What does dx actually mean in differential?

What does dx actually mean in differential?

What does it mean to "yea just multiply by dx and move it to the other side and..."?

Other urls found in this thread:

en.wikipedia.org/wiki/Infinitesimal
en.wikipedia.org/wiki/Non-standard_analysis
en.wikipedia.org/wiki/Hyperreal_number
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A differential is a map on the tangent space or something. I don't know if you can actually divide differentials like that but would be interesting if you can have the freshman "derivative as a fraction" thing make sense in higher mathematics. You can justify it with Radon-Nikodym derivatives or something but it doesn't really make it a proper fraction.

It's an infinitesimal.

>What does it mean to "yea just multiply by dx and move it to the other side and..."?
It only has meaning in non-standard analysis. But people who are willing to believe in such nonsense as cauchy sequences and dedekind cuts for some reason can't bring themselves to believe in non-standard elements because their brain is broken.

Physishits pls go.

You introduce an imaginary variable called d that, when multiplied by x, transmorphs it into an infinitely small piece of the greater whole.

It depends on the context, but it doesn't has a fixed meaning except on some number systems. It's better to take it as a heuristic.
Cauchy sequences and dedekin cuts are pretty easy to understand, I don't understand the struggle besides maybe being tedious. But non-standard analysis will kick your ass if you have no backround in a rigorous approach to math. Just quoting the "tansfer principle" , doesn't make it rigorous. Besides, you construct the hyperreals as an extension of R, so you still need to use those constructions.

What gets lost on a lot of people (and teachers, honestly) is that the differential operator [math]dy/dx[/math] is a fraction, or a proportion. [math]dy[/math] and [math]dx[/math] refer to "delta y" and "delta x" or a small change in y and x respectively. This "small change" is arbitrary, and therefore referred to using the notation.

So the derivative is simply a proportion comparing how a given change in x changes the function's value y at said x. The best part about fraction notation is that you can multiply it by the denominator to manipulate it for algebraic purposes.

Here's an example.

[math]dy/dx = 5[/math] Here, is a derivative that has constant value at all x. The value is 5. In an ordinary differential equations class, you might want to solve this using a separation of variables. We can multiply everything by [math]dx[/math]:

[math]dx*[dy/dx = 5][/math]
[math]dx(dy/dx) = dx(5)[/math]
[math]dy = 5dx][/math] It's "proper" to attach the dx after a coefficient just like an ordinary variable.

Now that the variables are separates, one is allowed to integrate:
[math]\int1dy= \int5dx[/math] Splitting up the differential operator is key here. the separation of variables allows you to close integral notation with the required deltas for the variable you are integrating.

Finally:
[math]y= 5x + c[/math]

So there you have it. This is what doing algebra with the differential operator is good for, and that's how it's done. I hope this helped you!

thanks

Its a small but finitely small change in x.

thanks!

wrong

is the only nonretarded post in this entire thread.

HAHAHAHAHA holy shit I hope these

are ironic

just approximating rate of change brop

(d/dx)y means derivative of y. you are basically saying what is the rate of change of the output with respect to the rate of change of the input aka instantaneous slope. brainlet puremath fags will say some weird shit and call everyone else retarded but just ignore them

lets say you have y = x, if you take the derivative of that you get (d/dx)y = (d/dx)x = 1.
y = x. dy/dx = 1. you can see that the rate of change of your output (whatever it is), over the rate of change of the independent variable is 1 for y. this is why you get 1

in other words, you can think of it as an infinitely small change in x

Its pronounced "dicks"

cannot unsee

An infinitely small change is no change at all.

Non-mathy user here

Explain

hes just bean a asshole

Physicists and engineers like to pretend derivatives are fractions, mostly because it sorta works and is the quickest way to get them crunching numbers without worrying about what the fuck they're actually doing.
df/dx(x) is the function defined as the limit of (f(x+h)-f(x))/h as h approaches 0. The notation as a fraction is just a relic from another age and as far as I can tell it's mostly coincidence that you can throw the components of the "fraction" around and get anything meaningful out of it.

I'm not an asshole, I'm also a physicist and if you think physicist don't have a rigorous course in differntial forms then you're a fucking brainlet.

Your limit explanation while correct lacks in width as you could generalize this concept to differential forms where indeed you might consider working with differentials "as with fractions" in a sense, but none of that is necessary as in separable differential equations the "multiply by dx and integrate" is literally just substitution.

