These fucking symbols

>it’s ANOTHER calculus thread

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....but it is just notation then.

differential form is the most intuitive interpretation tho

...and it helps you complete the task you set out to achieve

But the question is to understand if the object has a clear mathematical meaning, and your post implies it does but it ends saying actually no.

y = f(x)

since y is a function of x, it's derivative is with respect to x

D[y] = D(f(x))

dy = f'(x) dx

dy/dx = f'(x)

It isn't necessarily "math", rather the notation moves around as if we were multiplying both sides by dys and dxs. This isn't what's happening, it's just our notation

Yes. Do you?

Can you elaborate more?

still dont have a definitive answer as to whether dx is a notation or something i can work with algebraically

does multiplying both sides of an equation by dx make sense? does having dx as a component of a vector make sense?

In the first line of the table, the possible integer values of Δz. You can see that this is just a list of the integers, of course. Next is listed some integer values for other exponents of Δz. This is also straightforward. At line 7, it begins to look at the differentials of the previous six lines. In line 7, it is referencing line 1, just subtracting each number from the next. Another way of saying it is that it is looking at the rate of change along line 1. Line 9 lists the differentials of line 3. Line 14 lists the differentials of line 9. I think you can follow the logic on this.
1 Δz 1, 2, 3, 4, 5, 6, 7, 8, 9….
2 Δ2z 2, 4, 6, 8, 10, 12, 14, 16, 18….
3 Δz2 1, 4, 9, 16, 25, 36, 49 64, 81
4 Δz3 1, 8, 27, 64, 125, 216, 343
5 Δz4 1, 16, 81, 256, 625, 1296
6 Δz5 1, 32, 243, 1024, 3125, 7776, 16807
7 ΔΔz 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
8 ΔΔ2z 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
9 ΔΔz2 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
10 ΔΔz3 1, 7, 19, 37, 61, 91, 127
11 ΔΔz4 1, 15, 65, 175, 369, 671
12 ΔΔz5 1, 31, 211, 781, 2101, 4651, 9031
13 ΔΔΔz 0, 0, 0, 0, 0, 0, 0
14 ΔΔΔz2 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
15 ΔΔΔz3 6, 12, 18, 24, 30, 36, 42
16 ΔΔΔz4 14, 50, 110, 194, 302
17 ΔΔΔz5 30, 180, 570, 1320, 2550, 4380
18 ΔΔΔΔz3 6, 6, 6, 6, 6, 6, 6, 6
19 ΔΔΔΔz4 36, 60, 84, 108
20 ΔΔΔΔz5 150, 390, 750, 1230, 1830
21 ΔΔΔΔΔz4 24, 24, 24, 24
22 ΔΔΔΔΔz5 240, 360, 480, 600
23 ΔΔΔΔΔΔz5 120, 120, 120
from this, one can predict that
1/2

from this, one can predict that
24 ΔΔΔΔΔΔz6 720, 720, 720
And so on.

This is what you call simple number analysis. It is a table of differentials. The first line is a list of the potential integer lengths of an object, and a length is a differential. It is also a list of the integers. After that it is easy to follow the method. It is easy until you get to line 24, where it states, “One can predict that. . . .” Do you see how that conclusion was reached? By pulling out the lines where the differential became constant.
7 ΔΔz 1, 1, 1, 1, 1, 1, 1
14 ΔΔΔz2 2, 2, 2, 2, 2, 2, 2
18 ΔΔΔΔz3 6, 6, 6, 6, 6, 6, 6
21 ΔΔΔΔΔz4 24, 24, 24, 24
23 ΔΔΔΔΔΔz5 120, 120, 120
24 ΔΔΔΔΔΔΔz6 720, 720, 720
Do you see it now?
2ΔΔz = ΔΔΔz2
3ΔΔΔz2 = ΔΔΔΔz3
4ΔΔΔΔz3 = ΔΔΔΔΔz4
5ΔΔΔΔΔz4 = ΔΔΔΔΔΔz5
6ΔΔΔΔΔΔz5 = ΔΔΔΔΔΔΔz6
All these equations are equivalent to the magic equation, y’ = nxn-1.
In those last equations we have z on both sides, we can cancel a lot of those deltas and get down to this:
2z = Δz2
3z2 = Δz3
4z3 = Δz4
5z4 = Δz5
6z5 = Δz6
Now, if we reverse it, we can read that first equation as, “the rate of change of z squared is two times z.” That is information that we just got from a table, and that table just listed numbers. Simple differentials. One number subtracted from the next.

based quads

Why do you ask for concrete answers here? Open a math textbook ffs.

