Could someone help me figure out what the hell I can do to approximate $(10009)^{\frac{1}{4}}$ using taylor series?

@Owatch it is just 10009 to the power of 0.25

It's a constant.

" Approximate $\sqrt[4]{ 10009 }using a Taylor series. (Note that 4 ab = 4 a4 b ). "

So, I am just to reiterate it...?

10x10x10x10 is 100x100 = 10000

* $\sqrt[4]{10009}$

Approximate $f(x,y)$ at x=10, y=10

@Moses Hi.

That should be enough of a hint.

..

not sure what Alec wants to do. how about you @Owatch, what function do you think you'll be using a taylor expansion of?

single-variable

I really don't know what I'm doing with Taylor series. I'm trying to learn this right now, and I've never used them before.

So, my answer is: I don't know.

do you know what a taylor series is?

It's supposed to approximate a function. Or can be used to do so at a point.

$\sqrt[4]{10009}=10\sqrt[4]{1+0.0009}$ equals 10 times $f(x)=\sqrt[4]{1+x}$ evaluated at $x=0.0009$

@anon there are multivariable taylor series and 1/root(2)(1,1) is a unit direction.....

multivariable is inappropriate and overly complicated for someone who has never used taylor expansions to approximate before

it even seems overly complicated for someone with experience - I don't see what there is to gain from that choice

Then what is the point of the hint?

the point of what hint?

see first equality

I don't know where you obtained f(x) from.

Well we can agree to disagree. It could be a simple exercise in either context.

lol

I wouldn't know how to, or even what to do with a constant.

Why do I need f(x).

@Owatch $10\color{Red}{\sqrt[4]{1+0.0009}}$, you don't see the function $\sqrt[4]{1+x}$ being evaluated?

No, I see it.

I don't know why you did that though.

this goes back to the original question: do you know what a taylor series is?

I don't even know what it's asking. ..

because we want to plug a value into a taylor series expansion of a function

have you ever seen this done before? surely if you have a textbook it shows you examples before asking you to do it?

same for classes or lecture notes

Yes, I'm looking in my textbook. But I've searched everywhere and cannot find anything that gives me a constant and asks me to find what you found.

search again

"Searched everywhere"? Do you mean "basically nowhere"?

specifically in the section/chapter right before where the exercise appears

I need to ask a stupid question.

Yeah, you sure got me. I didn't even open the book Alec.

*want

@FrankScience the bar has been set pretty low.

Suppose $A$ is a commutative ring, and $a,b\in A$.

Is it true that $(a:b)=\{x\colon a\,\vert\,bx\}$ is a line bundle?

that escalated quickly

what does a line bundle mean to you today

Invertible sheaf over $\operatorname{Spec}A$. If I don't misunderstand anything, in this context it's just saying for $M$ the existence of a module $N$ such that $M\otimes N\cong A$.

It's not in the book. The exercise comes from a past paper I'm doing. I'm trying to look through the chapter on Taylor series and Maclaurin series to find any similar examples, but they don't have that sort of question.

Oh right! The ":" is the "such that" not the pipe.

I'll look keep looking.

I recommend you search for "taylor series examples" or even "taylor series root approximation" or something.

I mean EVEN Bing can help you with that.

@Owatch Step (1): recognize algebraic expression as a function evaluated at a point. (pick a function with a nice taylor series). (2) compute the taylor series of that function up to a desired number of terms, then plug in said point. that's your approximation

How can I prove that Y is irreducible on C[X,Y]/(X^2-Y^3) ? If $Y=(P_1X+P_0)(Q_1X+Q_0)$ then atfer some algebra I get that $Y$ divide $P_1Q_0$but next?

fractional ideals are rank 1 projective modules over A, yes

is that true even when you're not, like, a dedekind domain or something?

I'm asking for general, not necessarily Dedekind.

also note that projective means you can sum to a free module; we're looking to tensor

@MikeMiller yes, I think so.

First let's assume that $A$ is a finitely generated $k$-domain.

hey :(

$A = \Bbb Z/4$, $a=2$, $b=1$ is what I wanted

but you've cruelly taken this away from me.

well, nonzero fractional ideals which are invertible are certainly rank 1 projective.

sorry.

assuming $A$ is a domain.

