Anonymous at Mon, 9 Sep 2024 05:06:48 UTC No. 16369231

I can treat it like a ratio and it still works so I dgaf.

Anonymous at Mon, 9 Sep 2024 05:32:30 UTC No. 16369251

Idgaf if I need it to be an operator it's an operator.
If I'm doing separation of variables it's a ratio.

Anonymous at Mon, 9 Sep 2024 05:40:40 UTC No. 16369259

ratio of differential forms

Anonymous at Mon, 9 Sep 2024 05:45:36 UTC No. 16369264

>>16369225
The definition of the derivative is literally the limit of a ratio.

Anonymous at Mon, 9 Sep 2024 06:09:46 UTC No. 16369282

>>16369225
Sorry, your pedantry is the old pedantry. It's now in fashion among babby calc textbooks to say that dy and dx are rise and run, but measured along the tangent line instead of the curve.

Anonymous at Mon, 9 Sep 2024 06:29:14 UTC No. 16369299

it is limit of ratio y(t)-y(x) : t-x
when t -> x

Anonymous at Mon, 9 Sep 2024 06:46:25 UTC No. 16369315

>>16369225
that's a fraction

Anonymous at Mon, 9 Sep 2024 07:02:35 UTC No. 16369328

>>16369225
>This is an operator ("d/dx") acting on a function (y)
Literally a fancy way of saying "you divide this by that", undergrad. You'll find out some day.

Anonymous at Mon, 9 Sep 2024 08:07:37 UTC No. 16369369

>>16369225
Yes? Somewhat.

Its just a notation anon, my personal favorite being [math]{\partial}_{x}y[/math], but it doesn't really matter.

What you are meant to infer from that notation anon is that we are looking at what is essentially [math]\frac{{\Delta}y}{{\Delta}x}[/math] at a specific point of a non-linear function. You could even argue that [math]\frac{{\Delta}y}{{\Delta}x}[/math] is an operator on the function y, but it doesn't mean much to say "this is an operator on y" and call it a day.

> you can not calculate an "infinitesimal"

That's why we have the limit anon.

Anonymous at Mon, 9 Sep 2024 11:26:08 UTC No. 16369517

>>16369225
>current year
>doesn't understand infinitesimals
You likely don't understand infinities either, or properly grasp calculus in general. If you are a kid, you might still learn; but if you're an adult, you are unlikely to prosper in mathematics, and in any field where mathematics is applied

Anonymous at Mon, 9 Sep 2024 12:07:14 UTC No. 16369579

>>16369225
Midwit take, I can picture the bell curve meme and you fit perfectly
It's close enough to a ratio that you can treat it as one for the vast majority of applications

Anonymous at Mon, 9 Sep 2024 13:34:33 UTC No. 16369717

Oh no, if I treat it like a ratio it will only work 100% of the time. I don't care what you say. I'm pretty sure the last time they told us not to treat this like a fraction was in undergrad multivariate calc.

Anonymous at Mon, 9 Sep 2024 14:14:35 UTC No. 16369775

>>16369225
then why can I do this?

Anonymous at Mon, 9 Sep 2024 15:00:56 UTC No. 16369821

>>16369225

Anonymous at Mon, 9 Sep 2024 15:29:00 UTC No. 16369854

>>16369225
it's just the ratio of two 1-forms. Sue me.

Anonymous at Mon, 9 Sep 2024 15:36:48 UTC No. 16369863

>>16369225
Never seen a convincing argument for this take

Anonymous at Mon, 9 Sep 2024 23:23:28 UTC No. 16370736

>>16369225
undergrad here, currently 4 weeks into differential equations. on day one prof asked us which derivative notation was our favorite and i said dy/dx because you can separate them if you need to. he mentioned that "technically it's not a fraction for some complex reasons" but that we would be treating it like one for most of this class. what did he actually mean by this? why does it work like a fraction if it isn't one? i'm genuinely asking.

