/sci/ - Science & Math

[Anonymous No.10600483]

Can anyone here explain to me the real meanining of the "dx/dy/dz/ds" at the end of integrals? My math classes have kept moving forword without explicitly defining it and its giving me troubles now that ive reached "Calculus of variations".

[Anonymous No.10600518]

>>10600483
it's literally just notation to evoke the idea of the riemann / lebesgue definitions of the integral

[Anonymous No.10600524]

something something riemann sum something something delta X

[Nigger Faggot No.10600527]

>>10600518
Yep. It’s a notational convention. Also important to note: the separation of variables technique treats the differentials as algebraic objects, even though they are not. Don’t get too caught up on the meaning though OP

[Anonymous No.10600536]

>>10600527
>>10600518

[Anonymous No.10600558]

>>10600527
>what is a differential form

[Anonymous No.10600583]

>>10600483
you havent gotten to the calculus of variations without knowing this basic shit you larping faggot

[Anonymous No.10601051]

>>10600518
>>10600527
>>10600524
OP here:
Ok, so for example, look at the pic related equality for a surface integral (yes its a phone pic fuckoff). Would I be incorrect in saying that dS = |ru x rv| dA
If not: Why?
If so: How can I conceptualize their relation, or understand what's happening more deeply?

>>10600583
Nigger why the fuck am I gonna lie about this and if its so basic then give me the basic explination faggot. I know it comes from the change in x of the definition of the integral from the riemann sum already.

[Anonymous No.10601060]

>>10600518
>>10600527
>>10600524
OP here:
Ok, so for example, look at the pic related equality for a surface integral (yes its a phone pic fuckoff). Would I be incorrect in saying that dS = |ru x rv| dA
If incorrect: Why?
If correct: How can I better conceptualize their relation, or understand what's happening more deeply?

>>10600583
Nigger why the fuck am I gonna lie about this and if its so basic then give me the basic explination faggot. I know it comes from the change in x of the definition of the integral from the riemann sum already.

[Anonymous No.10601066]

>>10600518
how can anyone be this confident in their ignorance?

dx is a fucking differential form

[Anonymous No.10601085]

>>10600483
https://youtu.be/2ooWs_8hzxQ?t=6m50s

[Anonymous No.10601556]

>>10601066
how can anyone be this confidebt in their ignorance?

dx can be both a differential form and a notational convention. It depends entirely on the framework used.

[Anonymous No.10601575]

>>10600483

A very little difference in x/y/z value. Also integral is just "sum".

[Anonymous No.10601584]

>>10600483
dx is an infinitesimal amount of x. An integral is the representation of summing something up infinitely. You're summing up infinite infinitesimal x values to get a finite answer.

[Anonymous No.10601589]

>>10600483

it depends on the context

can be a measure
can be a density
can be seen as a differential 1-form

[Anonymous No.10601736]

>>10600558
>>10601066

The differential form is used in the context of linear approximations. If you are dealing with integrals or differential equations then the “differentials” are just a notation. Anyone who says the dx in an integral is a differential (i.e. literally a variable) is a fucking moron. Case closed.

[Anonymous No.10601744]

>>10601556
Finally someone who knows what the fuck they are talking about. The retards here actually believe dy/dx is a quotient of differentials and not just a notation emphasizing differentiation. These fuckin people man

[Anonymous No.10601751]

>>10600483
it is a differential form. be patient, you will learn about this in a differential geometry class.

everyone who says that it's an "infinitesimal" is a total brainlet

[Anonymous No.10601753]

>>10601589

best answer in this thread

[Anonymous No.10601758]

>>10601589
lol

[Anonymous No.10601761]

>>10601736
brainlet

[Anonymous No.10601765]

>>10600483
a lot of people in this thread don’t understand calculus

[Anonymous No.10601766]

>>10601589
Op here:
Can you briefly explain those 3 examples? In my case I'm mostly dealing with spatial problems. Surface integrals, vector fields, optimization, stuff like that. Which of those 3 fit that context?

