Mathematics - 2017-09-09 (page 2 of 4)

10 Replies

2:00 AM

I was given a 'hint' to set u^2 = 5x. I tried that and got the wrong answer

nitsua60

What, exactly, happened when you tried $u^2 = 5x$?

10 Replies

I may have done something wrong

quid

@BalarkaSen thanks for the info. Maybe I'll see him next time.

See you all and good luck with the integrals!

Balarka Sen

Hope to see you around again.

10 Replies

I subbed in the U, then I integrated using U substitution (using the letter B) where B = 1+u

dB = 1

and then I have int u^-1 * dB

which becomes ln(u)

nitsua60

2:04 AM

I'm wondering about your dx from the first "layer" of the substitution. (I'm pretty sure--though I'm just doing it in my head, no scrap at hand--that you never "did the u-substitution" correctly.)

Balarka Sen

@10Replies You mean dB = du, right?

Daminark

Yo yo yo

Balarka Sen

Hey hey hey

10 Replies

@BalarkaSen dB = dB i don't think I ever had a du which might be my problem

logical123

You need to set u=sqrt(5x)

My other hint is, remember 1-1=0

10 Replies

2:06 AM

U^2 = 5x is equivalent to that, and I was told to set u^2 to 5x

logical123

Don't do it that way. I just did it both ways and it's easier with the root

10 Replies

ok

Balarka Sen

Doing it with or without the root is honestly equivalent.

logical123

My second hint is honestly more important.

nitsua60

@10Replies Here's the problem: you've said you want $\int \frac{7}{1+\sqrt{5x}}$. But that's nonsense. You really want $\int \frac{7}{1+\sqrt{5x}} dx$.

10 Replies

2:08 AM

What does the dx mean?
I never had to deal with dx in calc I

It was just kinda there

And I ignored it, as I was doing now.

Balarka Sen

@logical123 It doesn't seem to be much of a hint, but, erm, sure.

nitsua60

@10Replies I want to punch your teacher. Possibly you, but definitely your teacher.

3

10 Replies

Oh I'm an awful person, you have two fists, might as well kill two birds with two stones

nitsua60

=)

(Thanks for taking that in the mostly-humor it was intended.)

logical123

Hahaha

Semiclassical

2:10 AM

@nitsua60 it is rather my kind of humor, indeed

aw, you edited out semi

Balarka Sen

You don't integrate functions; you integrate differentials. Must be a horrifying calc I class if it hasn't taught that.

logical123

When I do the u sub, I get $/int$

10 Replies

It was AP calc. I learned how to take a test, not how to do calc

logical123

When I do the u sub, I get integral of (14/5)*(u/(1+u))

nitsua60

The way I put it to my students is that the substitution isn't "complete" if you do what you've said: $\int \frac{7}{1+\sqrt{5x}} dx, u=1+\sqrt{5x} \rightarrow \int \frac{7}{u}dx.$

Daminark

2:13 AM

Ugh AP Calc... Pick up Spivak Calculus

Balarka Sen

@logical123 Oh I see what you meant by your hint.

That was cryptically written, but yeah.

logical123

Factor the constant out, then add 1 and subtract 1 to the numerator

I wanted it to be super cryptic

Back in Calc I pounded my head into the wall for at least minutes until I thought of that.

nitsua60

I don't know that you can blame AP for this bit... there are plenty of problems on that test where neglecting the differential in a substitution would bite you. The teacher, even if solely "teaching to the test" should have been hammering that home.

10 Replies

Where is du in 7/(1+sqrt(5x))? I can quite clearly see the U, but not the du

logical123

You need to solve for dx in your substitution

10 Replies

2:15 AM

@logical123 What does that mean?

nitsua60

@logical123 I'm not clever--I would have just long-divided the thing (and ended up in the same place).

logical123

I'm not super clever, but I love the clever solutions.

Balarka Sen

Subtracting and adding the same stuff is actually a pretty useful (and standard) trick.

logical123

In physics, that trick is routine

In your u = root(5x)

nitsua60

@BalarkaSen Yeah, just not one that I'd necessarily even think of when the obvious, two-line method that will always work is also staring me in the face.

logical123

2:17 AM

Differentiate both sides and solve for dx

nitsua60

@logical123 "And why should I do that?" he asks, rhetorically?