Anonymous at Mon, 9 Sep 2024 23:33:05 UTC No. 16370747

dy/dx = x
y/x = x
y = x^2

?? treating it as a ratio works fine ‽‽

Anonymous at Mon, 9 Sep 2024 23:34:07 UTC No. 16370750

>>16370747
FUCKING *KEK*

Anonymous at Mon, 9 Sep 2024 23:49:16 UTC No. 16370770

>>16370747
based algebraian

Anonymous at Mon, 9 Sep 2024 23:52:07 UTC No. 16370777

>>16370736
It is the limit of a fraction, which is why in almost every situation encountered it acts like one. Only autistic mathematicians care about the details, for everyone else it's simply a case of shut-up and calculate.

Anonymous at Tue, 10 Sep 2024 00:00:30 UTC No. 16370793

>>16370747
but the derivative of x^2 is 2x

Anonymous at Tue, 10 Sep 2024 00:49:17 UTC No. 16370841

>>16369225
OK, so what? It's the limit of a ratio, just a little abuse of notation. Or >>16369259

Anonymous at Tue, 10 Sep 2024 00:50:14 UTC No. 16370842

>>16370747
Very impressive, but can you solve this?

Anonymous at Tue, 10 Sep 2024 00:51:14 UTC No. 16370844

>>16370736
Im not op. It only works exactly like a fraction when you got something simple like [math] \tfrac{d}{dx} f = \tfrac{df}{dx} [/math] or [math] \tfrac{d}{dx} f \circ g = \tfrac{d}{dx} f(g(x)) = \tfrac{dg}{dx} \tfrac{df}{dg} [/math].
That last one is kinda weird though, and if anything is should be [math] \tfrac{d}{dx} f \circ g = \tfrac{dg}{dx} \left( \tfrac{df}{dx} \circ g \right) [/math], which certainly doesn't look like it preserves the fraction or the units.

Even with one variable it doesn't work for something like [math] \tfrac{d}{dx} f(x) g(x) = \tfrac{df}{dx} g(x) + f(x) \tfrac{dg}{dx} [/math]. Again, fraction aint preserved, but the units work fine.
Still, everything feels pretty simple, but all intuition goes out the door when you leave the first derivative. All higher derivatives have annoying pascal triangle-esque patterns like
[math] \tfrac{d^2}{dx^2} f(x) g(x) = \tfrac{d^2 f}{dx^2} g + 2 \tfrac{df}{dx} \tfrac{dg}{dx} + f \tfrac{d^2 g}{dx^2} [/math]
[math] \tfrac{d^2}{dx^2} f \circ g = \tfrac{d^2 g}{dx^2} \left( \tfrac{df}{dx} \circ g \right) + \left( \tfrac{dg}{dx} \right)^2 \left( \tfrac{d^2 f}{dx^2} \circ g \right) \stackrel{??}{=} \tfrac{d^2 g}{dx^2} \left( \tfrac{df}{dg} \right) + \left( \tfrac{dg}{dx} \right)^2 \left( \tfrac{d^2 f}{dg^2} \right) [/math]

cont.

Anonymous at Tue, 10 Sep 2024 00:52:18 UTC No. 16370846

>>16370844
When you go multivariable, if f = f(x,y), x = x(t), y = y(t), then you get
[math] \tfrac{df}{dt} = \tfrac{\partial f}{\partial x} \tfrac{dx}{dt} + \tfrac{\partial f}{\partial y} \tfrac{dy}{dt} [/math]
or that
[math] \tfrac{d}{dt} = \tfrac{\partial }{\partial x} \tfrac{dx}{dt} + \tfrac{\partial }{\partial y} \tfrac{dy}{dt} [/math].
The fraction isn't really preserved, and you can imagine the for if its inot just x and y but x,y,z,a,b,c etc. Units are preserved though.

