Anonymous at Fri, 21 Mar 2025 19:17:03 UTC No. 16625270

>>16625225
Because some of them are more convenient in some circumstances than others.

They all have their utility.

🗑️ Anonymous at Fri, 21 Mar 2025 20:02:48 UTC No. 16625297

>>16625225
Leibnitz notation for standard calculus
Lagrange notion for differential equations
Euler notion for treating differentiation as a linear operator in functional analysis.
Newton notion for cinematics.

Anonymous at Fri, 21 Mar 2025 20:03:49 UTC No. 16625298

Leibniz notation for standard calculus
Lagrange notation for differential equations
Euler notation for treating differentiation as a linear operator.
Newton notation for cinematics.

Anonymous at Fri, 21 Mar 2025 20:10:02 UTC No. 16625303

>>16625298
>Newton notation for cinematics.
kek

Anonymous at Fri, 21 Mar 2025 20:19:47 UTC No. 16625307

>>16625225
> Leibniz = normal person
> Lagrange = HURR I'M TOO BUSY TO WRITE THOSE D'S LOOK AT HOW IMPORTANT I AM
> Euler = DURR LOOK AT HOW LE OBSCURE AND MYSTERIOUS I AM
> Newton = HEH BET YOU DIDN'T KNOW WE USE DOT FOR TIME DID YA? *BASEDFACE*

Anonymous at Fri, 21 Mar 2025 20:39:18 UTC No. 16625324

>>16625225
Most math teachers aren't at the level to see what is best.
Leibniz is best but YOU MUST WRITE (dx)^n instead of dx^n. Learn Weyl algebra.
Newton and lagrange are for baby calc and physics retards who don't go beyond low degree.
Euler is good but it gets introduced in diffeq for the stupidest differential operators and EVERYONE misses the opportunity to learn Weyl algebra and how differential operators REALLY work.

Anonymous at Fri, 21 Mar 2025 21:03:26 UTC No. 16625348

>>16625225

Anonymous at Sat, 22 Mar 2025 04:50:25 UTC No. 16625593

I prefer [math] \partial_x [/math], thank you.

Anonymous at Sat, 22 Mar 2025 05:05:56 UTC No. 16625601

>>16625225
Newton.

Anonymous at Sat, 22 Mar 2025 07:12:30 UTC No. 16625652

>>16625298
Agreed, each one is quite suited to its usual area of application.

Anonymous at Sun, 23 Mar 2025 02:45:58 UTC No. 16626328

>>16625225
It's missing the [math]x_{,i}^j[/math] continuum mechanics tensor calculus co/contra variant distinction.

Anonymous at Sun, 23 Mar 2025 03:24:50 UTC No. 16626343

>>16625298
Euler notation is also far more intuitive for differential geometry (imo). I much prefer the operator notation used in Boumal's Optimization on Smooth Manifolds (as an example) over the Leibniz notation used in O'Neill's book.

Anonymous at Sun, 23 Mar 2025 09:11:40 UTC No. 16626480

Differential operator > everything
Of course it wasn't included here because you're all 18yo babby calculus students.

Anonymous at Sun, 23 Mar 2025 09:13:19 UTC No. 16626481

>>16626328
>not using [math] [math]\nabla_i x^j [/math] [/math]
thought you're gonna be smart and nobody knows basic tensor calculus here eh? The commas are so easy to miss it's a retarded notation.

Anonymous at Sun, 23 Mar 2025 09:47:09 UTC No. 16626487

>>16626481
>tensor calculus
A vector is 1D.
A matrix is 2D.
Is a tensor 3D?

Anonymous at Sun, 23 Mar 2025 10:11:39 UTC No. 16626493

>>16626487
Tensor is an NxN-dimensional generalization.

Vectors are "matrices" that are 1xN dimensional (column/row with N dimensions) ([math] V^\mu [/math] <- Notice how there's only one indice)
Matrices are... matrices that are 2xN dimensional (plane with N dimensions) ([math] M^{\mu\nu} [/math] <- Notice how there's only 2 indices)
Tensors are constructs that are NxN dimensional (N dimensional manifold with N dimensions) ([math] T^{\mu \nu \omega (...)} [/math]

Most common tensors are probably matrices, three indices are rare, though quite common because of Christoffel symbols. It's usually 2 or 4 (Riemann tensor). Mostly 1 (vectors).

So you can represent a 3xN dimensional tensor as a "3D cube" (many matrices making up a cube). But more than that you will run out of spatial dimensions.

