Mathematics - 2023-08-16 (page 2 of 2)

Ted Shifrin

18:00

It is sort of how I don't like the way almost every calculus book does multivariable calculus, including the order of topics in the differentiation chapter.

Semiclassical

the other opinion i have in that vein is that i kinda wish profs would focus kinematics problems on just those involving time

Ted Shifrin

I'm not sure I get that last wish. So what if I want to know at what height the point mass sliding down a sphere flies off?

Semiclassical

then you wait until you do conservation of energy

Ted Shifrin

Yes, it's force diagrams plus conservation of energy. So I guess I don't know what your objection/recommendation is.

Semiclassical

it's a matter of when those problems get emphasized in the course

do you do them only once you can put them under the rubric of energy conservation?

or do you already start to do that by including the formula $v_f^2=v_i^2+2a(x_f-x_i)$ in the list of major kinematic equations?

typically, courses do the latter

and expect you to already be able to apply that one in relevant cases

Ted Shifrin

18:05

Especially since students might apply such a formula forgetting the acceleration isn't constant.

Semiclassical

right

you can derive that formula from $x_f=x_i+v_i t+\frac12 at^2$ and $v_f=v_i+at$, so it's not like it's logically unsound to do in the context of constant-acceleration kinematics

but in terms of teaching i feel like there's enough of a hurdle getting them comfortable with the concept of "what information do i know at which time"

Ted Shifrin

When I taught Calc I and did a bit of differential equations at the end, I always tried to make sure there was an acceleration function where you couldn't just throw away $t=0$ from the antiderivative.

user 726941

My high school physics textbook decided to switch the order of presenting dynamics before kinematics, they said it was more "natural for students."

Semiclassical

bleh

Ted Shifrin

Students (and people who post on MSE) seem to forget part of the FTC because so often it turns into $0$.

I'm not sure I understand the distinction between dynamics and kinematics.

Semiclassical

18:08

in the context of intro physics, kinematics is "how do i solve the differential equation da/dt=0 with particular conditions"

whereas dynamics is how the acceleration in a problem is dictated by list of forces acting on an object (and vice versa)

Ted Shifrin

I guess the K/K course did pretty much only the latter. The former sounds like plug-and-chug formulas with no physics in it.

Semiclassical

well, i'm saying it in the math way. the physics content is understanding how position/velocity/acceleration are defined in the first place, which isn't trivial for intro students

particularly if they're not exactly calc-masters

user 726941

They said students have experience with forces and tend to get bogged down in kinematics.

Semiclassical

buuuuuuuuuullllllll

i won't dispute the latter part of it

Ted Shifrin

shoves Semiclassic against the wall — see, familiar with forces

Semiclassical

18:12

but to the extent that students have 'experience' with forces, it's with plenty of misconceptions

in particular, free-body diagrams are not trivial

user 726941

It sounded like a time management decision on where to focus attention.

Getting bogged down is a time sink.

Semiclassical

the issue with starting with kinematics is that, because we start by explaining the constant-acceleration formulas for position/velocity, the problems you get have a tendency to devolve into solving the quadratic formula

Ted Shifrin

Almost every course bogs down in places. It takes a lot of teaching experience to avoid traps, and some good teachers don't want to leave any students in the dark, so bog down unavoidably.

Semiclassical

which students hate

Ted Shifrin

I hate it too.

At what time did the ball hit the ground? So exciting. :D

Semiclassical

18:15

lol

i think you gotta bite that bullet, because understanding the definitions of position/velocity/acceleration is so critical to the rest

copper.hat

I think 'bogging down' happens because some stuff just requires a lot of grind.

Semiclassical

where i will agree is that i think profs aren't good at picking test problems on kinematics

Ted Shifrin

In the end, in foreign languages, math, physics, chemistry ... difficulties always devolve from a lack of mastery of the prerequisite materials.

user 726941

Grind? In what sense @copper.hat

copper.hat

Dealing with detail, meta reasoning, etc.

Ted Shifrin

18:17

Having earned a C (hopefully) in precalculus does not mean the student has enough mastery to be successful in beginning calculus, beginning chemistry (for sure!), or beginning physics (also for sure).

