Professor uses Leibniz notation for differentials

the patrician notation is [math]D_i f(p)[/math] for the [math]i[/math]-th partial derivative of [math]f[/math] at [math]p[/math], and save the [math]\frac{\partial}{\partial x}[/math] for tangent vectors.
>t. differential geometer

Yup engineering master race. I'll also solve that shit numerically, cause it's more practical.

It's not really that it gives you a wrong answers but heurestics are tools. So they are not going to teach you what's actually happening. For integration, there are a lot of problems when you start to consider integration over submanifolds, and for diferentiation, you can make a lot of mistakes (like chain rule and implicit diferentiation) if you don't understand what a function is and what does it mean to compute it's derivative. In a physics class, a profesor wanted to compute the formuña for the surface area of a surface of revolution, and everyone got confused when he used diferentials, and I got and properly defined the surface as a parametric equation and after some plug and chug I got the answer much more faster and everyone understood. For derivatives, trying to get the expresion of a smooth curve in its local Frenet frame (without being parametrizied by arclenght) will make no sense using differentials even if it's shorter.

>So they are not going to teach you what's actually happening.
lrn2nsa pleb and don't fucking give me this shit, nothing is actually happening since math is made up

>there exists a positive number that somehow is smaller than every other positive number and it's 0 when it's squared but not all the time because taylor
Or
>Let's define what it means for objects to get closer in a clear and intuitive way and let'd show that all the properties we had from geometry hold and general inuition hold but that will in the future give means to generalize the concepts to more abstract spaces.
What do you prefer brianlet? en.m.wikipedia.org/wiki/(ε,_δ)-definition_of_limit

>we can make up new number types all the time to solve problems BUT NOT THAT
hurrrrrr

en.m.wikipedia.org/wiki/Hyperreal_number never said you couldn't my point is that's conceptually easier to understand delta epsilon and then contruct these sort of number systems using algebraic approaches. If you present someone with differential forms they would call you bullshit if no other methods were previously used. Faggot.

>"multiply both sides by dt" is an abuse of notation
So when doing a substitution while integrating is an abuse?
x=f(t), dx=f'(t)dt
You mention parametrically defined surfaces.
Is writing ds^2 = dx^2 + dy^2 an abuse as well?
God forbid the fucking metric tensor.

My point is that the abuse of notation isn't actually an abuse because it can abstract away details that aren't relevant to the computation while retaining the details that are relevant.
IMO it is useful because it is a successful abstraction.

It's abuse if you have never taken differential geometry. Understanding the metric tensor is much harder and there was no neeed for students to know about that because it isn't a required class. Also, just because it's useful as a heuristic that doesn't mean it's not abuse. There's no problem in using them and it really specific cases whete you could run into some problems, but it's still "wrong" and the problem is that physicists sometimed take them by heart because it's more "physical" or more "geometric".

I had a baller professor for DiffEQs who used Leibniz notation. The first lecture they started off by really driving the point home that you are not multiplying the differentials or anything like that and went on for twenty or thirty minutes or so basically how sexy the notation was and why he liked it and why it could also be a snake in the grass.

You keep saying that it is an abused heuristic but I haven't seen any examples where this "heuristic" has been "abused" by applying it outside of its domain of applicability.
Isn't most formal mathematics about defining heuristics along with their valid domains of applicability?
Theorem: Given domain of applicability D, the heuristic H is valid.

The person you're replying to isn't me.

Formally, d/dx is an operator on f (x) such that d/dx applied to f (x) gives f'(x). It's comparable to other operators like +-×÷.
You can find theorems that validate things like you said: dy = f'(x)dx. Instead of stating those theorems every time, we just write the result. The abuse comes in when you treat the infinitesimal as independent objects.

Just like -x is equivalent to 0-x and -1*x. But -x is really a unary operator. Being lazy can cause issues, such as the misconception that -x^2 = -x^2. If you don't use the unary, this would be true, but strictly speaking, -x^2 =x^2.

The equivalence of objects doesnt make them the same. It's just that most of the time, it doesnt matter. But it's important to remember that we're doing it out of convenience, not formalism.

>Leibniz notation is the sign of an idiot
>He never said it gives the wrong answer, but saying things like "multiply both sides by dt" is an abuse of notation, and isn't possible with dot notation.
You two are fucking idiots. Leibnizian notation encourages precisely the right intuition about derivatives. Newton's notation for differential calculus set back English mathematics for years in comparison to the continent and this is well documented.

youtube.com/watch?v=48Hr3CT5Tpk
All of these rigorous plebs are confusing the post-rigorous people with pre-rigorous people.

Only because the intuition was rigorously proven. If you blindly use it on the basis that it makes sense, you're bound to run in to issues.

I won't argue what does or doesn't count as intuition, but there's some functions that wouldn't seem to have an integral/derivative and yet they do.

it's the most efficient way to do it. you only need leibniz when you can't use dot.
Case in point, the Legrange equation:
dot notation is used on the parameter q, but leibnez is used for everything else. You may not like it, but this is what peak efficiency looks like.

Also dot indicates a total derivative, so it's not the same as ∂t.