@anon Hi, How can I prove that Y is irreducible on C[X,Y]/(X^2-Y^3) ? If $Y=(P_1X+P_0)(Q_1X+Q_0)$ then atfer some algebra I get that $Y$ divide $P_1Q_0$but next?

yeah, the same idea should provide counterexamples whenever you're not working with a domain

I don't know whether invertible sheaves are exactly described by modules.

over Spec A I think so... don't quote me on it thoug

but with a domain, the multiplication map $(a:b) \otimes \frac{b}{a}R \to R$ should be an isomorphism, I think, where $\frac{b}{a} R$ is an $R$-ideal living in the field of fractions.

But $(b/a)A\cong A$ as modules?

i'll go back into my hole

i don't think it's true that every fractional ideal of a domain is invertible.

who keeps starring so many messages?

I'm thinking about the question that when the coordinate ring is UFD.

no idea. but paying attention to that would just increase star-craziness, not otherwise.

There was a theorem on Mumford's book that that of smooth $\mathbb C$-variety is a UFD if and only if its Picard group is trivial.

It follows from the criterion that if $(a:b)$ is principal for all $a,b$, then $A$ is a UFD ($A$ is a Noetherian domain)

@TedShifrin hi!

hi @Balarka ... How's the health and homework?

I wonder how it could be generalized to arbitrary Noetherian domain without terminologies and theorems from algebraic geometry, especially those of scheme theory.

Not good. :(

puts @Balarka in a hermetically sealed cage

I've been very ill. Can't do much math.

This is crazy. You're ill more than you're healthy. Some doctor needs to figure out what's wrong.

Well, insofar as anything is necessary.

It's the weather. I have some sort of cold allergy, or so the doctor said.

Maybe you should move to Mercury, where you'll be warmer.

Okay, you need to drink a lot of water.

No, Mercury isn't warmer.

Oh ...

Venus, then. But then Balarka will incinerate very quickly.

@FrankScience Drinking water is pretty much the only thing I can do in terms of food/fluid intake.

Sore throat and swollen tonsils doesn't help.

Perfect!

In what sense?

Ah, I had my tonsils removed when I was younger than you because I was always getting sick. Someone should have thought of that for you.

Me too.

But there's no panacea. Usually for 1-2 weeks.

hey @TedShifrin

@TedShifrin Removing tonsils won't change my topology. Different kind of surgery is needed.

That's an expensive word, @Frank. Congrats.

Is the hamiltonian always conserved under transformation of a particular reference frame?

@Balarka: Well, if I wanted to change your topology, I'd do something more radical. Smart ass.

@TedShifrin I saw it on Stein's books for undergrad analysis.

I doubt it, Karim.

LOL, @Frank. Start reading more French :)

yeah I don't think for example friction would be an example I dunno

I published a paper in French, @Frank. Maybe I'll send it to you :P

Can someone translate that to math words for me?

I think you know it should be the case for non-dissipative forces

Friction? What do you mean, Karim? Transformation of the frame doesn't change the forces that are acting.

I mean

Okay, there is another stupid question:

for example

suppose you have some system that moves under friction

i.e friction is acting on it as a force

Spectral sequence argument for Leray-Hirsch theorem.

then no matter what reference frame you take hamiltonian won't be conserved.

That's in Bott-Tu, I think, @Frank.

flees at the name of spectral sequences.

They don't use spectral sequences, I don't think, @Ted.

With all the fleeing you've done lately, @Balarka, we shouldn't see you for a month.

Bott & Tu DID use spectral sequence.

Their arguments are all spectral sequence arguments, @MikeM ... double-complex, etc.

But for de Rham cohomology.

So, Karim, you're asking if you can make energy be conserved by choosing an appropriate non-inertial frame?

yeah

not energy

but hamiltonian energy

The hamiltonian is total energy.

not always

I cannot remember what's wrong with that, but it seemed to me that it cannot be generalized to general case, since modules aren't necessarily free.