Semiclassical

kinematics problems tend to be somehow both overly formal and plug-and-chug

so it ends up feeling to students just like algebra exercise

copper.hat

because the focus is on rote learning rather than understanding

Ted Shifrin

@Semiclassical Well, that's because it really is.

Semiclassical

yep

my way of getting around that to some extent is to emphasize drawing relevant graphs

so that at least there's that level of geometric content

"where do these curves cross" etc

Ted Shifrin

You mean, pull out the graphing calculator and let it do it.

Semiclassical

18:19

eh, not really. i typically emphasize the velocity vs time graph

at which point it's usually something piecewise linear

copper.hat

they have their place, but grinding through often sheds light on stuff

Semiclassical

if you do the position vs time graph, i agree

(and acceleration vs time is usually a pretty useless graph)

Ted Shifrin

Actually, it's probably worth having them find distance traveled graphically by finding the areas of those triangles and rectangles.

Semiclassical

exactly

Ted Shifrin

Where was it highest? Why?

That sort of thing came into the "new" calculus curriculum of the 90s a lot. I sort of like that.

Semiclassical

18:20

the only bit you can't so easily include in a velocity vs time graph is the initial positions of the objects, but i think that's a reasonable tradeoff

Sine of the Time

@PM2Ring Obviously I didn't create such an account, I'm not interested in rep points

Ted Shifrin

Words are allowed, Semiclassic.

Semiclassical

ya

copper.hat

i need rep to get my jump suit.

user 726941

So do we stick with tradition and present kinematics first?

Semiclassical

18:23

that said, one deficit from a lab point of view is that you can only work with situations where the acceleration is not just constant but moreover known

so if you want to have them analyze a cart rolling down a ramp, then you run into the issue of "why is the acceleration $a=g\sin\theta$ down the ramp?"

which is a bit annoying. but you can still pose it as a question of analyzing the motion of the cart, discovering that it's constant acceleration motion, and then emperically work out that said acceleration is some fraction of $g$

user 726941

Full disclosure: I never used the textbook that switched the order :P

Semiclassical

lol

the more defensible re-ordering is doing work/energy before doing forces

i dunno if i'm sold on that one either, but there is something to be said for putting energy front-and-center

user 726941

How are you going to motivate that?

Semiclassical

good question. i'm not so familiar with it so i don't actually know

you focus on conservation of energy as fundamental, i guess? which is actually closer to how it works in analytical mechanics due to the Lagrangian and Hamiltonian formulatiosn

also, more focus on scalars than vectors to start with

user 726941

You could try using I.B.L.

Semiclassical

18:37

well, i'm typically working in a context of large-student intro classes

so you run into problems of scale

user 726941

Right.

Students do have a lot of misconceptions about energy.

Semiclassical

yeah

and an energy-first approach really relies on them getting comfortable with integral and differential calculus

which is not necessarily a deal-breaker: if you expose them to it more, then that will give them more experience with it

but it's not a trivial choice

user 726941

Do you allow them to write their own formula sheet?

Semiclassical

ah, that old question

i'm in the "we provide the formula sheet" camp

user 726941

Ok, np :-)

Semiclassical

18:43

firstly because, in a big course, it's hard to meaningfully police whether the formula sheet they generate is "appropriate"

but moreover i think it just puts everyone on a clearer footing. they're all given the same 'weapons' and it's up to them to use them

Ted Shifrin

@Semiclassical Not sure how you put work before forces.

Semiclassical

by taking energy conservation as the foundational principle, basically

i'll include a snapshot from one of the main papers on this

Ted Shifrin

@Semiclassical I did that, too, in diff geo, which is the only time I allowed formula sheets.

I get energy first. Just not work. Work done by the external forces? The who?

Semiclassical

yeah, it's really energy-first and not work

Ted Shifrin

Yes, of course. Hamiltonians before Newtonian is cool.

Semiclassical

18:49

i go back-and-forth on it as an actual curriculum

Ted Shifrin

I think it is math heavier and more abstract.

