[HN Gopher] On Leibniz Notation
___________________________________________________________________
On Leibniz Notation
Author : colonCapitalDee
Score : 139 points
Date : 2024-01-20 02:51 UTC (20 hours ago)
(HTM) web link (math.stackexchange.com)
(TXT) w3m dump (math.stackexchange.com)
| zero-sharp wrote:
| 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.
| volemo wrote:
| > 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?
| bsaul wrote:
| 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.
| LudwigNagasena wrote:
| dx,dy,dz are differential 1-forms. It's like i,j,k in a
| vector field.
| bsaul wrote:
| 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.
| LudwigNagasena wrote:
| 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.
| bsaul wrote:
| 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.
| galaxyLogic wrote:
| A simple conceptual description of a derivative is "speed".
|
| To measure the speed of a moving object you must divide the
| distance moved by the time it took to move that distance.
|
| So how can you measure what the speed is at a given
| location? In a sense you cannot, you can only measure it at
| a given interval over the period of time it took to move
| that distance.
|
| So it is kind of confusing. dx/dy represents the limit of
| measuring the speed over increasingly small distances and
| durations around a given point in space and time. If you
| take dx to 0 and dy to 0 it does not make sense because 0/0
| is ill-defined. Therefore we need some notation that
| implies we are really not talking about a single point, but
| an increasingly small distance, and duration.
| laingc wrote:
| > 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 a gradient, if you like. These are very
| important in Differential Geometry.
|
| You can use differential forms to do all sorts of things,
| but one example you may be familiar with is to compute area
| or volume forms over arbitrary manifolds. It gets a bit
| hard to define things on HN without TeX support, but using
| differential one-forms and the related exterior derivative,
| you can define a generalised Stokes' theorem that works for
| any smooth, oriented manifold.
|
| I used this in my PhD, and implemented it directly in a
| numerical method, so this has very practical engineering
| uses also.
|
| "Elementary Differential Geometry" by Barrett O'Neill is a
| pretty beginner-friendly introduction to some of these
| topics if you're interested, though there are many other
| good texts also.
| zero-sharp wrote:
| >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?
| lifthrasiir wrote:
| 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.
| zero-sharp wrote:
| 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.
| lifthrasiir wrote:
| 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_.
| laingc wrote:
| I replied with what the thing is called, explained what
| it can be used for, and recommended an introductory text
| to learn more.
|
| If you can come up with a more helpful reply in as many
| words, then please do so.
| Qem wrote:
| > 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
| bsaul wrote:
| 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.
| shunyaekam wrote:
| 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.
| volemo wrote:
| > What exactly does "dy/y = dx" mean?
|
| As I understand, dy/y = dx means that the derivative of 1/y
| with respect to y equals to the derivative of 1 with
| respect to x.
| shunyaekam wrote:
| The derivative of 1 with respect to x is zero, since the
| derivative of any constant function is zero.
|
| So you must be saying that the derivative of 1/f(x) with
| respect to x is zero for any f(x), where f is a
| differentiable function, with non-vanishing derivative
| near x (for it to be defined in the first place).
|
| That doesn't make any sense.
|
| Please don't respond to this. It's getting absurd.
| volemo wrote:
| I see my mistake now, sorry for being stupidly blind.
| LudwigNagasena wrote:
| If you take x as a stand-in for a number, then df/dx simply
| makes no sense, df/d(5) is either nonsense or division by zero.
| Your approach doesn't seem to make more sense out of Leibniz
| notation, unless I misunderstand what you mean by f(x) being an
| element of the range.
| bmacho wrote:
| If you know a language with function types, then you can
| think of it as f : R -> R
|
| that is, f having a type signature as a function, and
| f(x) : R
|
| having a type signature of a float (or: being a float). And
| so on.
|
| I've been using this approach to type-check calculus
| equations. Unfortunately the article in the top comment [0]
| says that
|
| > Note that f means something different on the two sides of
| the equation!
|
| so the Leibniz notation might be more ad-hoc than I thought,
| and I can't just type-check them, actually have to reverse
| engineer the intent of the authors. I have to think about
| this. For example I remember someone giving me 3 exercises
| from Stewart calculus, and I gave 2 back that those equations
| don't even type-check, maybe there are some notation abuses I
| am not aware of.
|
| [0] https://mitp-content-
| server.mit.edu/books/content/sectbyfn/b...
| rocqua wrote:
| 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.
| nmca wrote:
| 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.
| anderskaseorg wrote:
| Named arguments are great, but mathematical notation doesn't
| have them. x(t) is not a named argument to f(x(t), t), so
| notation like [?]f(x(t), t)/[?]x(t) is nonsense.
|
| If we did have real named arguments like f(a=x(t), b=t), where
| a and b are fixed by the definition of f rather than arbitrary
| names to be made up on each invocation, then maybe it would
| make sense to write something like ([?]f/[?]a)(a=x(t), b=t).
| Though that's still pretty far from Leibniz notation, where
| [?]f/[?]a is somehow supposed to be a numerical value depending
| on a, not a function to which arguments must be supplied.
|
| But what I think is really lacking in mathematical notation is
| explicit lambda abstraction: we should be able to write
|
| (lt. f(x(t), t))'(t) = (la. f(a, t))'(x(t)) x'(t) + (lb.
| f(x(t), b))'(t),
|
| which reuses the ordinary one-variable derivative and has none
| of these ambiguities.
| BruceEel wrote:
| 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.
| enriquto wrote:
| > 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 clear. What
| does D_2(V) mean, though? It does depend on your favourite
| ordering of the letters!
