 10 minuts
 
Yes, @Sab, many of my students do that, too, when I keep reminding them that exams are not about the hardest stuff.
 
Yep.
 
@TedShifrin $F$ is still assumed continuous in both variables, as I understand it. So the integral over $\partial\Delta$ depends continuously on $t$.
 
Well, @Sab, learn from the experience and see if you can do better the next course with modified learning/studying strategies.
 
r9m
8:45 PM
@G.T.R @TedShifrin I didn't know the inequality had a name ... its apparently a slight variation of Block's Integral Inequality :-)
 
@Ted that's the only thing I should do I guess. It's my first semester and I did some mistakes. I guess I will have to grab the 65% I'm expecting and move on
 
Wirtinger? @r9m ... also a case of the Poincaré inequality that shows up in diff geo and PDE.
 
This drives me nuts.
 
But, @Sab, I would still encourage you to take a course in something else you think you might be interested in ...
 
I'm taking applied maths, physics and computer science
Thing is physics is really loads of calculus
I ace it
applied maths is just easy
and so is com. scie
 
8:47 PM
Well, applied maths is still maths ... :)
 
r9m
@TedShifrin ya .. all these are bros and sis of Poincare Ineq :)
 
Well, computer science is a good career choice in this day and age.
 
Yeah, but I'm pretty sure I got 95%+
The only thing I'm sure about is I took pure* maths lightly
 
Well, work on Spivak during your holidays and see if you still enjoy the challenges, @Sab.
 
I'm covering the whole thing this month. :)
 
8:48 PM
I can send you more of my exams, too :D
 
Sure :D
Spivak is next to me right now, borrowed it from library for the month
I'm also gonna study stats and take the module next semester
 
Stats!
 
Stats is awesome.
 
How I wait to finally be at that exam. Though I may be sorry for that waiting.
 
Did I send you anything with integrals, @Sab, or just derivatives?
 
8:49 PM
@Ted You sent some integrals. :)
 
Oh, yeah, I did send stuff from home, so I see what I sent. OK, I'll send the rest of second semester stuff.
 
I would appreciate if you have some questions which makes me apply my proofs
 
Well, there are a number of proofs in those exams. Did you do them all?
 
Not all. I did what was on my syllabus
 
And Spivak's book has plenty for you ... more than plenty. Some are too hard. But some are appropriate for you.
 
8:51 PM
Yeah. I love Spivak's approach
I wish I knew about it wayyy before
 
OK, sent.
 
Got it. Thanks :)
I see Taylor Series
I did that in applied maths.
 
But I ask for proofs that things are valid :P
 
It was fun, but we didn't do it in depth as we will do in pure
 
As well as applications, of course.
@DanielF: Why not ask the OP if he/she knows some alternative ways to show a function is analytic, rather than just the definition? [Rather than saying what you want.]
 
8:54 PM
I love the exam on integrals. 10 points per integration :D
We got 4 points per integration lol
 
@TedShifrin I guess I'll do that, and see how it evolves.
 
But I don't mean to butt in, @DanielF ... :)
 
@Ted So the book you use for your class is Spivak, right?
 
@TedShifrin But, didn't you forbid "but"s yesterday?
 
Yes, @Sab, but I hand out supplementary homeworks on things like volumes, work, differential equations.
Indeed, that was my goal (mon but), @DanielF :D
 
8:57 PM
Aha, so I guess Spivak is the way to go. And I'll use Stewart as a supplementary.
I understand better from Spivak, it just makes more sense.
 
I wouldn't use Stewart at all, personally
not sure what I would use
I never read Spivak, but any calculus text I've looked at has been terrible
 
Stewart's an OK standard book. I liked Edwards & Penney a fair amount in their early editions (colleagues of mine here). I like Rogawski (now dead, sadly) from UCLA. Very interesting book. A few non-standard approaches.
Remember, @Mike. Calculus texts are not written for math majors.
Spivak, more or less, was.
 
Stewart doesn't give the "real" definitions.
And it omits some proofs.
 
"Real" definitions are beyond the average calculus student ... in today's world.
Right, if you want proofs, do Spivak or Apostol's Calculus. And do mine for multivariable/linear algebra :D
 

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