Semiclassical

ya

they do address that a few paragraphs later: "It is clear from Table I that our new energy-first curriculum relies more heavily on both differential and integral calculus than the traditional force-first curriculum. For example, PHSX 211 students are tasked with using calculus routinely when covering 1D and 2D translational motion,
circular motion, rotational motion, and oscillatory motion, whereas PHSX 210 students are not. We hoped that this repeated use of calculus through the curriculum would help students improve their fluency with applying calculus to solving physics problems."

and there is something to be said for "you don't get expertise by avoiding practice"

Ted Shifrin

It avoids force diagrams and trig?

leslie townes

mm, lenin advocated a force-first framework. "To defeat capitalism in general, it is necessary, in the first place, to defeat the exploiters and to uphold the power of the exploited, namely, to accomplish the task of overthrowing the exploiters by revolutionary forces; in the second place, to accomplish the constructive task, that of establishing new economic relations, of setting an example of how this should be done. "

Semiclassical

initially, yes

leslie townes

18:54

your counter-revolutionary approach is doomed to fail

Semiclassical

@leslietownes ah, but Marxism is itself an energy-first approach. it relies on understanding where power exists in class society!

Ted Shifrin

Power is work on steroids.

Semiclassical

(do not take me at all seriously, to be clear)

Ted Shifrin

LOL

Were we taking leslie seriously?

Jack Smith might subpoena us?

Semiclassical

"the use of vectors is delayed until after classical mechanics has been developed using energy-based and calculus-based approaches. The presentation of these concepts follows the traditional pedagogy of free-body diagrams and force decomposition, but the students are taught first to associate forces with a change of energy through work, rather than with an acceleration using Newton’s 2nd law."

Ted Shifrin

18:57

It’s less geometric, for sure. Leslie should approve.

Semiclassical

presumably the emphasis of trig is hand-in-hand with the introduction of vectors

leslie townes

mm, less geometric? okay. all power to the soviets.

Jakobian

Is there something like mild ketchup and spicy ketchup in English?

I'm asking because in Poland we usually have two types of ketchup according to if it's spicy (like with paprika) or more mild

so I'm wondering if it's the same in English and how you call those types of ketchup in English

Semiclassical

not really? in America i think we'd classify the latter as some form of "hot sauce"

leslie townes

jakob not really. a certain kind of restaurant will distinguish itself by making its own ketchup, and that ketchup may not taste like heinz ketchup, but i don't think we have a word for it.

Ted Shifrin

19:09

No. There is chili sauce, a more interesting version of ketchup, but not spicy.

Semiclassical

Heinz ketchup is the default here

they do sell different versions of Heinz, but the baseline is the standard one

leslie townes

oh yeah. also lots of adjacent things once you leave the realm of ketchup, but none of them would be called ketchup.

Jakobian

I think the type of more spicy ketchup I'm refering to is called "hot ketchup"?

leslie townes

the committee investigating ketchup vs. catsup hasn't even come out with their final report. we're way behind on this, as a language.

Semiclassical

there's also this, which isn't anything culturally relevant but does sit rent-free in my brain since this summer: foodandwine.com/…

leslie townes

19:11

i could understand why you might call it that, but if someone asked if i wanted "hot ketchup" i would ask what they meant by that before saying yes. it doesn't have a standard meaning in my mind.

semi: i would totally try that but not on a hot dog.

Jakobian

I see so it's not a thing to distinguish between two types of ketchup in America

Semiclassical

yeah. we do talk about "hot sauce" a lot tho

@leslietownes well, there is a recipe for it out there...brit.co/pepsi-ketchup-recipe

Jakobian

D.C. the III

@TedShifrin you didin't go over the proof of the equivalence of the three definitions of a manifold in your lectures? (specifically parameterized form to level set).........I'm hurt ...and a little confused. 😢

Jakobian

for example here you have "mild ketchup" on the label

I guess this is specifically because it's sold in Poland

Semiclassical

19:15

regional marketing differences yeah

PNDas

Hi! My book writes: If $w\in L^{2n/n+2}(U)$ then the linear functional $f(u)=\int_U wu\,dx$ (where $u\in H^1_0(U)$) is blah blah. I want to know why this is well defined. Here $U$ is bounded.

