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On Leibniz Notation

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Re: On Leibniz Notation

#31

Post by MatteoFrigo »

kergonath wrote: post #24
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".

This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…

FWIW, they start counting function arguments from 0, so _2 is indeed the velocity.

But I do agree with your main point that the order of arguments is irrelevant, and it is a mistake to make it a first-class citizen of the notation.

Re: On Leibniz Notation

#32

Post by zero-sharp »

laingc wrote: post #16
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…

> 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. You will be delighted to discover that they are in fact not magical or garbage. > What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ? dx is a differential one-form. You can think of it as a generalisation of…

>dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry.

This really doesn't help beginners. At all.

There are formal contexts where we can reinterpret division by zero and have it make sense. Should I start telling students that division by zero is allowed? Should I start teaching intro calculus students that 1+2+3+...=-1/12?

Re: On Leibniz Notation

#33

Post by Buttons840 »

I've learned calculus a few times in and out of school. I learned it best from a home-schooling focused text book. The author published a paper describing his notation [0]; see section 3 which is very readable.

The paper explains:

> Most calculus students glaze over the notation for higher derivatives, and few, if any, books bother to give any reasons behind what the notation means. It's important to go back and consider why the notation is what it is, and what the pieces are supposed to represent.

> In modern calculus, the derivative is always taken with respect to some variable. However, this is not strictly required, as the differential operation can be used in a context-free manner. The processes of taking a differential and solving for a derivative (i.e., some ratio of differentials) can be separated out into logically separate operations.

> In such an operation, instead of doing d/dx (taking the derivative with respect to the variable x), one would separate out performing the differential and dividing by dx as separate steps. Originally, in the Leibnizian conception of the differential, one did not even bother solving for derivatives, as they made little sense from the original geometric construction of them.

> For a simple example, the differential of x^3 can be found using a basic differential operator such that d(x^3) = 3x^2 dx. The derivative is simply the differential divided by dx. This would yield d(x^3)/dx = 3x^2.

Is this significantly different than what is normally taught in schools? Using the notation described here is the first time it has felt like a tool rather than a formality to me, and it's quite different than the way I was thinking while taking calculus in high school. I'm not really sure how unusual this notation is though?

[0]: https://arxiv.org/pdf/1801.09553.pdf

Re: On Leibniz Notation

#34

Post by brewmarche »

kergonath wrote: post #24
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".

This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…

As a mathematician I like the positional notation, but once we assign meaning to the slots it gets weird, so I had the same objection.

One could add some new notation to distinguish the element _T_ from the slot _T_. For example let’s write slots as _[T]_ (I’m not creative enough to come up with something good), then we can talk about ∂_[T]F or ∂F/∂[T].

One can think of the [.] operator as mapping from “semantic symbol” to slot number in a way.

Re: On Leibniz Notation

#35

Post by lifthrasiir »

zero-sharp wrote: post #32
laingc wrote: post #16

Earlier quoted context omitted.

> 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. You will be delighted to discover that they are in fact not magical or garbage. > What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ? dx is a differential one-form. You can think of it as a generalisation of…

>dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry. This really doesn't help beginners. At all. There are formal contexts where we can reinterpret division by zero and have it make sense. Should I start telling students that division by zero is allowed? Should I start teaching intro calculus students that 1+2+3+...=-1/1…

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.

Re: On Leibniz Notation

#36

Post by cygx »

Note that in context of differential geometry, the name in the denominator is not associated with the function, but a coordinate system on its domain:

Let M be a differential manifold, eg M = ℝ² and φ a chart, eg cartesian coordinates φ: p ↦(x(p), y(p)) where x,y: M → ℝ. Then, ∂/∂x denotes the holonomic vector field tangent to the coordinate lines t ↦ φ⁻¹(x(p) + t, y(p)) through any p ∈ M.

It is convenient to identify vectors and directional derivatives (this is in fact one possible way to define tangent vectors on manifolds), which for a function f: M → ℝ yields

(∂f/∂x)(p) = ∂/∂x|ₚ f = lim_{h → 0} ( f(φ⁻¹(x(p) + h, y(p))) - f(p) ) / h

Re: On Leibniz Notation

#37

Post by zero-sharp »

lifthrasiir wrote: post #35
zero-sharp wrote: post #32

Earlier quoted context omitted.

>dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry. This really doesn't help beginners. At all. There are formal contexts where we can reinterpret division by zero and have it make sense. Should I start telling students that division by zero is allowed? Should I start teaching intro calculus students that 1+2+3+...=-1/1…

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 interact.

Re: On Leibniz Notation

#38

Post by seanhunter »

Firstly I think that answer is amazing and I learned a lot from it even though I disagree with a lot of what they are saying. Math stack exchange is really high-quality sometimes and I think this is a great example.

For me the two notations are a lot like positional argument ordering vs named function arguments in a language like python which offers both. I personally prefer Legrange’s notation most of the time but think they both have their place. Legrange notation emphasises the ordering of the arguments and makes their naming going into the function irrelevant. This makes a lot of sense to me in a lot of situations where you have some pure abstract function with abstract arguments and makes it a lot easier to reason about certain things (like the example given or the chain rule which the author uses as an example where the common Leibniz formulation literally only makes sense if you do an explicit substitution in the function which is often not really explained).

On the other hand Leibniz notation makes a lot of sense to me in contexts like physics where the names of the function arguments are really important. You aren’t just picking your favourites as the author claims. For a very simple example, if I say a = \frac{d^2s}{dt^2}, every physicist knows a bunch. I am calculating accelleration as the second derivative of displacement with respect to time. “s” isn’t just my favourite letter today. So “named arguments” make a lot of sense here.

Secondly, Leibniz notation gives us the “derivative operator”[1], which is often very useful. Say I’m trying to find some derivative that involves a bunch of intermediate working out steps. It’s tremendously convenient to be able to say “ times d by dx of ” and come back and differentiate “some other expression” in a later step. In Lagrange notation I would need to let that second expression be some named intermediate function so I have something to hang my tick off, which is definitely a lot more of a pain.

[1] I’m not far enough along in my calculus journey to know whether I’m saying this right so forgive me if my terminology is incorrect - hopefully you get the intent.

Re: On Leibniz Notation

#39

Post by cygx »

brewmarche wrote: post #34
kergonath wrote: post #24

Earlier quoted context omitted.

This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…

As a mathematician I like the positional notation, but once we assign meaning to the slots it gets weird, so I had the same objection. One could add some new notation to distinguish the element _T_ from the slot _T_. For example let’s write slots as _[T]_ (I’m not creative enough to come up with something good), then we can talk about ∂_[T]F or ∂F/∂[T]. One can think of the [.] operator as mapping from “semantic symb…

Or use the convention from differential geometry, where the letter denotes a coordinate on the function's domain.

Re: On Leibniz Notation

#40

Post by bsaul »

Qem wrote: post #26
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…

> i've stumbled a few years ago on a math book that described the original concept of "infinitesimals" and how a whole different way of doing calculus exists. Was this book "Elementary Calculus: An Infinitesimal Approach", by Keisler? It's an awesome book. It's free to download at https://people.math.wisc.edu/~hkeisler/calc.html

yes that's the one. I didn't read or understand all of it, but it helped me a lot realize what i considered as weird (the dx object and the whole notation around derivative) was not due to comprehension problems on my part, but due to the fact that the field was still a work in progress.

Unfortunately, that realization came 25 years too late.

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