Oh, so you're making up a non-physical hamiltonian?

hamiltonian is total energy if the potential isn't derivable from velocity depedent forces

and the coordinates don't depend on time

OK, Karim. Fair enough. I don't know the answer.

@TedShifrin: I just looked at it again, and they have a proof before they ever say what a spectral sequence is, in the same flavor as the proof of Poincare.

@MikeMiller Do you remember spec seq argument?

I understand, @MikeM, but it really is the spectral sequence argument :)

oh ok

Fine.

Sort of like Munkres introduces standard procedures before he names them in his alg top book.

I heard most classical mechanics can has symplectic geometry as frame work of it

@evinda: Your original problem was on $[0,1]$, so who cares what happens outside?

anyway I shouldn't be wasting time now I will be back after my exams are done

this chat is sometimes addictive haha

Bott & Tu starts with Čech-de Rham bicomplex.

Karim: the basic Hamiltonian equation set-up is symplectic geometry. I don't think your question is handled, but I've never really studied this. You might look at Spivak's new book on Mechanics (but he assumes you know a first graduate course in manifolds).

yeah I would like to understand these stuff @TedShifrin hopefully by 2017 I will have understood some manifolds

then I can look at such stuff

@Frank: I would assume a standard place to look is Godement's book on sheaf theory.

those stuff seems very cool to me

I want to first make my analysis strong next semester so I can start looking at manifolds and stuff

in the summer

anyway I will cya guys l8er

cya @TedShifrin

On $[0,1]$ you're just making the function look more complicated, but it's the same function, @evinda. You know that.

Cya, Karim.

I planned to read Iversen's book, however.

Yes, I know... So what I wrote is correct , right? @TedShifrin

I didn't read it, @evinda. I'm tired of this.

Ok... no problem... Thanks for answering!!! @TedShifrin

@Frank: I actually did keep a copy of Godement. It looks like he doesn't name the theorem but does prove it in his spectral sequence chapter.

In which chapter?

Théorie des faisceaux, apparently. I mean, sub-chapter.

Oh, maybe not. Hold on.

Yeah, not sure. I'll have to look more carefully. Doesn't Hatcher prove it in his book?

You mean, that unfinished book for spectral sequences?

Not with a spectral sequences argument.

Ah, right, @MikeM.

page 51 of Hatcher's book, an exercise

If you google, there are all sorts of lecture notes doing it, @Frank. Plus John McCleary's book on Spectral Sequences (which I'd forgotten about).

Is it hard to prove that any smooth map between smooth manifolds has a regular point?

Is it even true?

Regular point or regular value?

That doesn't parse.

"near which the map is a submersion..." Give me a formal statement.

Oh, I see!

I see why the comment claims that it's a homework exercise.

$f : M \to N$, smooth map between smooth manifolds. Is there a point $p \in M$ such that $Df_p : T_pM \to T_{f(p)} N$ is a submersion?

Think linear algebra, @Balarka?

Consider constant function...

The arrangement is just the same as that on Hatcher's book.

@FrankScience: I think it's better to let someone else struggle than to give them the answer, no matter how trivial.

@MikeM: Frank hasn't learned to be as much of a strict sourpuss as you and I.

Oh, right, constant map. :(

Now tell me what a regular value of a map is. Your $p$ lives in the domain, so is called a regular point.

Values live in the codomain.

There's a reason I asked this, Balarka!!

I need to be a maid and clean my house. Unless someone volunteers.

Hearing no volunteers, see you all later!

@MikeMiller Trying to recall.

I think a regular value is a point $p \in N$ such that for every $x \in f^{-1}(p)$, $Df_x$ is a surjection?

Oh, yes, I remember now. Regular value theorem said if $f : M \to N$ is a map between smooth manifolds, $p \in N$, $f^{-1}(p)$ is a submanifold if $Df_x$ is a surjection for all $x \in f^{-1}(p)$.

Of course - this is what happens for maps between euclidean spaces.

hELLO.

Hello.

How do you do? @BenLim

How do I combine a $[$ with a $\Vert$?

@evinda rest of what problem?

@AlecTeal [\![stuff]\!] is a fix, \llbracket stuff \rrbracket in the stmaryrd package

$\llbracket$ $[\![$

Yeah that works pretty well, thanks.