Suppose you got a plane surface described by the set of points that satisfy
[math] 2x + 3y - z = 0 = f(0) = f [/math]
Doesn't really matter what f is equal to, long as it's constant. Notice this equation is the same as
[math] 2x + 3y = z [/math]
Suppose you want to find the slope [math] \tfrac{\partial z}{\partial x} [/math]. Then you find that [math] df = 0 = \tfrac{df}{dx} = \tfrac{\partial f}{\partial x} + \tfrac{\partial f}{\partial z} \tfrac{\partial z}{\partial x} [/math]. This means that
[math] \tfrac{\partial z}{\partial x} = - \tfrac{ \partial f / \partial x }{ \partial f / \partial z } [/math].
Certainly this doesn't at all preserve fractions.

🗑️ Anonymous at Tue, 10 Sep 2024 00:55:16 UTC No. 16370851

>>16370846
ignore the f(0) part, it should be removed

Anonymous at Tue, 10 Sep 2024 01:07:50 UTC No. 16370872

>>16370846
ignore the f(0) part.

Just for completion, if you have a general function f = f(x,y,z), and given a level set [math] f^{-1}(c) [/math] for some constant c (in the last example, the plane surface was [math] f^{-1}(0) [/math]), then to you can find the slope of one variable with respect to another with [math] \tfrac{\partial a}{\partial b} = - \tfrac{ \partial f / \partial b }{ \partial f / \partial a }[/math]

Anonymous at Tue, 10 Sep 2024 01:14:34 UTC No. 16370881

>>16369225
>infinitessimal Lipschitz constant
what

wouldnt it be more accurate to say that its the infimum of the set of Lipschitz constants for a given function evaluated at a point?

maybe that doesnt work...
its a fucking fraction dude. nobody except for permavirgin retards think its anything other than an operator or a fraction.
maybe its a differential form too.
and an inner product.
and a function on the plane.
something else too...

Anonymous at Tue, 10 Sep 2024 01:17:10 UTC No. 16370886

>>16370842
N=1
Where's my million dollars?

Anonymous at Tue, 10 Sep 2024 01:20:39 UTC No. 16370888

>>16370886
holy fuck, bro did it

Anonymous at Tue, 10 Sep 2024 02:07:34 UTC No. 16370934

>>16369225
But Op, it is DEFINED as a ratio

A ratio is by definition what you described
A finite ratio would just be a decimal number

You have smol brain

Anonymous at Tue, 10 Sep 2024 02:14:44 UTC No. 16370942

>>16370844
How do you type like that with math symbols?

Anonymous at Tue, 10 Sep 2024 02:16:41 UTC No. 16370945

>>16370942
you have to get a special keyboard that has all the math symbols on it, they usually go for about $2000 on amazon

Anonymous at Tue, 10 Sep 2024 02:29:25 UTC No. 16370958

>>16369225
True, but as long as it's separable you can treat it as a ratio to save yourself about 8 steps of calculus.

Anonymous at Tue, 10 Sep 2024 02:39:03 UTC No. 16370972

>>16370945
>you have to get a special keyboard that has all the math symbols on it, they usually go for about $2000 on amazon
Thats sounds retarded arent there just programs you can download for free that give you a little window where you can click on the symbols with your mouse?

Anonymous at Tue, 10 Sep 2024 02:40:52 UTC No. 16370975

>>16369225
Ok anon so now what? What was the point of this unwarranted rant?

Anonymous at Tue, 10 Sep 2024 03:58:57 UTC No. 16371062

>>16370975
To give himself a false sense of superiority.

Anonymous at Tue, 10 Sep 2024 04:19:23 UTC No. 16371090

>>16370942
In /sci/, when you post a comment in the top left corner there's a TeX button, which opens up an environment where you can write out TeX code.

Anonymous at Tue, 10 Sep 2024 05:17:45 UTC No. 16371215

>>16369225
>This is not a ratio
It's the limit of a ratio. If the function is nice enough it behaves like a ratio.

Anonymous at Tue, 10 Sep 2024 23:17:00 UTC No. 16373377

>>16370972
>arent there just programs you can download for free that give you a little window where you can click on the symbols with your mouse?
No, you need the special keyboard so the computer can get the special inputs

🗑️ test at Thu, 12 Sep 2024 01:20:42 UTC No. 16375292

>>16371090
[math]Minuet in G[/math][eqn]a(y)[/eqn]