[Anonymous No.10601775]

>>10601766
he’s just making shit up

[Anonymous No.10601778]

>>10601775

wrong

[Anonymous No.10601803]

>>10601775
How to spot the CS student : 101

[Anonymous No.10601877]

>>10601761
>unironically calling someone a brainlet and not refuting anything they say

[Anonymous No.10601906]

>>10600483
Dont overthink it at this level. Also Lagrangian mechanics (which i assume you're doing) is not by any means "getting into calculus of variations". For this usage, just treat variations as small differences of functions (or linear "corrections"). The chain rule also holds nad is basically the only property you'll use (and also that the variations commute with derivatives): [eqn] \delta \mathcal{L} (q,\dot{q}, t) = \frac{\partial \mathcal{L} }{\partial q} \delta q + \frac{\partial \mathcal{L} }{\partial \dot{q|} \delta \dot{q} + \frac{\partial \mathcal{L} }{\partial t} [/eqn]

As for your question, the other anons pretty much cover it: they are differential forms (or antisymmetric 1-tensors) as far as physicists are concerned.

[Anonymous No.10601913]

[eqn]
\delta \mathcal{L} (q,\dot{q}, t) = \frac{\partial \mathcal{L} }{\partial q} \delta q + \frac{\partial \mathcal{L} }{\partial \dot{q}} \delta \dot{q} + \frac{\partial \mathcal{L} }{\partial t} [/eqn]

>>10601906
sorry, bad TeX

[Anonymous No.10601933]

>>10600483

There are some pretty good answers in this thread regarding the mathematical definition of this notation, but I'd like to add some physical intuition might help OP:

The "dx/dy/dz/ds" can be thought of as telling you that the integral is a sum over tiny chunks of a variable. The letter following the "d" tells you which variable is to be subdivided into small chunks.

If there is a "dA" in two dimensions, then that means that the chunk has units of area. You could choose to represent this as a product of "dx dy", i.e. two terms with units of length. Alternatively, you could use polar coordinates, but in that case you would also have to include a factor that tells you that a unit of physical area dA corresponds an area r dr d\theta in the polar coordinates. That additional factor is the Jacobian, and that may explain your confusion about the different representations.

[Anonymous No.10601997]

>>10601060
If I remember correctly, that relationship between dS and dA comes from a change of variables.

[Anonymous No.10602167]

>>10601736
[math]f(x)\mathrm{d}x[/math] is a differential form, or more precisely 1-form. It has nothing to do with your silly approximations.

[Anonymous No.10602187]

>>10601589
"density" is just a measure. [math]\mathrm{d}\mu[/math] is used just as a convention, to make it similar to the customary [math]\mathrm{d}x[/math] and [math]\int_X f \mu[/math] where [math]X[/math] is a space and [math]\mu[/math] is a totally valid, though uncommon notation.
In [math]\int_X f \mathrm{d}x[/math] the whole [math] f \mathrm{d}x[/math] is a differential form.

[Anonymous No.10602191]

>>10602167
>f(x)dx and not f’(x)dx

[Anonymous No.10602203]

>>10602191
>every 1-form is exact
Lmao. You're not this retarded right?

[Anonymous No.10602204]

>>10602191
What's your problem? [eqn]\sum f_{i_1,...,i_n} \mathrm{d}x_{i_1}\wedge...\wedge \mathrm{d}x_{i_n}[/eqn] is a differential form, so I can say [math]f \mathrm{d}x[/math] is as well.

[Anonymous No.10602218]

>>10602203
>he still doesn’t understand the history of the differential form and it’s meaning in the context of elementary calculus

[Anonymous No.10602221]

>>10602218
Stop being retarded. I have no time for the likes of you.

[Anonymous No.10602242]

>>10602221
>I don’t really know what I’m talking about so I’m gunna say ur retarded

[Anonymous No.10602251]

>>10600483
Infinitesimally small x/y/z/s

[Anonymous No.10602979]

>>10601933
Op here:
Thanks bro. I think that's actually what I was looking for. So in the pic related equality, or when changing coordinate systems, the extra terms that arise, (EX: dS "becomes" |ru x rv| dA or dx dy becomes r dr dtheta) are arising from the Jacobian transformation?

I was thinking there was some way to easily derive what d(variable) should become when changing intergral types.
But, if it comes from the Jacobian aint no one got time for that.

[Anonymous No.10603050]

>>10602167
Ok what the fuck? How much of a brainlet are you? Linear approximations using differentials form the basis of the physical meaning of the FToC. There is nothing wrong with defining it in that way because that is the historical view of differentials.

