The h Bar

Sir Cumference

12:29 AM

> Integration inverses differentiation

Trying to resist starting an integral vs. antiderivative debate again...

@NovaliumCompany Basically integration generalizes the concept behind multiplication

Jake Rose

@SirCumference oh calm down it’s not that bad ;)

@NovaliumCompany yeah but a way of understanding the integrals is that they’re the opposite of derivatives. The derivatives make really obvious intuitive sense but the integrals take more thought.

Sir Cumference

Well that'd be antiderivatives...

@JakeRose Yeah I know, it's just a petpeeve of mine. :P Imo it's often necessary to separate the two concepts, especially in cases where the fundamental theorem doesn't apply

That, and the fundamental theorem becomes a lot more intuitive when antiderivatives and integrals are distinguished

@NovaliumCompany This is an excellent explanation on how integrals can be interpreted

bolbteppa

12:46 AM

'repeated multiplication' in the sense that area is base times height i.e. area

Sir Cumference

The entire concept behind integration is that we're adding products. It's just that we're taking the number of products to infinity

Each one differs slightly from the other

So for example, let's say a man is initially moving at 5m/s, and then after 3 seconds he instantly changes to 10m/s and stays that way for 7 more seconds. How far did he travel in those 10 seconds?

To get it in meters, we'd take $3\cdot 5 + 10 \cdot 7$.

If instead he starts changing many times in that interval, we'd end up adding many products

If he is continuously changing, we'd basically be adding infinitely many products

Integration is basically like multiplying one number by another, except we bear in mind that one of the numbers changes while the other does

bolbteppa

But you can't multiply those 5 m/s by 3 bananas and get the right answer

Sir Cumference

@bolbteppa I never understood why integration is so often associated with area. I mean yes it obviously gives you the area, but in elementary school you are not told "multiplication is defined as the area of a rectangle"

It's indeed the area of a rectangle, but it limits the way you can think of its uses

bolbteppa

There's a reason integration is literally a subset of measure theory, measure = measurement i.e. area etc

How does this notion of multiplication explain probability and integration over those insane spaces

Or things like integrating that function which is 1 on the rationals and zero on irrationals

Sir Cumference

I'm giving a basic introduction on the motivation behind the Riemann integral

It's not as precise as more modern definitions, but it's helpful in gaining an intuition on the applications

At the very least it's much more general than "opposite of derivatives"

bolbteppa

12:57 AM

Yeah, but why are you multiplying

It's because of area = base times height

Sir Cumference

Y...yes. But you never think in terms of area when using multiplication in physics

Either way, the Riemann integral is still basically the limit of adding a bunch of products

@bolbteppa You get 15 bananameters per second :P

bolbteppa

Integration as opposite of derivatives is good in that $\int \frac{dy}{dx} dx = \int dy = y_2 - y_1$ makes sense i.e. you sum up the changes with respect to $x$ in $y$ as it changes with $x$ which is just the change in $y$ which is basically saying the change of the area composed of one side $dy/dx$ and the other side $dx$ after cancelling out $dx$ will just leave you with the changes in the $dy$ side with all intermediate changes cancelling out $(y_2 - y_1) + (y_3 - y_2) + ... = y_n - y_1$

Sir Cumference

@bolbteppa Well if we're gonna be precise you don't "cancel" the $dx$. I'd hesitate to rely to much on the classical notation

But yeah it's the sum of changes

Nonetheless it only matters if your integrand is the derivative of another function. Far, far more functions are integrable than antidifferentiable

The explanation in terms of multiplication is still more general

bolbteppa

Referring to 'The limit of a sum of products where one term is the value of a function within some subinterval and the other term in the product is the difference of two functions at the endpoints of that subinterval over the subintervals as the subintervals shrink to zero' (Stieltjes) as just a sum of products, you might as well just call it a sum of functions and forget the multiplication

Sir Cumference

@bolbteppa Well yes you wouldn't be wrong to say that :P But you'd lose the intuition on when it's useful

At any rate my point is that the "opposite of derivatives" explanation technically only holds in extremely specific situations, i.e. when the fundamental theorem is applicable

bolbteppa

1:09 AM

It's way way too general to say it's just a limit of a sum of products this is literally the definition of an infinite series the difference is that the terms in an integral are supposed to behave in a very specific way which a general infinite series need not satisfy

So in that sense you're right, because an integral is a limit of a sum of partial sums it is an infinite series and so you can view it as a sum of products but you're using the intuition of infinite series which extends beyond integrals so you're missing the thing that distinguishes an integral from an infinite series by taking this POV

Sir Cumference

@bolbteppa Yes, I should have been more specific when saying "sum of products". But the concept of "multiplication when one quantity changes with respect to the other" is far more useful when realizing when to apply integrals, compared to "area under the curve"

It's a looser and less precise explanation, but it's much more intuitive and nonetheless very general

