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On Leibniz Notation

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Re: On Leibniz Notation

#41

Post by bsaul »

LudwigNagasena wrote: post #11
bsaul wrote: post #7

Earlier quoted context omitted.

I don't know about the person you're responding to, but those operations over what usually appears at the denominator of the derivative purely for notation purpose has always looked to me as complete magical garbage. What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ? I think this notation is single handly the reason why i've never been comfortable with calculus. PS: i've…

dx,dy,dz are differential 1-forms. It's like i,j,k in a vector field.

Thanks for the term, but it still doesn't help me understand what i'm allowed to do with it.

Vectors is a great example: as soon as you're introduced to vectors, you immediately starts to be given definitions on how to multiply / add them together and with regular numbers.

dx remained a mystery even during my first 2 years of calculus in university. I used them purely as a notation tool, but really didn't understand them properly.

Re: On Leibniz Notation

#42

Post by lifthrasiir »

zero-sharp wrote: post #37
lifthrasiir wrote: post #35

Earlier quoted context omitted.

For teaching purposes you are definitely allowed to lie, as long as that lie can be resolved eventually (not necessarily in this semester ;-). That's how we have been generally taught about integer divisions and negative square roots. But behind the scene, the `dx` notation can be fully generalized and made rigorous with differential forms, or that was what I have been told.

This is definitely not an apples to apples comparison. Integer division is something everybody is expected to learn. Also we don't teach imaginary numbers to middle schoolers as soon as they learn about square roots. To some extent we have to speak to our audience. I consider that part of effective communication. I don't think "assume the person you're speaking to is/will be a mathematician" is an effective way to in…

I meant that, yes you are right. You are not expected to teach differential forms to non-math students at all because it's not effective. The existence of differential forms only means that it can be eventually made rigorous if you push hard.

Re: On Leibniz Notation

#43

Post by bmacho »

Not just Leibniz notation, but for example https://en.wikipedia.org/wiki/Partial_derivative is total bonkers, every single equation on it means the total opposite than in the rest of the mathematics.

Wtf is

> the partial derivation of a function f(x,y,z...)

or deriving a function respect to a variable? Functions don't have variables, named variables, that's Python, not math[0]. In math expressions can have free variables, and function arguments are indexed with numbers. You can derive x+y by x, or you can derive f : (x,y) |-> x+y by its first variable, but you can't mix them. That only leads to things as df(x,x)/dx and such abominations.

Multivariable calculus people really should just adopt the function notation of the rest of the mathematics fr.

[0] : https://en.wikipedia.org/wiki/Function_(mathematics)#Definit...

Re: On Leibniz Notation

#44

Post by BruceEel »

enriquto wrote: post #30
BruceEel wrote: post #5

It's probably because I'm an ignorant idiot, particularly when it comes to calculus, but this read like a revelation: "the concept of a function shouldn't depend on what your favourite letter is!" Very helpful answer, thanks for posting.

> the concept of a function shouldn't depend on what your favourite letter is On the other hand, you can also argue that the concept of a function shouldn't depend on your favourite ordering of its variables. Thus, if you have a potential that oscillates in time, such as: V = (1 + sin(t))/sqrt(x^2+y^2) You may want to take derivatives with respect to each variable, such as dV/dt, dV/dx and so on. Their meaning is cle…

ah, interesting dark corner there. Also, I was thinking there must be merit in allowing for some discrepancies in use of different notations when dealing with "read" use, as in learning a new concept, vs. "write" use, as in working on something, like the SO OP..

Re: On Leibniz Notation

#45

Post by nemoniac »

tobinfricke wrote: post #10

The discussion in Sussman and Wisdom's "Structure and Interpretation of Classical Mechanics", about how the Euler-Lagrange equations don't literally make sense as traditionally written, has long resonated with me: https://mitp-content-server.mit.edu/books/content/sectbyfn/b... They also adopt a notation where partial derivatives are taken with respect to "argument slots".

A positional notation was introduced previously by De Bruijn but presumably Sussman and Wisdom came up with theirs independently since they don't cite him.

https://en.wikipedia.org/wiki/De_Bruijn_index

Re: On Leibniz Notation

#46

Post by zero-sharp »

robinhouston wrote: post #25

I don’t like the accepted answer here, because it criticises Leibniz notation without explaining what it means – and apparently without actually _understanding_ it, since the author several times writes words to the effect of “this is nonsense” in reference to expressions that in fact make perfect sense when interpreted correctly. The salient issue that the author of that answer seems not to have understood, which we…

>I don’t like the accepted answer here, because it criticises Leibniz notation without explaining what it means – and apparently without actually _understanding_ it, since the author several times writes words to the effect of “this is nonsense” in reference to expressions that in fact make perfect sense when interpreted correctly.

Did you look at the posting history at all? I think the poster understands what it means.

Re: On Leibniz Notation

#47

Post by LudwigNagasena »

bsaul wrote: post #41
LudwigNagasena wrote: post #11

Earlier quoted context omitted.

dx,dy,dz are differential 1-forms. It's like i,j,k in a vector field.

Thanks for the term, but it still doesn't help me understand what i'm allowed to do with it. Vectors is a great example: as soon as you're introduced to vectors, you immediately starts to be given definitions on how to multiply / add them together and with regular numbers. dx remained a mystery even during my first 2 years of calculus in university. I used them purely as a notation tool, but really didn't understand…

Well, I don't know much about the way calculus is taught in the West, but I remember that Zorich's Mathematical Analysis (ch. 5 Differential Calculus and ch. 8 The Differential Calculus of Functions of Several Variables) was pretty clear about everything.

Re: On Leibniz Notation

#48

Post by tlb »

> "the choice of letters SHOULD NOT affect the meaning of a mathematical statement"

I mean, that ship sailed a long time ago. You can't understand any modern math or physics (or ML, or CS) paper without depending on variable naming conventions. Attempts (like Sussman & Wisdom) to have a properly lexically scoped notation end up being quite verbose.

Re: On Leibniz Notation

#49

Post by 082349872349872 »

movpasd wrote: post #17

I hope you will forgive me for this wall of text, but this is a topic that is quite close to my heart and that I've gone back and forth in many times before settling on my current perspective. I appreciate that especially for mathematicians and programmers, making a clean distinction between a function and its evaluation is a key conceptual point, and Leibniz notation obscures this fact. However, there are good reaso…

On HN, we love walls of text. Thank you for yours!

Any chance you might be able to brainstorm an example or three of what a well known equation would look like in that formalism?

Re: On Leibniz Notation

#50

Post by xamuel »

One can make a very good argument that in elementary calculus, we actually use what logicians would call "terms", but refer to them as "functions". For example, if f is a unary function symbol and x is a variable then f(x) is a term, different from f(y) if y is a different variable. It's possible to develop everything quite rigorously using this machinery and when the dust clears, you get a rigorous version of what is actually done in practice in the elementary calculus classroom. As an advantage, certain things become much clearer, for instance, there's a very nice abstract multivariable chain rule which I describe in this paper: https://philpapers.org/archive/ALEFDV.pdf
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