@FrankScience not much.

@anon hey

For $\operatorname{Spec}A$, is it true that $A$-module $M$ corresponds to an invertible sheaf if and only if there exists $N$ such that $M\otimes_AN\cong A$?

(to be sure that I translate the language correctly, since I know rare about schemes)

$A$ is a Noetherian ring.

@anon That's the only solution I ever found.

Hello.

@BenLim yo

$f(x)=-10sin(2x)cos2x$, where x is greater than or equal to 0 but smaller than or equal to pi.Write down an expression for f(x) in the form of $k sin 4x$

where k is a constant

4k rep, hooray.

@anon if you're interested : math.stackexchange.com/questions/1563207/…

oh, I see you already commented.

:)

I am kinda lost on this question. Does anyone have any tips?

@anon you are good at math. Can you please help me with the question above?

@Michael go backwards from k*sin(4x) to -10sin(2x)cos(2x). what can you do to sin(4x)?

@anon 4sinxcosx

no

-10sin2xcos2x=-20 sin(x)cos(x)

-20/4=-5

I think that gives the right answer

but that is completely wrong

@anon

sorry I dont know

you can't just make up stuff, you have to follow rules

like, 1+1=2, I'll add 1 to the left side, and I get 1+1+1=2. can't do that. one valid rule is we can add something to both sides, so I could add 1 to both sides and get 1+1+1=3. lesson: don't make up stuff.

again, what is sin(4x)? what rule can you use on it?

@anon double angle identity

right, so k*sin(4x) equals what?

it equals k*(2sin(2x)cos(2x)) right?

yes

oh

I get it

thank you

I can be a little dense sometimes

@anon

what does arg mean?

Heyo.

@anon what you been up to?

@Michael argument / phase

What is the point of the liar's paradox? Does it just show that binary truth values can't be assigned to English sentences? Like I don't even know why it's worth talking about.

@BenLim well, took the putnam again. thinking vaguely about some lie theory stuff.

@anon Ah I see. The students had the putname here yesterday too.

I'm thinking about some moduli theory crap.

it's interesting and people want to explain it @Anthony

arg(wz)=arg(w)+arg(z)

where does this come from?

@anon Sure, so I guess I should phrase that better, what is the resolution? Like... Can't you just conclude that English statements aren't true or false?

@Michael do you know an expression for sin(2x)

Oh I was scrolled way up.

@AlecTeal mister anon helped me

Actually, alright, I guess I get it. There is an issue, and there are multiple ways of resolving it. What's the canonical way?

@Anthony and that doesn't strike you as counterintuitive? there are all sorts of caveats that explain why some sentences aren't true or false - they are made relative to a given set of assumptions, they are context-dependent, they are subjective valuations, they are not well-defined - and we would want to know which caveat is why this doesn't have a truth value

iunno

ask a philosopher

@anon ever had the need to deal with etale cohomology?

@Michael learning about complex numbers?

@BenLim nope

@TedShifrin mathoverflow.net/questions/225407/… any help appreciated here.

@anon I just started a few hours ago.

heya @Anthony @Michael

@TedShifrin hello professor ted. Is there a good place for me to learn Complex numbers?

@anon Sure that makes sense. I think what happened is I learned some things about math before I knew what math was and then I got these dumb issues ingrained in the back of my skull. Thanks.

@TedShifrin Helllllllo!

@Michael They're not incredibly difficult. Just grab any precalculus book.

I'm actually on my way back out though, gotta keep writing apps.

Talk to you all later.

A precalculus book will do. I also have a chapter about complex numbers and the geometry of them in my abstract algebra text, @Michael.

Bubye, @Anthony.

(Oh and I was considering reading your book, @TedShifrin)

:P

@TedShifrin What is the definition of infinity?

There isn't one :P

@Anthony I learned the stuff from precalculus for dummies.

Oh I meant his differential geometry book, whoops.

oh sorry meant idiot's guide to precalculus

@BenLim: That's too much scheme/algebra stuff for me.

@TedShifrin ok.... sigh

Yeah, @Anthony, you'll enjoy some of the diff geo stuff.