>>10602979
Historically the differential was defined by the derivative of a function at a point and some change in x, called dx. This produces a change in the function denoted by dy. Basically dy is a variable dependent on dx, which can be any real number. That’s all there is to it. The basic concept can be extended to multi variable functions. In integrals it is just a notation defining what you are integrating with respect to.

[Anonymous No.10603094]

>>10600483
Maths people love overloading symbols.
Dx can be just a notation, a reminder of your measure or weight function (in the context of Lebesgue or Stiejenes integration), a diferential form, an infinitesimal, the tangent vector of your path of integration, the normal of the surface you are integrating on, etc.
The fact that a lot of those thing are related, or that books might switch meanings in the very same page just helps to make it all confusing.

[Anonymous No.10603183]

>>10601933
>The "dx/dy/dz/ds" can be thought of as telling you that the integral is a sum over tiny chunks of a variable. The letter following the "d" tells you which variable is to be subdivided into small chunks.
So is this >>10601584 actually correct, and this >>10601751 misleading?

[Anonymous No.10603190]

>>10603183
no >>10601584
is wrong, infinitesimals are bullshit
>>10601751
is right but incomplete
it is a differential form, but thats too advanced and it can have more meaning than just a d form
and >>10601933
is as right as you should need for understanding

[Anonymous No.10603229]

>>10603190
The last guy refers to "tiny chunks". Are those not infinitesimal?

[Anonymous No.10603236]

[math]{\d}x[/math]?

[Anonymous No.10603245]

>>10603229
no, they're finite. you use the tiny chunks as an approximation and as the chunks get smaller the approximation gets better, but they arent infinitesimal, theyre always some non zero size. its one way of approaching how to describe it, i personally dont like it much but it works as an introduction.

[Anonymous No.10603249]

>>10603229
Yes, they are.

>>10603245
Yes they are

[Anonymous No.10603290]

>>10600483
To be fair, You need at least 3th grade of pure mathematics in university to truly understand a meaning of dy/dx.
The differential form is extremely subtle, and without a solid grasp of differential geometry most of the theories will go over a typical high schooler's head.
Just imagine one of those addlepated simpletons scratching their heads in confusion as they cannot understand the true nature of Stokes' theorem unfolds itself on our human brain. What fools.. how I pity them.

[Anonymous No.10603555]

>>10601066
stupid geometrykid. us measure-chads are so much fucking smarter than you it's not even god damn funny. your "forms" are a complete meme. measure is the only meaningful basis for integration.

[Anonymous No.10603565]

>>10600483
How on earth your math classes teach calculus of variations without teaching what an integral is? It sounds like you didn't pay attention.

[Anonymous No.10603568]

>>10603555
Holy fuck! Based measure chad. Hot damn sir you told that faggot the FUCK off!

[Anonymous No.10603573]

>>10603555
>This brainlet doesn't know about geometric measure theory

[Anonymous No.10603587]

>>10603565
You don't even know enough about proofs to understand why the question I'm asking isn't as simple as it looks. Sounds like you went to a shit school.

[Anonymous No.10603593]

>>10602242
u r tho

[Anonymous No.10603597]

>>10603593
no u

[Anonymous No.10603598]

>>10603587
What the fuck are you talking about? Your shit question is as simple as it looks if you know what calculus is.

[Anonymous No.10603612]

>>10603598
>If you know what calculus is
Ok, no way this is not a troll.

If and only if this is a troll then,

gr8b8m8

Such that,

I actually thought you were retarded for a second.

Therefore,

I'm not taking your bait anymore.

[Anonymous No.10603617]

It just stands for an infinitely small increment of x. So if you think of a Riemann sum, the height of each bar is the function and the width is the change in x, which in this case is the infinitely small 'dx'.

[Anonymous No.10603631]

>>10603612
Mate don’t worry about him. What is your mathematical background so far? Before Calculus of Variations

[Anonymous No.10603640]

It’s the width/area/volume/etc of the Riemann sum slice

[Anonymous No.10603641]

>>10603631
So you agree with him that is more complicated because proofs? Its still a riemann integral in calculus of variations you fuck.