Someone more awake than me should be capable of explaining my idea more accurately and still keeping that intuitive concept

@bolbteppa The most important thing imo is separating between the notions of integration and antidifferentiation. You give the intuitive explanation behind the FTC, but most people dismiss that and simply consider the two to be the exact same thing, aside from one having a constant of integration (which...should be called a constant of antidifferentiation)

This is how you get terminology like "indefinite integral"

Terminology in analysis is a complete mess tbh

bolbteppa

1:27 AM

Yeah I get it but isn't there more to it than just being an infinite series - the notion of area, approximating areas of curved regions using non-curved approximations which become accurate only in the limit to infinity, distinguishing it from arbitrary infinite series. I don't really see how it's a useful way to view integration, e.g. you're basically multiplying one thing with another thing which actually goes to zero in the limit :p what's the spark to it vs. area

Sir Cumference

@bolbteppa It's just that if I was your elementary school teacher and the only thing I told you about multiplication is that "multiplying 3 with 5 gives you the area of a 3x5 rectangle", you wouldn't intuitively realize its uses in physics

You'd dismiss it as something primarily used by geometers

Even derivative is given a more intuitive explanation. The first time you learn about it, it's in terms of rates of change, rather than strictly the slope of a tangent line

I'm obviously not saying the area definition is wrong, just that it isn't sufficient for the sake of learning practical calculus

bolbteppa

1:42 AM

Your example is not just multiplying apples and oranges though, you're multiplying things with the right units that let you use an 'area' to end up with an interpretation of one side of that area. If velocity is change of position per unit time then summing up change of position with respect to time times time leaves you with summing up changes of position, so that all intermediate differences cancel out leaving you with the overall change.

The sum of products idea is like summing areas and without specifying that one of the terms in the product has units that the other term has in it's 'denominator' so that the units 'cancel out' you'd be led even further away from physics :p

I agree it's not obvious and is unintuitive until you get it

Sir Cumference

@bolbteppa Your integrand doesn't need to be a quotient or derivative though

bolbteppa

This is why people didn't discover calculus properly for about 2000 years, simple things like this

Sir Cumference

@bolbteppa I'm not really sure I understand what you're saying by "use an 'area' to end up with an interpretation of one side of that area"

bolbteppa

"let's say a man is initially moving at 5m/s, and then after 3 seconds he instantly changes to 10m/s and stays that way for 7 more seconds. How far did he travel in those 10 seconds" in this example you're summing up the velocity times 'change in time' i.e. rate (= change in position over change in time i.e. a quotient) times time, leaving change in position, or rather change in position times the identity still giving you an area.

Sir Cumference

Again, I don't really understand what you mean by "change in position times the identity still giving you an area."

Oh wait, I think I get what you're saying

bolbteppa

1:51 AM

Like you're cancelling the units, but formally that side of the area is still there, the units were never there to begin with mathematically

Sir Cumference

Wait, now I'm not sure I'm following. Is the problem that I specifically chose a case where FTC holds?

Or is it that my example was physical, so it used units?

bolbteppa

In physics the idea behind this example is going to hold since velocity is always the derivative of position w.r.t. time i.e. we're always integrating a derivative...

Sir Cumference

Welp I'll be back in a few hours when I finished my essay.

bolbteppa

I'm 95% sure that things like divergent series completely destroy the notion that integration is like a sum of products as well btw

e.g. the whole summing $1 - 1 + 1 - 1 + ...$ to any number

Luyw

4:51 AM

there's one odd way to think about integration when it comes to interpreting it as a sum. suppose for a second that $x$ is in meters, you could think of distance as an infinite number $n$ of "points" in space, $n\to\infty$, then in this case $f(x)\Delta x$ would mean that you now have $n*f(x)$. now, as usual, you think of $\Delta x$ as vanishing to $0$, and thenthink $f(x)dx[\text{meters}]$ as just one point times $f(x)$, and so summing all these would get you the sum of all $f(x)$s

in meters that is, so if the integral equals $5.36$, you consider it as the number of points in $5.36$ meters.

this is how I actually like to think about flux, work etc

"as just one point times $f(x)$", nearly that is, approximation.

"number of points in $5.36$ meters" infinite of course, but it's a way of "summing" infinity.

if you think of it this way, it kind of also makes sense that the average of $f(x)$ is $\frac{1}{b-a}\int_a^b f(x)dx$, you're diving by the "number of points" in $b-a$ meters, which corresponds to the number of $f(x)$s. again, infinite of course, but we're just manipulating infinities using names.

Novalium Company

6:43 AM

mornink

Slereah

7:33 AM

samuel-lereah.com/db/problemdb

Look at all those dumb ass problems

now to fill them with the worst possible formalisms

ACuriousMind

8:24 AM

@Slereah ew, your URLs have spaces in them!

Slereah

As god intended!