Can someone explain to me the different fields of mathematics? What they do and stuff

@TedShifrin I read the definition of a wedge product for the first time since sophomore year, and finally I understand the tensor product definition.

Tears of joy streamed down my face.

It's not that scary, @Anthony :)

@Anthony wait till you get to algebraic stacks.

@BenLim Oh no, I can't afford to fall into Wikipedia today.

Alright, toodles everyone!

@TedShifrin Man I need to learn deformation theory at some points, you know all the crap about prorepresentable functors etc

That's far from the deformation theory I learned. I learned it from the calculus/sheaf theory perspective.

@TedShifrin I see.

@Michael: That's a bit time consuming for chat.

They're probably the same.

ok sorry. What about this quod.lib.umich.edu/s/spobooks/5597602.0001.001/53:2.16/…. The Idea of polar coordinates aren't clicking in my head, and this is a really basic book.

I don't know that book. Jasper and I tend to have very different taste in textbooks. Polar coordinates are easy: Just draw the picture of the ray from the origin to $(x,y)$.

@TedShifrin there are some really smart ppl here.

At school with you, you mean, @BenLim?

Ok, this is another weird one (sorry I'm still in highschool). Prove the identities $sin(x+y)*sin(x-y)=sin^2x-sin^2y$ It looks like $(a+b)*(a-b)$

I am having a feeling that the OP is not really familiar with cellular homology, hence my answer hasn't been of much help. I think there should be a way to do this using the long exact sequence, but my approach is not really panning out.

Want to help, @TedShifrin? :)

@Michael

@Michael just expand with Sum property.

@zickens sum property? IB never taught that...

But it isn't really, @Michael. You have to use the formulas for $\sin(x\pm y)$.

I am trying for the long exact sequence of $(X, S^n)$. This piece : $H_{n+1}(X)\to H_{n+1}(X, S^n) \to H_n(S^n) \to H_n(X) \to 0$ is the relevant piece.

The boundary map up there is multiplication by $m$, as it sends relative cycles to their boundaries.

So if I can just show that $H_{n+1}(X) \to H_{n+1}(X, S^n)$ is zero, I'm done.

@Michael Sin(a+b) = sin(a)cos(b) + sin(b)cos(a)

Remember that trying to do cellular homology requires all sorts of fiddling with homology of a triple, @Balarka. You aren't going to write it all out in an answer. Or I would never consider doing so.

@zickens Wow. How does that work? Again I apologize, my curriculum ignores alot of things...

@Michael, go to my profile page and find my email and email me. I have a very brief trig review sheet I wrote for my Honors calculus students years ago. It has brief proofs of pretty much everything for you.

BUT you should get a Schaum's outline or some standard trig text.

@TedShifrin Can you elaborate on "trying to do cellular homology requires all sorts of fiddling with homology of a triple"?

Anybody have some good multivariable calculus books? And by good I mean dark and full of proofs.

We talked about this a year ago, @Balarka. Pretty much all the proofs involve homology of a triple.

The two standard American references are my book and Hubbard and Hubbard, @zickens.

Proofs of what?

Hubbard and Hubbard? Never heard of it...

To be mathematically meaningful, one needs to incorporate linear algebra, and most books don't do that @zickens.

Google.

Because of the Jacobian stuff, right?

No, because the whole point of a derivative is that you're doing best linear approximation. So what is a linear map?

I haven't got there yet. I just finished a the first Calculus course at my UV; but right now, the teachers are on strike, so I'm gathering a little bit of everything. Haven't digged about Linear Maps, but I recently read about Taylor Expansion and Linear Approx's.

Where are you, @zickens?

And, by the way, for a "dark" treatment of single-variable, full of proofs and gorgeous exercises, see Spivak's Calculus.

Venezuela; USB.

What books do you folks use for calculus?

@TedShifrin Sorry for being annoying, but I'd really like to hear about the homology of triple thing you have in mind. I don't recall this discussion from the previous year (sorry!). If you don't care, feel free to not answer - I understand.

I am using this one quod.lib.umich.edu/s/spobooks/5597602.0002.001/…

@Balarka: I'm pretty sure it's in Hatcher.