10 Replies

du = (5/2)x^(-1/2) .... dx?

logical123

Yep, now get dx alone.

10 Replies

(5/2)*du*sqrt(x) = dx

Balarka Sen

Can you express the left hand side as a function of u?

nitsua60

2:19 AM

(And I say "rhetorically" in a specific manner: intended to advance the rhetoric, not the current colloquial meaning of "not expecting an answer.")

logical123

The way I think about why you solve for dx is, you want to see 'how big ' the old differential is

In terms of the new one

Perfect

Wait, no, (5/2)*sqrt(5x)du=dx

Now, recall what your u is.

10 Replies

sqrt(5x)

Balarka Sen

@nitsua60 I guess so.

logical123

U=root(5x), which is on the left side right?

Plug in u there

nitsua60

@BalarkaSen (nvm--I was thinking of the wrong line--I was the one thinking three steps ahead--you were absolutely correct. Sorry!)

logical123

2:21 AM

Now you have (5/2)udu =dx

10 Replies

You can plug u into du?

logical123

You substitute u for root(5x)

You follow me?

Balarka Sen

@nitsua60 (I thought you were quibbling amount the use of the my word "function", and maybe I should have said "plug in" instead.)

10 Replies

yeah

Balarka Sen

(Also why are we using parenthesis to communicate)

10 Replies

2:23 AM

(not sure but i feel left out)

logical123

Cool, so now plug in the entire left side into dx into the original integral

As well as u in for root(5x) into the denom

nitsua60

@BalarkaSen (No, I was thinking ahead to the algebra that'll have to be done on the u-space integrand and assumed you were setting up a shortcut there, rather than noticing that you were just prompting him to complete the substitution from x-space to u-space.)

Balarka Sen

(iseewhatyoumean.jpeg)

nitsua60

@BalarkaSen (I blame SE: it all started with "(removed)")

Balarka Sen

(Parenthesis people singing to SE: We didn't start the cancer / It continues growing with the memes you're posting)

10 Replies

2:27 AM

(7/(1+u))-5/2udu

logical123

No minus sign lol

10 Replies

Should be "*", right?

logical123

And I flipped the fraction, should be 2/5 right?

Corect

10 Replies

yeah

wait, why is the fraction flipped?

because we devided both sides of du by (5/2)x^-.5 ?

logical123

Yes

My bad, on my phone so not being hyper careful lol

Now, just factor out the constant in front of the integral, and what we want to integrate is u/(1+u) du

10 Replies

2:36 AM

Ok, so now I have u/(1+u) du (and I took out all the constants)

logical123

Perfect

Now my thought process is, if only we got 1+u / 1+u, then that would just be the integral of 1...

What do I have to do to retain equality of an expression if I just choose to add something? You better subtract that something as well!

'Make' the integral (1+u-1)/(1+u) du

10 Replies

Sure. We'll pretend that I understand why this works...

So (u+1-1)/(1+u)

nitsua60

@10Replies The alternative is polynomial division... most people prefer logical's path =)

10 Replies

I don't remember polynom division so I guess we are taking the other way

logical123

No

Do not pretend

What makes you think that isn't 'legal'

10 Replies

2:42 AM

It is perfectly legal, it just is the same thing as it was before lol

as you said, 1-1 =0

logical123

Ok, I understand what you meant. Not sure why this get the answer quicker

Hopefully it becomes obvious

Next bit would be to separate out the expression like ((1+u) - u )/ (1+u), then expand as (1+u)/(1+u) - 1/(1+u)

nitsua60

I frequently tell my students that a surefire method to tackle a problem you don't know how to solve is "make it look like a problem you do know how to solve." That's what logical's walking you through right now. You don't know how to integrate $\int \frac{u}{1+u}du$ off the top of your head (we assume), but you would totally know how to integrate $\int \frac{1+u}{1+u}du$. (With some setting-aside of a sticky bit for a moment.)

logical123

Which is 1-(1/(1+u))

Yes! Nitsua gets it. The idea is to fudge stuff to make it like things you know how to deal with, and unfudge it 'later'

10 Replies

How am I "separating" out the expression?

logical123

Distribute the denominator over the expression on top

Like

Dodsy

2:47 AM

hello

got some strange messages

logical123

(1/(1+u))((1+u)-1)

10 Replies

I'm still not quite understanding this operation.

logical123

If you have (7+x)/5

nitsua60

@Dodsy Without really knowing you, I'm going to quote Bachman Turner Overdrive and advise you to "slide, slide, slide, let it slide."

logical123

Isn't that 7/5 + x/5 ?

