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On Leibniz Notation

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Re: On Leibniz Notation

#2

Post by zero-sharp »

Briefly, the way you make sense of this is by being consistent with notation and being aware of definitions. f is a function. f(x) is not a function. f(x) is an element of the range. Unfortunately, I'm sure growing up your teachers probably referred to f(x) as being the function. If you just simplify the function composition and clearly label objects, I'm pretty sure you won't have this kind of confusion. It really shouldn't be this complicated.

While we're at it, it's probably not the best idea to represent derivatives as fractions either (for the sake of notational consistency). But that notation will never die.

Re: On Leibniz Notation

#3

Post by volemo »

zero-sharp wrote: post #2

Briefly, the way you make sense of this is by being consistent with notation and being aware of definitions. f is a function. f(x) is not a function. f(x) is an element of the range. Unfortunately, I'm sure growing up your teachers probably referred to f(x) as being the function. If you just simplify the function composition and clearly label objects, I'm pretty sure you won't have this kind of confusion. It really s…

> While we're at it, it's probably not the best idea to represent derivatives as fractions either (for the sake of notational consistency).

Do you mean dy/dx? Why isn't that a good idea? Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx? What are the alternative you'd prefer: prime notation, D-notation, or else?

Re: On Leibniz Notation

#4

Post by rocqua »

The idea that the letter choice shouldn't matter fully makes sense to me. And in general, point free notation can be nice (point free meaning operating on the function rather than introducing an arbitrary point and operating on the function evaluated at the arbitrary point.

Despite this, I still believe Leibniz notation is superior for multi-argument functions. For multi argument functions, named arguments are much clearer than just depending on the order of arguments. Essentially I advocate for dropping point-freeness for clarity on the difference between function arguments.

Besides that, by expression equivalence it is clearly the case that the only correct interpretation of the derivative is the 'composition' where the change in t also counts for the change in x. Because replacing the f(x(t), t) with g(x) (where g is the composition, should not change the outcome of the derivative.

Re: On Leibniz Notation

#5

Post by BruceEel »

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.

Re: On Leibniz Notation

#6

Post by nmca »

rocqua wrote: post #4

The idea that the letter choice shouldn't matter fully makes sense to me. And in general, point free notation can be nice (point free meaning operating on the function rather than introducing an arbitrary point and operating on the function evaluated at the arbitrary point. Despite this, I still believe Leibniz notation is superior for multi-argument functions. For multi argument functions, named arguments are much c…

This is a noble take for programming languages too, where point-free style in e.g. Haskell is a pain in the neck, and named arguments are a blessing.

Re: On Leibniz Notation

#7

Post by bsaul »

volemo wrote: post #3
zero-sharp wrote: post #2

Briefly, the way you make sense of this is by being consistent with notation and being aware of definitions. f is a function. f(x) is not a function. f(x) is an element of the range. Unfortunately, I'm sure growing up your teachers probably referred to f(x) as being the function. If you just simplify the function composition and clearly label objects, I'm pretty sure you won't have this kind of confusion. It really s…

> While we're at it, it's probably not the best idea to represent derivatives as fractions either (for the sake of notational consistency). Do you mean dy/dx? Why isn't that a good idea? Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx? What are the alternative you'd prefer: prime notation, D-notation, or else?

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 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. And it seems to me those kinds of computations over "dx" come from there. But the end result of mixing concepts really looks like trash.

Re: On Leibniz Notation

#9

Post by shunyaekam »

volemo wrote: post #3
zero-sharp wrote: post #2

Briefly, the way you make sense of this is by being consistent with notation and being aware of definitions. f is a function. f(x) is not a function. f(x) is an element of the range. Unfortunately, I'm sure growing up your teachers probably referred to f(x) as being the function. If you just simplify the function composition and clearly label objects, I'm pretty sure you won't have this kind of confusion. It really s…

> While we're at it, it's probably not the best idea to represent derivatives as fractions either (for the sake of notational consistency). Do you mean dy/dx? Why isn't that a good idea? Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx? What are the alternative you'd prefer: prime notation, D-notation, or else?

The answer to this question is answered in the FA. Leibiz notation is confusing except in the most simple cases.

For example:

> Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx?

What exactly does "dy/y = dx" mean? What is on the LHS and what is on the RHS?

It acts like a mnemonic scribble for an intermediate step. It doesn't have any mathematical meaning.

Re: On Leibniz Notation

#10

Post by tobinfricke »

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".

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