IIRC, Hatcher talks about homology of a triple in only 1 place in whole chapter 2.

@zickens Anyway, linear algebra is a must. Here's a book.

@Michael, that's because Jasper told you to. I looks like a pretty standard course, but with some linear algebra in it. Apostol is a classic for that sort of course. My book and Hubbard and Hubbard are much more proof-oriented, with higher dimensions as well as interesting applications.

@TedShifrin Is it bad? Is Jasper's advice not reliable?

Hmm, well, probably it uses it in the proof that $H_n(X, A) \cong H_n(X/A)$ for pairs $(X, A)$ with HEP.

I don't know if it's good or bad. As I already said, Jasper makes recommendations very liberally, often without knowing the book well or even having read it. His taste and mine do not agree, most generally.

But I don't remember any other mentions off the top of my head.

@Balarka: I have to leave, but I don't know how to do the cellular stuff without homology of a triple. But I haven't thought about it in centuries.

If by triple you mean homology long exact sequence of $(X, A, \emptyset)$, I agree with you :D

No, you know that is not what I mean.

I did not keep all my old alg top books (like Vick, Spanier, Greenberg, Munkres).

How would you find the area between the graphs of $e^{3x}$ and cos$(2x)$ when x is greateer than or equal to -pi and smaller than or equal to pi/2.

Yeah, otherwise I don't know how cellular homology can be done using homology of a triple. Seems interesting, though.

My derivation of cellular homology uses degree theory.

Cellular stuff could be derived from spectral sequences, if I remember well.

Heh, I think of SS as generalization of cellular homology.

Spectral sequence is just an algebraic machinery.

Cellular homology = homology of homology. SS of a filtered complex = pages consisting of homology of homology of ... of homology (n times) for each n.

@Balarka: The diagram Hatcher has on p. 138 reminds me of a triple, since he has two connected pairs.

Hmm, makes sense. I haven't thought about it this way.

@TedShifrin The common book is Purcell's; I used a lot of books including that one. I think that at some point I used Leithold's and Apostol's...

Oh, ugh. Purcell is terrible. Apostol's Calculus book is very good.

@TedShifrin Sorry for bugging you. Since the book Jasper recommended isn't great, is there a good book that would start from the very basics of multi-variable calculus that would also cover linear algebra?

I noticed that Purcell's is very "solve-this-like-that" oriented; didn't read it usually.

@Zickens, if you want to be a mathematician, you should learn computational calculus well, but you should also start learning more of the proofs and doing lots of proofs, even while you're learning calculus, single- and multi.

@Michael: I have no idea whether it's great or not. But it's a good start for you. You don't need to be doing a super-advanced course yet.

Anyhow, @Zickens, I need to leave. But see if you can find Spivak's Calculus ... It's great.

Will do; see ya @TedShifrin

@Michael I suggest that you study linear algebra independently.

@TedShifrin How much integration knowledge do you need to proof that the volume of a sphere is $\frac {4}{3} pi*r^3$

Bubye @TedShifrin. Thanks for the triple thing, I should think about this carefully tomorrow.

@FrankScience Would khan academy work?

@Michael Solid of Revolution level.

@zickens oh wait I know that. I think I just made an error somewhere in the proof.

@FrankScience Can you help me out with this? I can't prove why $H_{n+1}(X) \to H_{n+1}(X, S^n)$ should be $0$.

@Michael For general calculus and mathematical analysis, I suggest Zorich's Mathematical Analysis. For linear algebra, I don't know much but maybe Halmos' book is good.

@BalarkaSen Sorry today I have no time.

Aw. Thanks, though.

@FrankScience Oh my goodness it is beautiful, much better than Jasper's book. THank you!

@Michael It's hard to compare books, since tastes are different.

@FrankScience I guess you are right. Does it have proofs?

@Michael But generally speaking, you can believe that books written by masters aren't bad, and usually insightful.

@Michael You mean, Zorich or Halmos?

@FrankScience zorich

Yes, and a number of exercises, which are valuable.

@FrankScience math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf Can you explain the symbols on page 21/597? It is equal to page 2 of the paper copy of the book.

If you have time ofcourse.