10 Replies

2:50 AM

Ok

logical123

Same operation as in the actual problem, distribution

But you can choose to group however you want in the numerator

10 Replies

(u/(1+u)) + (1/(1+u)) - (1/(1+u))

Oh I think i get it

Dodsy

@nitsua60 kk

logical123

Remember the associative property? That's what allows us to group into (1+u) and -u in the numerator

Bing bang boom

10 Replies

(u/(1+u)) + (1/(1+u)) - (1/(1+u)) = (u+1)/(u+1) - (1/(1+u)

right?

logical123

2:52 AM

You're starting to see it, you got this. Finish it off. Last hints are you'll get a log, and don't forget to back sub what u actually is

nitsua60

@Dodsy But you know the history better than I--this is "casual bystander" talking, not "diamond" talking =)

logical123

You got it mate.

Dodsy

@nitsua60 I don't know anything, been gone for so long.

=)

happy to have a place to talk about math

nitsua60

@logical123 (And double-check the domain. Never hurts, IMO.)

@Dodsy Starting a class on Chaos/Dynamical Systems on Monday. Excited about that =)

Dodsy

oh nice!

nitsua60

2:57 AM

But I've got a weird artifact of the schedule where my students'll have only one meeting in 12 days at one point. It's around the time I'd usually assign a partner- or group-project, but assuming they'll be traveling some I really should make that one a solo. But that means I need to re-think some other one or ditch one or something, and I've not yet found any combination that makes me happy =\

logical123

The domain of what? The denominator?

10 Replies

(14/5)*(sqrt(5x)-ln|sqrt(5x)+1|) + c

logical123

Wammy

Full marks.

Lol

10 Replies

Thanks!

logical123

Next problem lmao

nitsua60

2:59 AM

@logical123 The result vs. the integrand. It's a habit I try to instill in my students.

logical123

Oh absolutely

I like to think about what the restriction on u is give the restriction on x

X is bounded below by 0, and so is u

10 Replies

Calculus 1 was easy. No wonder people said calculus was hard. Calc II is awful.

logical123

Because sqrt is a strictly increasing function, you know you're all good and no tomfoolery may occur where u goes negative when it isn't supposed to

Yeah calc the second bites everyone in the bottom it seems.

nitsua60

@10Replies enjoy diff eq.

logical123

Calc 3 is easier I think.

Diffy q is love, Diffy q is life

10 Replies

3:03 AM

@nitsua60 I'm hoping that wasn't sarcasm, but, if it was, I'm planning on not taking more calc as I want to major in CS.

logical123

My degree is in physics, odes and pdes are in my sleep

nitsua60

Nothing'll make sure you understand I and II better than the method of froboenioueus =)

(I'm pretty sure I spelled that right.)

logical123

Well, for a good cs program, you need calc 1-3, Diffy q, and linear algebra

Plus a discrete mathematics course

10 Replies

I have looked at the requirements and the ones I want to attend don't need diffyq or calc 3

logical123

If your program doesn't require that, do it anyways.

2

10 Replies

3:04 AM

I already took linear algebra. That was a fun class.

logical123

That's my bottom line advice.

10 Replies

Lets see if I pass calc II first.

logical123

If you have a cs major and can't whip up a sim of the strange attractor in an hour, you suck.

nitsua60

@logical123 I get to teach Griffith's intro QM this term--I'm super-excited =)

logical123

No need to understand the strange attractor, but that project is super easy.

I love that text.

Daminark

3:05 AM

Differential equations? Eh... I've had about 2 weeks of them and I'm not terribly fond lmao

nitsua60

@logical123 That was the one that took me beyond "this is my subject" to "these are my volk."

10 Replies

I wrote a 3D engine with linear algebra. I'll have to google what a strange attractor is... after I finish my calc II homework thats due in 1 hour

Balarka Sen

I am supposed to be learning ODE's

logical123

Well, my ap calc teach in high school literally said the universe is one big differential equation.