Anonymous at Sun, 23 Mar 2025 11:00:42 UTC No. 16626509

>>16626493
>>16626487
I realized I blundered here on second look with my definition of a dimension.

Vector is N dimensional.
Matrix is NxN (2 dimensions) dimensional
"3D cube" matrix equivalent would be NxNxN (3 dimensions) dimensional.
Tensors are generalized to be NxNx....N dimensional.

Anonymous at Sun, 23 Mar 2025 11:13:39 UTC No. 16626514

>>16626493
Your image doesn't show a 7D tensor.

Regarding an m×n×o tensor:
If m = 1, then tensor = west-east matrix.
If n = 1, then tensor = south-north matrix.
If o = 1, then tensor = down-up matrix.
If m = n = 1, then tensor = altitude vector.

Regarding an m×n matrix:
If m = 1, then matrix = column vector.
If n = 1, then matrix = row vector.

Anonymous at Sun, 23 Mar 2025 11:16:58 UTC No. 16626517

>>16626509
>I blundered
Omg, you blundered!

Anonymous at Sun, 23 Mar 2025 12:29:56 UTC No. 16626553

>>16626481
Well so far it's just me and you kiddo.

Anonymous at Sun, 23 Mar 2025 13:59:08 UTC No. 16626590

>>16625225
The calculus book my university used had Leibniz notation but with the delta symbol in place of d. I've never seen another textbook since use that variation. I had to drop out for family reasons and when I went back about five years later, they were using the standard notation (and I had to buy a different book).

Anonymous at Tue, 25 Mar 2025 00:28:25 UTC No. 16627825

>>16625225
In practice, usually Leibniz. Unless I'm doing linear differential equations. Then Euler

Anonymous at Tue, 25 Mar 2025 02:49:50 UTC No. 16627931

>>16626493
A tensor isn’t just any matrix. It has to be an invertible linear transformation ie an element of GL(n). According to you, Christoffer symbols are tensors, even though they aren’t invertible and every basic GR course teaches you that.

Anonymous at Tue, 25 Mar 2025 04:05:00 UTC No. 16627978

Cool thread, I've been very curious about something, if anyone could help me out I'd appreciate it.

Does anyone know what these people are on about when thy talk about the superiority of Newton's notation?

>It has been noted that despite the convenience of Leibniz's notation, Newton's notation could still have been used to develop multivariate techniques, with his dot notation still widely used in physics. Some academics have noted the richness and depth of Newton's work, such as physicist Roger Penrose, stating "in most cases Newton’s geometrical methods are not only more concise and elegant, they reveal deeper principles than would become evident by the use of those formal methods of calculus that nowadays would seem more direct." Mathematician Vladimir Arnold states "Comparing the texts of Newton with the comments of his successors, it is striking how Newton’s original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibniz."[61]

I know nothing about math btw, I only ever had to take Calculus II. This is for a novel. If it would be difficult to explain in a way that makes sense pointing towards prerequisite concepts to research would also be helpful.

Anonymous at Tue, 25 Mar 2025 09:32:54 UTC No. 16628083

Never understood why americans write with Leibniz notation when Lagrange is so much more easier to write, who the fuck can be bothered to write that whole shit out every time?

Anonymous at Tue, 25 Mar 2025 15:30:31 UTC No. 16628253

>>16625225
anything other than Leibniz is mental illness

Anonymous at Wed, 26 Mar 2025 06:34:58 UTC No. 16628812

>>16625225
I use (dx)\D_x for operators. I know the dx is always there so I usually drop it.
I use Lagrange for outputs.

The dx is supposed to be on the left of D_x
This gives the correct expansion when changing variables.
[ (dx)\D_x f(x) ]_{x=g(a)} = f'(g(a))
OR
change variables (like in integrals)
x=g(t), dx = g'(t)dt.
[ 1/g'(t) (dt)\D_t f(g(t)) ]_{t = a} = (1/g'(a))*g'(a)f'(g(a)) = f'(g(a)).

I will do the next one with tex
[math][D_{x}^n f(log(x))]_{x=e^a} = [(e^{-t}D_t)^n f(t)]_{t=a}\\
= [e^{-nt}(D_t-(n-1))\cdots(D_t - 1)D_t f(t)]_{t=a} = e^{-na}\sum\limits_{k=0}^{n}(-1)^{n-k}\left[{n \atop k}\right]f^{(k)}(a).[/math]