I seriously wonder if engineers ever get some proper math education, otherwise what the fuck are they doing in their calc courses? (Not like I'd know, we're having real fucking analysis courses not like them brainlets)

I learned that substitution thing but I never used it, I just pretend it's a fraction

Good post user, this is what I would have wanted back in AP Calc. For some reason it didn't get pushed into my head that dx was a variable that can be used in the same types of ways other variables can be.

Well, don't write the step y/dy = ... without an integral, because that's just amateur (unless you explain how you're working with differential forms in that sense).

>The notation as a fraction is just a relic from another age and as far as I can tell it's mostly coincidence that you can throw the components of the "fraction" around and get anything meaningful out of it.
someone on stack exchange tell you that?

you just defined dy/dx as a fraction yet it's just a coincidence that it can be treated like a fraction? it literally is a fucking fraction you fucking donut pure math faggots

It's not a fucking variable, jesus fuck. What the fuck are people being taught these days?

Ok it's not a variable I spoke badly, but it's a "thing"

I'll let myself out

yeah when you move the dx over that's actually a substitution
I saw it in class like once

Nigger, you're not allowed to ask that question until math classes 500 level and beyond.

It's not a fucking fraction, it's a real number - the limit of the slope of a secant between two points, you fucking retard, you can't treat it as a fraction and if you do, good luck proving that your numerical solution to a PDE system is stable.

Idk man i only know basic derivative stuff regarding calculus but it makes sense to me. It's just rise over run nigga. You can't have rise over run when there's no change, but you can't introduce change without it not be inaccurate for a curve. The less change, the more accurate your calculation will be.

This is not correct. It's a reference to a buried function.

wow which pde class did you take where you didn't treat it like a fraction?

i didn't know variables were defined as real numbers

It's not substitution, it's just the FToC, [math] \int
\frac{df}{dx}(x)dx=f(x)+C[/math]
Also pretty sure any definition of derivatives comes down to this limit, you define differential forms on some manifold as the derivative of curves or something.

Defining it as a limit of fractions doesn't make it a fraction itself.

dy/dx is usually a function, not a real number.

It's actually much more complicated than that, you could formulate that the solution of a differential equation of a type y' = G(y) is given by the formula y = G^-1(x+C), where G is the primitive function of 1/g(y) that is continuous on some interval (a,b), then you could say that y is truly the maximal solution to the equation of this type, but the proof is not really trivial as you have to show that

a) existence and uniqueness of G^-1
b) y is indeed the solution of the given problem
c) maximality on R x (a,b)
e) uniqueness of y
and finally that the solution exists for every intial conditions lying in R x (a,b).

This is the only rigorous way I know of that proves that the "multiply by dx" bullshit actually works, but learning these bad habits in intro classes like calc is playing with the devil.

The most fundamental definition of a derivative is the derivative of a function at a point, from which we generalize to the derivative of a function at any point (a function).

Fuck calculus it's fucking gay

My man.

Study fucking analysis from day 1 in university or paint some trains, cause you might as well be in kindergarden if you enroll in a calc class.

Just because you're a brainlet engineer doesn't mean that you don't need to understand how shit works, that's what medschool is for.

if these threads teach me anything, it's that nobody really knows anything

even the seemingly most intelligent people in these threads are ambiguous and never give definitive answers

The way I have understood it is that the d in dx implies that you intend to look at the limit as x (or any other independent variable) approaches zero.

So the ratio dy/dx would give you the ratio between a change in y and a change in x, as x gets smaller and smaller.

I'm not sure but I think if you look at it that way, most of the trickery that is performed with moving dx etc around, makes sense.

what they should teach you is that everything is either calculation or bullshitting

mathematics post-Euler really went downhill

What the fuck do you study? Did they really tell you this?

It comes from a geometric definition of the derivative.

Why, was it incorrect? Or incomplete?

I don't study mathematics so my terminology is lacking

one thing that always gets me is that apparently no one in math has realized that the verb and noun forms don't agree.
Integrate -> Integration -> Integral

Differentiate -> Differential
no

Derive -> Derivative
no

Differentiate -> Differentiation -> Derivative

>implying differential calculus isn't required for proper analysis
>implying vector calculus isn't the most useful form of math

That annoys me too. They do agree in Norwegian at least, I'm guessing in other languages as well

Get with the times man, the 1960 were over 50 years ago.
en.wikipedia.org/wiki/Infinitesimal

Why are they nonsense according to you?

