Back in Calculus the professor was adament about dy/dx being a notation, like y', and not a quotient itself...

Back in Calculus the professor was adament about dy/dx being a notation, like y', and not a quotient itself. Why then do we use "seperation of variables" in differential equations, taught by cross-multiplying. Is it not actually that process? Even in my book, it reads that to solve dx/dt = -kx, multiply both sides by dt/x. This assumes dx/dt is a quotient and not a representation of the derivative. I understand it works, can someone help me understand why?

[math]\int \frac{dy}{dx} dx = \int dy [\math]

[eqn]\int \frac{dy}{dx} dx = \int dy [\eqn]

there is a rigorous real analysis-y process I recall my professor going over in lecture which justifies the whole thing. but I forget. For me, it is often sufficient just to know that it can be justified, rather than to know the justification.

Meh. Divide by x and integrate. The dt by itself doesn't make sense without the integral, but people write it anyway for some reason. I think it's sloppy.

[math]\frac{dx}{dt}(t) = - k\, x(t) [/math]

[math]\frac{1}{x(t)}\frac{dx}{dt}(t) = - k [/math]

[math]\int \frac{1}{x(t)}\frac{dx}{dt}(t)\,dt = - \int k\, dt [/math]

[math]\log(x(t)) + C = - k\, t[/math]

Also, let

[math]f(x) = 1/x [/math]

then the integral on the left is

[math]\int_{t_0}^{t_1} f(x(t))\, x'(t)\, dt [/math]

which by the change of variable formula is

[math]\int_{x_0}^{x_1} f(x) dx [/math]

which is

[math]\int_{x_0}^{x_1} \frac{1}{x} dx [/math]

I wondered the same thing a couple weeks ago but still don't have a straightforward answer. How would I solve y'=y^2/x without using Leibniz notation or separation of variables? I also wondered why we can use the definition of the derivative (the difference quotient) to find the power rule but we can't use the definition of the integral to find the reverse power rule [x^(n+1)/(n+1)]. I still have trouble understanding why integration even works the way it does. In fact, I have no clue why any of the things I'm doing in my differential equations class work. I just do them and get A's on my tests.

The reverse power is something along the lines of Integration by parts. Image related, where dv/dx is v' and same goes for u.
I'm lost also on OP's question.

It's just the chain rule
[math] \frac{df}{dt} = \frac{df}{dx}\dfrac{dx}{dt} [/math]

Oh, so what you're doing is multiplying both sides by a (dt) and taking the integral of that, after dividing by xt? Wouldn't the t have to become (t^2)/2 after the integration from step 3 to 4?

Your professor is right that d/dx is NOT a quotient.

That being said, infinitesimal calculus has been rigorously developed, and in some cases you can abuse notation and treat a differential as a quotient. I personally have never formally studied infinitesimal calculus, but am familiar with what I need to study to feel more comfortable about the technique. Look it up to see where it's applicable (1st order separable ODEs being the one place I know about).

Using them without understanding their justification is precisely why you are where you are OP. Don't feel bad, everyone is taught that way.

Yeah that's how I was taught too. I'm just trying to say you can't arrive at the reverse power rule using the definition of the integral, which I believe is the limit of an infinite sum.

you don't need infinitesimal calculus, it's just the chain rule
if you have

dy/dx = f(x)/g(y)

then you go (1/g(y)) dy/dx = f(x)

and then you integrate with respect to dx

[math]\int_x \frac{1}{g(y)} \frac{\partial y}{\partial x} dx = \int_x f(x) dx[/math]

at this point, by the change of variables theorem,

[math]\int_x \frac{1}{g(y)} \frac{\partial y}{\partial x} dx = \int_y \frac{1}{g(y)} dy [/math]

which is what you wanted

>Back in Calculus the professor was adament about dy/dx being a notation, like y', and not a quotient itself

Because your professor is old and doesn't know about Robinson's Infinitesimals

>Because your professor is old and doesn't know about Robinson's Infinitesimals

published 50 years ago

Yo, someone posted a couple of weeks ago that infinitesimal calculus did not extend to multiple dimensions, and while I see a correct approach may not be "obvious" I suggested that surely someone had tried before. Got no response. Searched more on wiki and wasn't able to find the right links. Would you know about that?

Yes, in fact Robinson's infinitesimal calculus can be extended readily to fractal dimensions using some of Grothendieck's old results on nuclear spaces.

Is the nonstandard analysis formally accepted into the scientific community? It seems to be big, I wouldn't understand why it wasn't taught. Or are they taight parallel to each other?

It isn't taught much, or used much. I think people believe it doesn't have much advantage.

Bumping, still kind of confused by the answers. If anyone can provide a link to the reasoning or anything, I would be more than content. Google isn't helping me out.

I learned it both ways, and different professors I've had in math and physics have a preference, or sometimes use both in different contexts.

Dy/dx is useful for vectors and partials, y' is useful for systems of equations and numerical approximations.

When you start to code to find numerical solutions, you use dy/dx almost exclusively for motivation because you have to convert the infinitesimal into discrete approximations.

>In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size.
Mathematicians are just trolling the shit out of everyone right. The fuck is wrong with your guys vocabulary.

Here's a reasoning: it works (separation of variables, related rates), it's easy (just a fraction), and until you piss off some Autist working for Big Math, it doesn't have problems.

It's already been fully explained.
You don't need to treat it like a fraction to solve differential equations, but in practice we do because there's no point in writing out the same step dozens of times when we all know what it's taken to mean

Look up differential forms kiddo

If you have a problem with Liebniz notation, why are you enrolled?

Yes, its the change in x in the numerator and the change in time in the denominator. Those values can be moved arond from on side of the equal side to the other. One reason is because integration eliminates the denominator of the liebniz notation of the derivative.

>not treating solutions of differential equations as elements of an affine space
>limiting yourself to finding the general solutions of a problem

Silly Engineers, never change.

>topological vector space

Multi dimensional space equipped with some understanding of how points connect

>finite-dimensional vector spaces

Basis of said space is finite

>seminorm

A way to measure distances in your space. In this case some things have zero lenght outside of a zero vector.

>unit balls

Balls in your space of the norms equalling.


Read a fucking book sometimes it's not hard.

>equalling one

my bad

>real analysis-y
try to be not so faggoty, pls fgt

We could take a paragraph to express that simple notion, or we could use some vocabulary. I personally prefer the latter.

Why do calc teachers go so far to say dy/dx is not a fraction if it can be treated like one, behaves like one, and has many properties of fractions? Why say it isn't if you can't explain why it isn't? Makes no sense to me.

I think he was talking about the fact its called a nuclear space.

From what I gather, dy/dx notation hassomething to do with making Stokes Theorem easier where you can treat is as a fraction, but I haven't gotten there yet.

My high school calc teacher insisted on f', but in college my professor used dy/dx

Because
>has many properties of fractions
is not the same as
>behaves like one

There are similarities and exceptions.

Ex: (dy/dx)(dx/dy) = 0

>Ex: (dy/dx)(dx/dy) = 0
(dy/dx)(dx/dy) = 1?

>Dy/dx is useful for vectors and partials, y' is useful for systems of equations and numerical approximations.
The two forms of notation have little more to do with infinitesimal vs limit calculus beyond the views of their inventors somewhat aligning with each. Please don't suggest writing dy/dx is using infinitesimals.

I personally have distaste for y' notation, as it implies there is only one dimension worth differentiating over. I generally prefer dy/dx but somewhat like the operator form D_x y.