[Anonymous No.10603642]

>>10600483
Lol dude I thought I could answer your question because I have taken elementary calculus and then I searched up “calculus of variations.” Holy hell dude that crap is confusing as hell. I will study it eventually but damn bro I feel sorry for you haha

[Anonymous No.10603647]

>>10603642
Calculus of variations can be developed with ordinary calculus. The fundamental lemma is all you need.

[Anonymous No.10603648]

>>10603631
As far as "basic" calculus goes I know through multivariable calc except for second order differential equations. Surface integrals, stokes theorem, vector fields, curl and divergence to give the gist.
Idk why everyone in this thread keeps
saying "omg! calc of variations how can you be doing that and not understand the dx?!" I'm trying to figure out how to better understand the derivation of integral type transformations and coordinate changes, its not something they make explicit in school, they just dump formulas on your head. Which is why faggots like >>10603641 end up thinking they "know what calculus is" because they saw (reimann summ = intergral) in a textbook.

>>10603642
Bruh, I just fucking euler-lagrange that shit till the surface is proven minimal breh.

[Anonymous No.10603649]

>>10603641
Nibba don’t even pretend for one second that you have never been confused in school before. Sometimes we have gaps in our knowledge and all it takes is a little bit to “get it.” Obviously there is something op just hasn’t “gotten” yet. So why be a dick about it if we’ve all been there? Unfortunately I don’t know anything about calculus of variations or anything about analysis past elementary calculus so I can’t help him.

[Anonymous No.10603672]

>>10603648
So you claim to know multivariable calc and you dont know about the change of variable theorem? You just asked "guyz wtf dx is" which has no depth beyond explaining notation. You want to understand exaclty what? Yes intuitively it's a differential lenght that when multiplying with f gives you a differential area. It is more modernly just telling you which variable you are integrating and it's just notation for the limit of partial riemannian sums, it could also mean integration wrt the lebesgue measure on the real number line and it could mean it's a 1 form, and many more integration concepts. It's still just a fucking integral, it geometrically gives you the area under a curve and it's related to differentiation through the FTC.

[Anonymous No.10603674]

>>10603649
Reddit is leaking again.

[Anonymous No.10603680]

>>10603672
Don’t listen to this fuck face OP. I had the same issue in an engineering class of mine. It felt like everyone understood it but I eventually found out it was because nobody understood it so they couldn’t explain it well. I finally figured it out on my own but I have yet to find anyone that can give a satisfactory explanation besides what’s in my own head. Apparently everyone on the internet is a genius lol

[Anonymous No.10603681]

>>10603674
Nah you’re just an ass. Human struggle is not exclusive to reddit fags, although they do bitch way too much

[Anonymous No.10603684]

>>10603680
>Engineering
>fuck face
>thinks no one knows what an integral is

[Anonymous No.10603690]

>>10603681
Nah m8, you just type as a redditor faggot.

[Anonymous No.10603692]

>>10603684
>acts like a genius hard ass online
>is not actually like that irl

[Anonymous No.10603695]

>>10603672
See
>>10601060
and
>>10602979
Do you actually read the thread or just jump straight into the shitposting? It's like you're trying to math flex on me or something. ●Have sex●. Multivariable calc is not that difficult or rigorous. Also, that entire post reads like you just ripped it off of wikipedia so you could math flex on an anonymous Leibniz fanfic enthisiant fourm.
>>10603680
Appriciate the support bro but honestly whoever >>10603684 is has some serious insecurity issues. I'm picturing an angry man behind a computer screen yelling "He doesn't know multivariable calc! Only supreme gentleman 300Iq patricians such as myself are capable of handling more than 1 variable!" It's kinda funny honestly.

[Anonymous No.10603697]

>>10603690
How so? I can’t stand reddit. Anonymity doesn’t mean you have to be a jerk

[Anonymous No.10603704]

>>10603695
Lol yeah for sure. Well I’ll go through the same struggle when I learn about CoV so good luck

[Anonymous No.10603707]

>>10603692
>Still thinks you need a genius to understand fucking integration
>>10603695
>All this projection because people called you out
Everyone looms like a genius if you are retatded.

[Anonymous No.10603711]

>>10603704
Thanks bro good luck to you too. Just gotta take it one step at a time. We're all gonna make it.

[Anonymous No.10605626]

>>10603229
>>10603245
>>10603249
MAKE UP YOUR DAMN MINDS