If you are at a point in math where you don't know the answer to this question, then the answer is yes. When you are finally at a stage where it is no longer okay to do this, you will know it.

In thermodynamics, Δx is used for path-independent changes, and dx is used for path-dependent changes, even when it's a finite amount.

bait

How so? Honestly, just doing a whole lot of calc problems helps with this and I understand how extremely frustrating the whole dy dx why is it like a fraction thing is. There is no piece of knowledge that's missing, it's really just keep doing problems and the gaps kinda start to fill in and the methods make sense.

Calc 3 is where I learned partial derivates, gradient, all that jazz, and once you start working in calculus with variables other than x and y, and take the derivatives of multivariable function and do partial derivatives, the notation starts to come.

Infinity and infinitesimal don't really work, as they are limits/concepts. You can't really multiply or divide by them conventionally. Since the "with respect to" symbol (dy, dx) is an infinitesimal variable, you can't divide by it.

y = mx + b
D[y] = D[mx + b]
dy = m dx

dy/dx = m

This final step, I AM NOT, dividing both sides by dx. Here, it is simply a rewrite of the original equation.

dy = m dx : is in a form readily integrable

dy/dx = m : is in differential equation form

Jesus fucking christ you are all brainlets. Did all of you skip the limits section of calc I? "dx" is the limit of "[math]/Delta x[/math]" as it approaches 0.

Damnit, i meant [math] \Delta x [/math]

Got a problem with that, frogposter?

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Some of us didn't learn bullshit for engineers in our calc classes.

Maybe you should have because limits are used to define derivatives and integrals you fucking brainlet

It's a number smaller than every real non-zero number but greater than zero.

Yes, limit of quotients between the difference between a function with thd function evaluated at some point. and the difference of x as it approaches that point. Your limit is trivialy 0.

so it's 0 then ?

Delta x is used to represent a literal change in x that you can measure.
dx is notation that you should only see in front of an integral sign or in a derivative "fraction" until you do differential geometry and these become properly defined.
Most likely your prof was treating them as fractions as this is a consequence of the chain rule, that in most cases it will work out alright.

Thank you, finally i understood

Oh boy, this thread gave me cancer.
OK, what the fuck is the differential anyway?
We say that the function [math]y(x)[/math] is differentiable if [math]y(x+h)-y(x)=A(x)h+o(h)[/math]. This stuff [math]A(x)h[/math] is the differential [math]dy(x;h)[/math]. So, by the definition, the differential is a linear in [math]h[/math] function. It's not undefined magic notation or some nonsense, it's a function. Remember, children, the differential is a function.

So, what's the differential of the function [math]y(x)=x[/math]?
[math]x+h-x=h=dx[/math]

Now, it happens that [math]A(x)[/math] is actually the derivative [math]y'(x)[/math]. That's why we can write
[math]dy=y'(x)dx[/math]

Can we divide by [math]dx[/math]? Hell yes. It's just a function!
[math]\frac{dy}{dx}=y'(x)[/math]
So, the derivative is the coefficient of proportionality between the functions [math]dy[/math] and [math]dx[/math].

[math]\Delta x[/math] is a number.
[math]\text{d}x[/math] is a differential form.

delta x is for finite changes
dx is for differential changes

literally the only difference

delta x is just a difference. How stupid are you fucking people even? dx = x_2 - x_1

i always thought dx was the opposite of something infinitely large. Like 1/inf

length of a curve is therefore the sum of all ds
ds=sqrt(dy^2+dx^2)

That's just old notation that is used in diferential geometry. Also your algebra is useless for higher dimensional spaces, that's why the best definition is en.m.wikipedia.org/wiki/Fréchet_derivative

they're the same thing.E= VM
dE = dV/m which is also ΔE = ΔV/m