Are these two the same

>x + 1 divided by infinitesimal quantity x
>y + y^2 divided by infinitesimal quantity y

my diff eq class was miserable
high, algebra-intensive homework load
maybe I'm just a brainlet tho

Oh, so there are schools out there teaching their freshmen non-standard analysis or are we still in nonsense territory?

yeah the algebra is a bitch. I'm going to have to touch up on my algebra skills
not sure what you're trying to get at, leibniz notation is pretty standard stuff

Leibniz notation is just df/dx(x)=lim (f(x+h)+f(x))/h, it says nothing about treating the derivative as a fraction.

thr worst part is I feel like I didn't learn much, I did all my homework and got mu A but I couldn't solve a system of ODEs if my life depended on it and we never graphed anything we solved

so you don't reckon dy and dx represent anything on their own?
as long as you can set them up, who cares. we've got computers to do the dirty work.

i don't know, integrate both sides, then plug your solution into your original equation
does the solution work? if no, then no
if yes, then yes

but it's so fucking empty. it might as well have been an accounting class. and there wasn't a single moment where you could catch your breath and think about what you were studying

Again, it's not a fraction, it's notation.

I never said it was a fraction. I was simply using dy and dx to represent infinitesimal increments.

You have no framework to work with infinitesimals. It's nonsense from your perspective.

Algebra with integration at the end
Some classes also have linear algebra included at some colleges

>You have no framework to work with infinitesimals
why not?

Because you haven't been introduced to any machinery that deals with infinitesimals. If you think you do, tell me what infinitesimals are and how they work.

>machinery
what?
>tell me what infinitesimals are
infinitesimals are things so small that there is no way to measure them. that doesn't mean you can't use them in equations. I don't think you know what you're talking about. you cannot explain why an infinitesimal is invalid in that equation, other than I don't have "machinery", whatever that means

analysis autists really are the scum of the earth
if it was up to you people calculus wouldn't even exist

>what?
that says it all

t. not the user you were arguing with

> infinitesimals are things so small that there is no way to measure them
What kind of a thing? A number? A set? I'm guessing number since you're dividing a number by these infinitesimals.
Infinitesimals aren't a thing in standard analysis.

explain what the fuck machinery is then
yeah, a number in this case. as is typical of dy or dx.
>Infinitesimals aren't a thing in standard analysis.
guess we're doing non-standard analysis then

What he meant was a set of definitions and rules that allow you to work with infinitesimals in a meaningful and consistent way.

I don't know much about non-standard analysis, pretty much nobody does, including you.
Machinery is meant to refer to a collection of tools, in this case, all your rules of derivatives and integrals.
Well, calculus can be done well, but this high school level "derivatives are fractions" stuff really isn't math.

so can you explain what definitions or rules are violated by working with infinitesimals in the equations in the OP?

>to use the notation dy or dx you must be treating "derivatives" like a fraction
ah yes

>this high school level "derivatives are fractions" stuff really isn't math
hey, it was good enough for Leibniz

There is a small chance that he's referring to the... density of one measure with respect to another or whatever it's called, I think the notation is similar, but I bet against it.

Leibniz died in 1716, you realize how primitive math was at that point?
The problem is that what he is doing in OP is nonsense, it's nonsense that often works out. Maybe always if you do it right, but who knows what right is when you're dealing with nonsense?

>dude like we don't know nuffin... everything is made up
ah yes.
>it's nonsense that often works out. Maybe always if you do it right, but who knows what right is when you're dealing with nonsense?
we should disregard all theories then because that's basically what a theory is.
you could've just told me you were one of those types in your first reply as opposed to embarrassing yourself in showing you can't explain what definitions or rules are violated by working with infinitesimals in the equations in the OP.

You could just as well call them oranges and your argument would be just as good. You can't divide numbers by oranges because it's nonsense. Its not my job to convince you that you should eat oranges and not try to divide by them.

not really considering it is an actually useful method to solving differential equations. again, you might as well just disregard all theories with your thinking. I'm sure you'll go far. the reason why mathematicians use theories like these are because they work as far as we know and can be built upon and aid them in deeper math. whereas you will stop at theories like this and imaginary numbers, etc.

No, that's the point, you're not using any theory when you do these manipulations, you just do it and hope it works. There are well justified ways of doing this, but that actually requires knowing math.
Also, the mathematics of imaginary numbers is well established, unlike treating derivatives as fractions.

it's non-standard analysis, like you say. it is a theory.
theshapeofmath.com/oxford/physics/year1/calc/sepdiff

Well, I guess I just hope you've gotten a better understanding of what you're doing.

>you just do it and hope it works
the cases in which it doesn't work are pretty esoteric and obvious

One would hope, but if you actually think derivatives are fractions, you'd be at risk of not finding it obvious.
Also, there was that case of some fluid dynamics solution to a problem that had no solutions, probably with some questionable methods.

>In diff eq
>Cant cross multiply