What does infiteismally small number even fucking mean to you?
>inb4 a number really close to 0

We can't algebraically manipulate infinity, but we can algebraically maniupulate infinitesimally small numbers? Wtf

Educate yourself kiddo.
en.wikipedia.org/wiki/Non-standard_analysis

A number smaller than any real number but greater than zero.

I'm sure all the faggots who talk about non standard analysis haven't gone through the backwards and retarded mess that is actually constructing that system. And for what? To justify brainlet memegineers? Differentials aren't even used for that, it's mostly so that they can use general coordinate systems without actually having to learn differential geometry. Learning that shit was a complete waste of my time.

Sorry i meant cauchy seqs and dedekin cuts. Why are those nonsense?

>We can't algebraically manipulate infinity

Ordinals and Cardinals are a thing you know.

>infinity
everything is either calculation or bullshit

ordinal arithmetic is just another kind of calculation, it doesn't have shit to do with infinity since infinity doesn't exist

>just a coincidence
mathlet

If you make a Tangent (vector) on a certain point of graph and divide it on X and Y vector ( of coordinate system) dx will be vector pointing in X direction.

learn something new
en.wikipedia.org/wiki/Hyperreal_number

Agreed. Although Euler had no problem with infinitesimals.

numbers don't exist either
they're tools bro

What does Wildberger think of hyperreal numbers? Are they part of, like, a hyper-wrong system?

What do you think about them?
Have you read a book on them before you dismissed them?
Imaginary numbers were rejected by small minds before they were found to be useful.
Same shit, people don't change.

except wildberger has no problems with algebraic extensions of rational numbers retard

Why do you ride wildberger's cock so hard? Do you have no brain of your own?

Everyone I know who is right always agrees with me.

>the limit of the slope of a secant between two points,
Or, you know, the x component of some interval of that linear slope divided by the y component...

dunning kruger
confirmation bias
and a sad person

Now I'm sad, thank you user. Fuck you.

Actually, pretty big minds rejected imaginary numbers, just like with 0 or negative numbers. This is why we need to give every scientist at least a basic education in the history of sciences, so that they don't repeat the mistakes of the past and unnecessarily narrow their own possibilities.
By the way I'm the dude you replied to. I didn't dismiss them or anything, I actually just wanted to make a comical reference to Wildbergers "Nuh-Uh"-attitude.

not that user but what??? that's not the same thing

you need a limit, otherwise you aren't finding the slope at a point

But do the computers(engineers and physicists) ever do anything like that?
Isn't it usually df/dx(x)=g(x)? Then they say dy=g(x)dx so y=G(x)+C by "integrating".

dy/dx is as many have said the ratio of the change in y(x) with respect to a very small change in x.
Originally this change in x was thought to be "infinitely small but not zero", people had problems with this idea because it was not rigorous and led to some problems, so they developed the concept of limits.
Both approaches have their pros and cons. The infinitesimal approach is often more intuitive, though it lacked formalism. However in the 1960s they were formally constructed using hyperproducts. I recommend you read on the hyperreal line. It is a very interesting read, and ultimately it allows you to choose what you use and believe.

df/dx = g(x) is the derivative
df = g(x)dx is the line that approximates the function. It literally is a line.

Oh. Hey you're not bad. :)

No it's not, it's nonsense.

Yes it is, it's not nonsense.

Dont be dense dfx0 is identified as the linear function that behaves like f(x0+h)=f(x0)+dfx0(h) + R(h)h some function that has that R(h) goes to 0 as h tends 0 so you can talk about a change f(x0+h)-f(x0) =~ dfx0(h) if h is small enough, i.e. a fisto irder aproximation. and becaucse dfx0 es linear it can be identified with a matrix that in the case of single variable functions, itd a number which js called the derivative but in higher ordee functions is the jacobian.

Thank you for clarifying my post :) you're nice.

>the line that approximates the function
that doesnt even make sense

It does

Well yes, we do all have to go to highschool. Might as well say implying we don't need to take course in rationalising the denominator and learning the quadratic formula to do proper analysis.