Balarka Sen

@nitsua60 I've read a few pages of that book. Was nice.

logical123

3:06 AM

Just to see who would look into them

The world is associated with change. We model things with derivatives and equality, so naturally solving models of the world requires an understanding of these... differential equations

Semiclassical

@nitsua60 oh hey, me too

well, TA for it

nitsua60

@Semiclassical Sweet =)

Semiclassical

mostly i'm just happy about the workload

Silent

@BalarkaSen, will you please look at (this)[math.stackexchange.com/questions/2421663/… question?

nitsua60

Wait a minute... how many of us are actually physicists just hanging out in the math-lounge?

o/

Semiclassical

3:08 AM

lol

Daminark

I appreciate ODEs, don't get me wrong, just that it's not something I enjoy spending my time doing

Semiclassical

I'm definitely in that category.

nitsua60

I'll count that ^^ as a o/

Semiclassical

Secret does computational chemistry, which I'll count as close enough for this purpose.

nitsua60

@Semiclassical Oh yeah. The P Chem colloquia were held in the physics complex in grad school, because so many more physics-people were interested in attending than were the chem-people.

Daminark

3:10 AM

Oh huh, I knew he was into chem of some sort but I didn't know it was computational. That stuff sounds pretty nifty

Semiclassical

hah

well, 'computational chemistry' isn't very descriptive

I know he does DFT calculations, though

nitsua60

(Then someone set up a slackline between the two buildings. I don't know if that encouraged or discouraged the chem-folk from coming over?)

Balarka Sen

@Silent I don't think you need the increasing hypothesis. It comes to play in higher dimensions, where to do change of variables you must compose with an actual diffeomorphism, I think.

logical123

I honestly don't know what I am lol.

Daminark

Same lol

logical123

3:13 AM

I have a physics major and math minor but was reading grad level material on atmospheric sci my sophomore year, and right now I have a database dev internship

Balarka Sen

@Daminark a higher category theorist?

logical123

Don't bring categories... OR SHEAFS... into this place

Silent

@BalarkaSen yesterday a comment appeared saying that $g$ must be injective. Is that necessary?

Balarka Sen

I don't see any such comment.

You don't need injectivity or even monotonicity in one dimensional change of variables.

logical123

@10Replies the strange attractor is a fractal like Chaotic system

Silent

3:15 AM

@BalarkaSen That comment was taken down

Faust

@Daminark

nitsua60

@BalarkaSen @Silent I think you're both overthinking it.

Silent

@nitsua60 how?

nitsua60

That hypothesis allows the author not to futz about the ordering of endpoints of the interval in sentence 2.

Semiclassical

@Secret Am I right in calling you a computational chemist? I vaguely remember DFT stuff

logical123

3:16 AM

If the room is filled with physics people, mind if I ask what y'all got on the physics gre?

Secret

@Semiclassical Indeed I am a computational chemist because I have some experience with ab inito and DFT calculations

Faust

whats a gre

Semiclassical

I want to say 70th percentile

Balarka Sen

@nitsua60 Yes, but that's easy to deal with.

Semiclassical

but that was back in 2010

Faust

3:17 AM

?

Secret

though I usually just said I am a computational and organometallics chemist PhD

nitsua60

@BalarkaSen Yeah, but they don't. The monotone hypothesis lets them just slap it down and not get into the interval.

Balarka Sen

There wasn't much overthinking on my part; in one dimensions there is no need for monotonicity and the sort, just differentiability suffices.

logical123

I got 54th, but that was a 720/990 which is usually a decent score

Balarka Sen

in high dimensions you get rekt precisely because of what you said

Semiclassical

3:18 AM

monotonicity makes things easier, though.

Daminark

@Balarka I mean I like the categorical mindset for sure but I'm really closer to a number theorist, also I might end up getting into theoretical compsci

logical123

I basically knew nothing about atomic and particle physics on the test.

nitsua60

@logical123 tbh, very high 90s.

10 Replies

I'm still in highschool. Rip.