I mean why $\int_U wu\,dx<\infty$. Is it using some generalized Holder's inequality?

Ted Shifrin

@D.C.theIII Not enough time and too technical for a first course. I’m sorry you’re hurt.

D.C. the III

Oh.....so I shouldn't be losing my mind over understanding the proof then?

because that's what's happening...

Ted Shifrin

Not at this stage, no. Didn’t I do parametric to graph in the book? Then graph to level set is easy.

@PNDas Parentheses are needed. This seems more Sobolev inequality, but not my wheel house at this point.

D.C. the III

I "get" the gist of it and all, but it is the technical stuff that and the logical steps that are leaving me a bit perplexed. No in the book you go "Implicit" --> "explicit" --> "parametric" --> "implicit"

So I'll just go over some details of the proof again at this moment and won't lose my head becuase I don't understand it all, but make note of what I'm having trouble with

PNDas

19:29

@TedShifrin Thank you. You are right.

Ted Shifrin

oh, ok. Right. Understand the example, @D.C.theIII

D.C. the III

Yea, I'm working up to that. I'm thinking understanding the proof a lil bit more will help with the example.

Ted Shifrin

No, other way around.

D.C. the III

Ok. I'll come back and pester in a few hrs when I 'm working through the P-Set

leslie townes

it's kind of funny about manifolds, there's a certain amount of technical setup you need to discuss anything, and almost none of it is fun or interesting, so a lot of instructors will skip or minimize it. leading to ted having to remind people of the basics of embedded submanifolds of R^n, over and over, in the MSE comments.

as if it's all just one big prank, being pulled on ted

Ted Shifrin

19:36

Imagine if Ted were as paranoid as, say, Comrade Donny.

Or half as narcissistic.

Semiclassical

@TedShifrin heavens to murgatroyd

D.C. the III

Well I do understand the proof more after going over it again, it is highly techincal and I couldn't reproduce it, but I have a better idea of how you are using the previous ideas. Notably I see the way the inverse function theorem was applied. ....but we go on and future passes of it will elucidate more (big word of the day)

sunny

> Definition. If the power series $\sum_{n=0}^\infty a_n (x-c)^n$ converges for $|x-c|<R$ and diverges for $|x-c|>R$, then $0\leq R\leq \infty$ is called the radius of convergence of the power series.

I'm slightly puzzled by this definition. $R$ can equal $0$, and then there are no $x$ for which the series converges, not even $x=c$, right? Because $|c-c|=0\not <0$.

Ted Shifrin

20:01

Good point. The 0 series converges, so you must allow $|x-c|=0$.

leslie townes

the definition given is generally silent on what happens when |x - c| = R (as it should be). when R = 0 there will always be that one point of convergence.

sunny

@leslietownes hmm, how? The definition is explicitly saying the power series converges for $|x-c|<R$ and that $R$ can equal $0$. So which $x$ satisfy $|x-c|<0$? None, right?

leslie townes

R is best thought of as a rough measurement of the "size" of your set of convergence, but not as a specification of that set. so R being 0 just means there isn't a lot of convergence, not that the set of convergence is empty.

sunny: none, but that still doesn't say anything about what happens when |x - c| = 0.

sunny

true

leslie townes

when R is 0 one of the conditions never holds and the other says that the series diverges for x not equal to c. it is silent on what happens at c (although as you and ted and i have noticed, there is no need for silence in this case)

Ted Shifrin

20:07

The radius of conv is really a sup.

We just cannot define it that way in a calculus class.

冥王 Hades

Should I also post a ketchup bottle

Ted Shifrin

Japanese mayo would be better.

