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On Leibniz Notation

math.stackexchange.com

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Re: On Leibniz Notation

#71

Post by archgoon »

xigoi wrote: post #66
enriquto wrote: post #30

Earlier quoted context omitted.

> 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…

In what you have written, V is not a function, but a value calculated from some other values, so it does not make sense to differentiate it. If you instead write: V : ℝ³ → ℝ V(t,x,y) = (1 + sin(t))/sqrt(x^2+y^2) then it’s clear what D_2(V) means.

[dead]

Re: On Leibniz Notation

#72

Post by pinkmuffinere »

captainmuon wrote: post #70

Funnily, I disagree with almost every point in the most upvoted answer. For me (as a physicist by training), the f' notation is a shorthand, and df/dx is the more clear notation. Because especially in physics, you often deal with functions that are dependent on multiple variables. Do I want to differentiate wrt. time or position? You can use "f dot" for the time derivative, but in more complex cases you are out of lu…

Many of your objections are addressed in the stackexchange post:

1. "You often deal with functions that are dependent on multiple variables. Do I want to differentiate wrt. time or position?" -- the proposed solution is to put a subscript under the function name, to clarify whether you're differentiating wrt time, position, etc

2. "How can you distinguish between f(x) and f(t) which are very different functions?" -- the answer claims (and I agree) that using f(x) and f(t) to represent different functions is bad notation. If x and t are variables, surely f(some_variable) == f(other_variable). If x and t are specified values, then f(value1) may not equal f(value2), but f still _really really_ looks like the same function, just evaluated at different points. Better is to use different function names, like 'f' and 'g'.

3. "what if the variables depend on each other? You need a notation that distinguishes between treating "x" as a variable, and "x" as something that is dependent on some other variable" -- there is a proposed way to represent composition of functions. I don't want to dive into latex editing on hn, but you can see it in the post

4. "The nice thing about Leibnitz calculus [1] is that you can do things like reduce fractions with dx'es, you can flip things around and calculate dx/df, the chain rule is not a rule that you have to memorize but just an obvious expansion etc., and it mostly just works" -- it does indeed work sometimes, but this is not rigorous. When it works, it does so because you're using it in a domain where it just happens to work. "Cancelling" dx's is not reliable, and will sometimes lead to error. I'll admit I find the chain rule mnemonic convenient though

Re: On Leibniz Notation

#73

Post by jnwatson »

tesdinger wrote: post #59
bmacho wrote: post #43

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 varia…

> Functions don't have variables, named variables, that's Python, not math[0]. No, Python named arguments are a good analogy for Leibnitz notation.

The confusion is using the same namespace for argument slots and variables. It might have been clearer if, for example, Leibniz would have used Greek letters alpha and beta instead of x and y.

Re: On Leibniz Notation

#74

Post by enriquto »

xigoi wrote: post #66
enriquto wrote: post #30

Earlier quoted context omitted.

> 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…

In what you have written, V is not a function, but a value calculated from some other values, so it does not make sense to differentiate it. If you instead write: V : ℝ³ → ℝ V(t,x,y) = (1 + sin(t))/sqrt(x^2+y^2) then it’s clear what D_2(V) means.

> not a function, but a value calculated from some other values

The term of art is "an expression" [0]. And you can also differentiate expressions with respect to their variables. It's a perfectly supported construction in all symbolic computer algebra packages. No need to assign an (arbitrary) ordering to your variables in order to differentiate with respect to them.

[0] https://en.wikipedia.org/wiki/Expression_(mathematics)

Re: On Leibniz Notation

#76

Post by bsaul »

LudwigNagasena wrote: post #47
bsaul wrote: post #41

Earlier quoted context omitted.

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.

I’ve found the book online and will definitrly look into it. It’s going to be my first time with russian math teaching style, i’m really looking forward to it. From what i’ve seen browsing the first chapter it seems very down to earth and straightforward, i really like it. Also, it covers exactly the scope of calculus i want to get better at, so thanks again.

Re: On Leibniz Notation

#77

Post by amluto »

MatteoFrigo wrote: post #31
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…

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.

By that standard, if f is a function of x and y, then writing f(1,2) is syntactically bad because the argument slots don’t have a meaningful order. One could surely invent a valid mathematical formalism with exclusively named argument slots, but this isn’t how math is generally done.

(I admit it might be a lot easier to avoid losing track of which thing is a row and which is a column in a gnarly linear algebra expression if all dimensions were explicitly named, and this would come with a tradeoff of verbosity. Also, the interpretation of a matrix as a linear function from vectors to vectors would need some clarification as to which dimension is input and which is output, so maybe it would look a bit like Einstein notation with superscript dimensions and subscript dimensions?)

Re: On Leibniz Notation

#78

Post by cygx »

tesdinger wrote: post #61
amatic wrote: post #19

> "what's so special about x ?" Does f(x) mean one function, whereas f(y) means a completely different function? This looks like the difference between parameters of a function and arguments. In the definition, you have the parameter x, used internaly and, when calling the function you use an argument - located in the calling context. In python: def f(x): return 2*x ## x is a parameter x = 3 y = f(x) # x in an argume…

Never heard of this distinction between parameter and argument, to me they are homonymes. To me, argument or parameter refers to a term in the function definition as well as a term in the function call.

Cf https://en.wikipedia.org/wiki/Parameter_%28computer_programm...

Re: On Leibniz Notation

#79

Post by [deleted] »

MatteoFrigo wrote: post #31
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…

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.

[deleted]

Re: On Leibniz Notation

#80

Post by bikenaga »

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 can be a differential form, but in (elementary) calculus books it's exposited this way: Suppose you have a function y = f(x) taking real numbers to real numbers, so x is an independent variable, y is the dependent variable. You define an independent variable Δx and define dx = Δx. Also define dependent variables dy and Δy by

    Δy = f(x + Δx) - f(x)  and  dy = f'(x) dx.
Then f'(x) = dy/dx. This may look like a stupid hack to make the last formula work, but actually it's a little more. If you use nonstandard analysis, you define the derivative of a function f from reals to reals by

  f'(a) = st( (f(a + Δx) - f(a)) / Δx )
where st takes the standard part of a hyperreal number and Δx is a nonzero infinitesimal. This is like the usual limit definition, without limits. Then you can use the formulas above and "dy" and "dx" are numbers, albeit hyperreal numbers.

(The "dx as a differential form" vs. "dx as a number" is probably coming from the fact that the tangent space to the reals at a real number is isomorphic to the reals, so the dual space [where dx lives] is too.)

(Calculus via infinitesimals is pretty cool; a good resource for this is H. Jerome Keisler's "Elementary Calculus" and "Foundations of Infinitesimal Calculus", both available for free: https://people.math.wisc.edu/~hkeisler/)

I second the recommendation for Barrett O'Neill's book - I used it in my differential geometry class at MIT.

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