Daminark

@Faust GRE is basically the grad school SAT

Semiclassical

3:19 AM

@Daminark Learn this and teach it to me, then :)

Categorical quantum mechanics

Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Abramsky and Coecke. == Mathematical setup == Mathematically, the basic setup is captured by a dagger symmetric monoidal category: composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger...

logical123

I had and have a very nontraditional path in education lol

Balarka Sen

@Semiclassical Sure. The question was if I need it. I don't :)

Faust

do they do the same thing for math?

logical123

Working 35 hours a week plus 4 classes, studying for the gre was a bit on the back burner lol

Daminark

@10Replies when I say I'm a ____, I mean more like, I'm into it. I'm an undergrad, and we've got a few people here who are high schoolers as well

logical123

3:20 AM

Yes, there is a math subject gre

Chem and bio

Balarka Sen

@Daminark HoTT then

Faust

wierd

logical123

They axed the cs one about 5 years ago

Faust

wonder if they do it in canada

Daminark

@Semi that sounds cool as shit

nitsua60

3:21 AM

@logical123 We had pretty serious comps in undergrad (28 hours written and 2 hours oral all in one week), so senior year was basically six months of studying punctuated by brief trips to visit grad schools. And studying on the plane.

Semiclassical

I'll admit, I mostly find it interesting to read that stuff because it comes with diagrammatic calculations :)

logical123

I mean, you pretty much have to take the general gre to get into any grad program in the states

Faust

i see

logical123

The general is literally the grad school version of the sat/act

Faust

merica

Balarka Sen

3:22 AM

@Semiclassical We didn't start the cancer / It continued growing with the memes you're posting

Semiclassical

@Daminark You might find this interesting if you've got time: math.ucr.edu/home/baez/rosetta.pdf

logical123

@nitsua60 yeah I went to uiuc for one year, couldn't afford it, took a year off and worked a delivery job then paid for the rest of undergrad working and schooling, at a small unranked public on the north side of Chicago

I was the curve breaker in every single class, needless to say lol

Faust

oh yeah you guys have spencive schools

logical123

Lmao nice, rub it in.

Semiclassical

student debt is a harsh mistress

10 Replies

3:25 AM

any debt is.

Faust

im over half a mil in debt m8

Balarka Sen

did you click the link

Secret

I'd like a diagram that drew all feymann diagrams with the correct contribution all at once, just like how instead of drawing resonance contributors in chemistry, one can drew the overall structure with dotted line bonds

logical123

Yummy, I don't envy you. Sounds like my parents. I'm like 60k in

Semiclassical

@BalarkaSen I did. and then I backed away.

Faust

3:26 AM

well i didnt say what my nw was :P

10 Replies

I'm 0k, but, I also have no money, so theres that.

logical123

They made exactly enough to get no aid for my ed, which is a common occurrence in the lovely 'murica

Balarka Sen

@Semiclassical the n-point of view is mind blowing, indeed

Faust

i was thinking of maybe working down there when im done all my schooling

Semiclassical

tbh I mostly view QFT in general as black magic though

Balarka Sen

3:28 AM

I dunno how it works

logical123

Most of physics is black magic.

nitsua60

@Silent well, I posted my first answer here--hope it goes okay. (cc: @BalarkaSen)

logical123

Spontaneous symmetry breaking hooha coupling stuff to other stuff

Balarka Sen

I should learn some physicist's gauge theory

Semiclassical

eh, spontaneous symmetry breaking isn't bad

Balarka Sen

3:29 AM

There's not many sources that I know unfortunately

logical123

I know it isn't that bad, its just spooky

@10Replies finish the problem set yet? Calc 2 integrals are my favorites lol, so many tricks

10 Replies

No ;(

logical123

Go go go, 29 min left I assume

Balarka Sen

@nitsua60 I gave a +1

logical123

Stupid webassign

10 Replies

3:31 AM

Yup

nitsua60

@BalarkaSen Aw, shucks. I'm blushing. (You just can't tell under the wool.)

logical123

I got lucky and had an old school teacher for ap calc in high school, and my calc 3 and diffy q were honors level at uiuc so, absolutely no webassign

10 Replies

Homework is only 10% of my grade, so if I get like a 70% on homework it shouldn't be too bad.

I'm hoping the homework is super hard compared to the tests and quizzes

logical123

Yeah that wouldn't be a disaster but I'm not your teacher, idk how devilish they can be.