冥王 Hades

Funny. I had it as a topping with Okonomiyaki

Semiclassical

the disk of convergence problem i remember having a devil of a time with was the binomial series

not the interior of the disk, which is easy enough

but i remember the behavior on the circumference being surprisingly annoying

leslie townes

you don't have to be newton to analyze the binomial series, but it sure helps

Semiclassical

20:16

reminds me (admittedly randomly) that one Newton-thing i never really grokked was Newton polygons

leslie townes

yeah, those are weird. they come up outside of algebra, too, although you'd never know that from wikipedia. is that what physicists use them for?

Semiclassical

i don't think i've ever seen them in physics tbh

leslie townes

oh. they pop up in analyzing the behavior of certain oscillatory integrals.

Semiclassical

huh

i have seen them show up in the context of series expansions

leslie townes

varchenko is the guy i associate this with. www2.math.upenn.edu/~pemantle/2009-SS/varchenko.pdf is a copy of the paper i had heard of.

Semiclassical

20:22

which does match up in my brain with the connection to oscillatory integrals in an occult sort of way

huh, nice

leslie townes

i was going to give a shoutout to him for not including a single picture of a newton polygon in that paper, but then i saw his figure 1. :(

oh, he cites VI Arnold for a bunch of stuff in there.

figures it goes back to arnold.

Semiclassical

i know that oscillatory integrals in general connect up with sectorial series expansions, and thus with puiseaux. so not too shocking

(sectorial is the wrong word for it but i dont' remember the right one)

right, the fact that asymptotic series expansions typically only make sense in a sector of the complex plane

Jakobian

@sunny the series always converges when $|x-c| < 0$

leslie townes

sectorial expansion, such as expanding the concept of collective ownership of the means of production from the industrial sector to collective farms in the agricultural sector.

as always, lenin pointed the way

Semiclassical

i'd make some clever reference to Trotsky but i got nothing

leslie townes

20:26

i vote for semiclassical to be expelled from the chat for acknowledging the existence of trotsky

Semiclassical

also something-something Bukharin

leslie townes

if i were a room owner, these fictitious names would disappear and be replaced with clumsily airbrushed background.

Jakobian

How you prove that something holds is check for every case. Here we have to check that for $x$ such that $|x-c| < 0$, the series converges. But no $x$ satisfies $|x-c| < 0$, which means we already checked every case. Thus it must converge for every $x$ such that $|x-c| < 0$

leslie townes

it also diverges for all such x. it is a very special region of convergence.

sunny

@Jakobian I see, you meant this vacuously.

Jakobian

20:31

I don't like the word vacuously

it's just another meaning for empty quantifier which is just definition

Ted Shifrin

You need a separate definition for $R=0$. As I said, thinking of the radius if conv as a sup is a good idea.

Jakobian

@TedShifrin why is that? From what I see, the definition is valid, for $R = 0$ or otherwise

@leslietownes yeah. Thankfully this doesn't matter for our definition

Ted Shifrin

Only if you stop to say that it diverges for all $|x-c|>0$. Sure.

robjohn

22:25

@Jakobian From Halloween

Jakobian

oo spooky

robjohn

I have two bottles of Tomato Blood

Jakobian

does an algebraic closure of a local field need to be a local field?

Hm, so this isn't using complex analysis or anything, but as best I can tell it is still using particularities of $\mathbb{C}$ like local compactness? This still doesn't transfer to other settings? — Harry Altman 36 mins ago

this person is asking me if my proof of continuity of roots generalizes from $\mathbb{C}$, I'm guessing it does but it needs some tweaking
I never worked with local fields or anything like that (honestly I learned the word today)

leslie townes

you get into these weird non-complex-analysis arguments, jakobian, you find yourself having weird conversations with weird people. "local fields." shrug

Jakobian

I mean I'd care if I knew what it is

I guess the one commenting needs to tweak the argument himself if he wants a generalization

leslie townes

22:42

haha, look at the number of questions in the OP. submit a vote to close, needs more focus.

Jakobian

23:44

@leslietownes that's true actually. I feel like people would hunt me down with torches medieval style if I did that

leslie townes

the site was just different in 2011. i might non-jokingly vote to close if it were posted today and people had yet to answer.

robjohn

23:57

@leslietownes Yeah, I wasn't here for half the year.

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$Mathematics$ Mathematics