All the site is programmed by hand

Handcrafted!

ACuriousMind

That doesn't mean you can't encode your URLs :P

I mean, it means one cannot copy-paste your URLs: samuel-lereah.com/db/problemdb/Static%20point%20charge

huh

Firefox automatically did that encoding for me!

It would appear browser developers have already programmed around people like you :P

Hm, maybe you have encoded it and Firefox is just decoding the displayed URL nowadays. I retract everything I said...

Slereah

I think I did yeah

Lemme check the code

Nope!

So anyway

why is the basic epsilon in Latex the worst one?

Nobody likes using $\epsilon$

Everyone wants $\varepsilon$

Secret

8:50 AM

Some random thought:

Give me an example of a statement that does not use any of the following notions:

containment, hierarchy, category, describability, predictability, transcendence, limit, existence, relation, narrative, mysticism, logic

ACuriousMind

@Slereah Probably because Knuth liked it better

Slereah

Damn that man

You could make an argument for $\phi$ versus $\varphi$

And everyone prefers $\pi$ to $\varpi$

ACuriousMind

I mean, he wrote TeX in his own homebrew literate variant of Pascal. That he's a bit eccentric is well-established :P

Slereah

But varepsilon is objectively better

Also this is 2019

Computers have enough memory

You can add \Alpha and \Beta and such

ACuriousMind

You can't add that to TeX because it's finished, nor to LaTeX because the chance of breaking existing sources who also define these is rather high

Slereah

8:55 AM

Not with that attitude certainly

ACuriousMind

Maybe LaTeX3 will do it, if they ever release

Slereah

10:14 AM

hm

What's another cute little physics problem

Electric conduction, perhaps

Novalium Company

11:05 AM

Knowing all the angles of a triangle, you cannot know the sides, only their ratios?

Slereah

11:15 AM

Well yes

Since you can expand the size of a triangle while keeping its angles

At least in flat space

It's actually equivalent to Euclid's fifth axiom

"There exists a pair of similar, but not congruent, triangles."

Loong

However, that's seventh grade math. You should have learned that in school.

John Rennie

They teach conformal field theory in seventh grade? :-)

skullpetrol

►

Novalium Company

11:31 AM

@Loong This is exactly the things they don't teach. I mean, I figured it out on my own since I am reading a math book. Our school builds buildings on loose foundations.

@skullpetrol We've learned about the similarity topic, but as I've said, Bulgarian school system is bad

I basically go to school to finish school. Most of my studies take place in home teacher - me.

skullpetrol

@NovaliumCompany episode 1 will answer your question my friend

Slereah

@JohnRennie I mean they teach homothetic transformations, certainly

it's not rocket science

small shape becomes big shape

skullpetrol

11:48 AM

Brain Surgeon - That Mitchell & Webb Look , Series 3 - BBC Two

►

nor brain surgery :-)

PM 2Ring

12:31 PM

Oh great. JD is backing up safesphere physics.stackexchange.com/questions/492477/… I suppose I could just delete all my comments in that thread...

ACuriousMind

@PM2Ring Just flag cases where comment discussions spin off into not discussing the actual post they're on like that.

Sometimes users with idiosyncratic views like to debate in comments instead of posting answers because they can't be downvoted in comments. Don't take the bait.

PM 2Ring

@Loong If not earlier. I'm sure we were at least introduced to the topic of similar triangles in 5th or 6th grade. I'm pretty sure I remember proving that a triangle on a diameter of a circle is a right triangle by 6th or maybe 7th grade.

skullpetrol

moving it to a chat room where

> This user is suspended on the parent site and cannot chat for 192 days.

PM 2Ring

@ACuriousMind Yeah, sorry. I'm partly to blame, for continuing the discussion. I should've just posted the Wikipedia quote & bailed out.

skullpetrol

sounds like solitary confinement to me :-)

ACuriousMind

12:41 PM

It's not

PM 2Ring

@skullpetrol How does that work? Clearly JD can now post on Physics, which appears to be his parent site.

ACuriousMind

@PM2Ring If you get suspended in chat because you get suspended on your parent site, you can change your chat parent, but the suspension and message remains the same

I.e. that suspension does not come from a suspension on Physics

PM 2Ring

@ACuriousMind Ah, right.

So does that mean that JD is suspended from all chats?

Loong

all chats on chat.stackexchange.com

PM 2Ring

Thanks

skullpetrol

12:51 PM

so he is in a chat room where he can not chat

PM 2Ring

Excellent.

skullpetrol

it happens all the time

they can go over to the stack overflow chat rooms and chat

PM 2Ring

@Loong I see your Barium-133 question still doesn't have any responses. Oh well. I don't know much nuclear stuff, but it looks like an interesting puzzle.

@skullpetrol Right. I don't expect he'll try to harass me in the SO Python room. :)

skullpetrol

:-)

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