@nitsua60 what is your favorite chapter in Griffiths?

Faust

most of my crap nowadays is just hw weight

10 Replies

3:35 AM

It's a college class, I don't really interact much with the professor, so I hope he isn't evil

Faust

i got 2 classes with 60% hw weight

10 Replies

RIP in pieces

damn

Faust

and one of them has 3 take home midterms on top of that no final

nitsua60

@logical123 No recollection! It's probably been about fifteen years since I cracked it open! And I start class on Wed!

logical123

Hahahahaha brilliant.

nitsua60

3:36 AM

(Dear students in the room: you're not the only procrastinators in your classroom.)

logical123

I like the operators chapter, especially the raising and lowering section

Oh boy, I actually got into it with a prof once because of that

nitsua60

@logical123 I didn't even remember that Griffiths got that far--I think of Sakurai when I think of raising and lowering operators.

Faust

@nitsua60 i caught my analysis prof with a mistake today in proving the rational zeros theorem i know =)

logical123

Sounds lime you used the old edition.

nitsua60

Is it Griffiths' QM or E&M that gets heavy into Green's functions? It must be QM, right? If so, that was definitely my least favorite chapter at the time. (I'm fine with them now, but tackling Green's for the first time...? Ugh.)

logical123

3:38 AM

My prof raged at the class because of procrastination, and later I was cracking wise in another classroom to a few classmates about how he can yell at us for not finishing the p set but if he gets an exam back in 5 weeks it's a miracle.

E/M

Semiclassical

ah, green's functions...ugh

logical123

I don't recall any greens in qm at all in his book

Balarka Sen

what's green's functions again

Faust

there not that bad

@Semiclassical think christ aweful symbols thats bad

Semiclassical

$LG(x,x')=-\delta(x-x')$

logical123

3:39 AM

Yeah

Balarka Sen

oh so they're not really functions

Semiclassical

Green's functions show up in quantum mechanics when you do scattering theory

logical123

Solutions to the impulse response

Balarka Sen

gr8

logical123

Is the most basic answer I can give

Semiclassical

3:40 AM

(where $L$ is some differential operator above)

logical123

'Some'

Yeah greens are funky, but their use is far reaching.

It's not just qm, signal processing uses em

Semiclassical

what's funny is how easy the simplest green's function is

Balarka Sen

I don't really know how distributions work in real life

Semiclassical

$\nabla^2 \phi = \delta^{3}(\vec{x}-\vec{x}')$---oh hey, a point charge generates an electric potential

@balarka In the case of Laplace's equation in 3D without boundary conditions, Green's function is literally just the electric potential of a point charge.

logical123

Most indubitably

Semiclassical

3:43 AM

it's boundary conditions which make life miserable, alas

that's my recollection from Jackson anyways.

logical123

Matching solns at boundaries is jackson

Indeed.

nitsua60

Oh, Jackson. It's on my bucket-list to re-work Jackson on my own. Because I didn't pay it the proper respect the first time.

I have memories of rectangular wave-guides that largely block out anything else =\

(Oh, and memories of going skiing when I should have been spending more time with Jackson.)

Semiclassical

"The algebra is tedious but straightforward."

Balarka Sen

@Semiclassical I see. Doesn't the Gauss's law tell me that the PDE corresponding to the electric potential of a test charge considered as uniformly distributed over a Gaussian surface in the 3-space given by $\nabla^2 V = \rho/\epsilon_0$? I guess that's the thing there

Semiclassical

Right.

nitsua60

3:48 AM

In any case, sorry to hijack the physicists-pretending-to-be-mathematicians room with a physics discussion. It's been fun, but I'm off for the night. Be well, all!

Balarka Sen

Except you're making the limit of the radius of the Gaussian sphere to be 0.

@nitsua60 See you.

Semiclassical

well, you're taking the charge density to be a delta function. so yeah, can make the gaussian sphere as small as you want without ever getting inside the charge

Balarka Sen

Gotcha. I get triggered by delta function not really being a function :P

It's limit in the approximate identity sense I suppose

logical123

And because you end job with the delta in this situation, it is a greens function, because the delta is exactly the impulse he spoke of.

Balarka Sen

Makes sense.

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$Mathematics$ Mathematics