|
| Even if you prefer positional notation for derivatives (a la
| Spivak), it must be recognized that the naming-based notation
| has its merits, and sometimes is clearer.
| BruceEel wrote:
| 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..
| xigoi wrote:
| 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.
| enriquto wrote:
| > 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)
| LudwigNagasena wrote:
| A somewhat relevant article, it explores an alternative to
| Leibniz notation: Extending the Algebraic Manipulability of
| Differentials (https://arxiv.org/pdf/1801.09553.pdf).
| tobinfricke wrote:
| 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".
| kergonath wrote:
| 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
| and the orders of the arguments is completely irrelevant, but
| _what_ these arguments are is important. The Leibniz notation
| should certainly be improved or replaced, but not by this.
|
| I mean, they even fuck it up in the sentence after their
| equation:
|
| > The Lagrangian L is a real-valued function of time t,
| coordinates x, and velocities v; the value is L(t, x, v).
| Partial derivatives are indicated as derivatives of functions
| with respect to particular argument positions; [?]_2 L
| indicates the function obtained by taking the partial
| derivative of the Lagrangian function L with respect to the
| velocity argument position.
|
| From what they say at the beginning the second argument is the
| position, not the velocity, so [?]_2 L is [?]_x L, not [?]_v L.
| Which is a really easy mistake to make because every single
| equation needs to go back to the definition of L to make sense.
| MatteoFrigo wrote:
| 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.
| codethief wrote:
| The order isn't really irrelevant, though, is it? If you
| take the total derivative and represent it as Jacobian
| matrix, you hopefully won't argue that the order of the
| matrix entries won't matter. (Especially if you later on
| employ it in a chain rule.)
| MatteoFrigo wrote:
| I think that there are two cases of practical importance,
| which have incompatible requirements.
|
| The first case is where you have a N-dimensional vector
| space where all dimensions have the same units. The
| standard example would be the Newtonian 3D space.
| Depending on what you are trying to do, you can view it
| as a collection of coordinate-free abstract vectors, as a
| triple (x, y, z) of real numbers, or as an array X[i] of
| three coordinates in a given basis. In this case I would
| agree that X[0], X[1], X[2] is better than (x, y, z), the
| order matters, and you can define the Jacobian is a 2D
| array that represents a certain abstract derivative in a
| given coordinate system. I would argue that the formalism
| of Sussman and Wisdom (which they got from Spivak) is
| totally adequate to this case, and perhaps even the best
| possible.
|
| The second case is the one of the Lagrangian that parent
| mentioned, where L is a function of the triple (t, x, v).
| You could pretend that (t, x, v) form a vector space, but
| this definition won't get you far. I would regard (t, x,
| v) = t * (1, 0, 0) + x * (0, 1, 0) + v * (0, 0, 1) as
| meaningless because it is adding time, space, and
| velocity. You cannot really do rotations or general
| linear transformations in this space. You can define a
| Jacobian matrix if you want, but now all entries in the
| matrix have different physical units. In this case I
| would say the fact that v is the third element of the
| tuple is irrelevant, and that the tuple is better
| regarded as a map from symbolic names "t", "x", and "v"
| to real numbers. I would argue that the Spivak formalism
| is inadequate in this case, and it seems that many
| physicists on this thread think the same for essentially
| the same reason.
|
| This difference is kind of analogous to double X[3]; vs
| struct { double t; double x; double v; }; From one point
| of view they are the same, but in practice they have
| totally different meanings.
| kergonath wrote:
| > FWIW, they start counting function arguments from 0, so
| _2 is indeed the velocity.
|
| Dammit yes, you're right! Well, it's not a bit less
| confusing.
|
| The most frustrating is that they have a point: we need to
| be stricter about disambiguating functions and numbers, and
| derivation really should be an operator.
|
| But you don't need to go all the way to zero-indexing
| (which is definitely not a thing in the fields I know) or
| positional arguments. This is putting abstract notation
| purity above practical concerns. It's not surprising they
| like Scheme.
| tobinfricke wrote:
| > But you don't need to go all the way to zero-indexing
| (which is definitely not a thing in the fields I know) or
| positional arguments. This is putting abstract notation
| purity above practical concerns.
|
| Yes, it's definitely a perspective influenced strongly by
| computer science.
|
| > It's not surprising they like Scheme.
|
| In fact one of the authors, Gerry Sussman, is one of the
| original inventors of Scheme.
|
| https://en.wikipedia.org/wiki/Gerald_Jay_Sussman
| amluto wrote:
| 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?)
| brewmarche wrote:
| 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.
| cygx wrote:
| Or use the convention from differential geometry, where the
| letter denotes a coordinate on the function's domain.
| codethief wrote:
| Differential geometry is not exactly a great example of
| notational clarity. :-)
| nemoniac wrote:
| 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
| Tainnor wrote:
| > So, until you are able to perform this "translation" from
| precise to imprecise notation, I suggest you stick to the precise
| notation, until all the fundamental concepts are clear, and only
| then abuse notation.
|
| I agree with this part. I think in teaching, the precise notation
| should be taught first and everyone should understand it well.
| But later on, it will make communication and even just doing
| calculations easier if you can accept the imprecise notation, use
| it and translate it into something clearer when needed. This is
| particularly handy in e.g. differential equations where sloppy
| notation makes certain manipulations a bit easier.
| kzrdude wrote:
| When starting multivariable calculus it might be a good place
| to start with clarifying this, at the latest.
| movpasd wrote:
| 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 reasons why physicists use Leibniz
| notation, and this answer really glosses over that.
|
| The reason is that the particularities of the mathematical
| structures used to model a physical problem matter a lot less
| than the relationships between the actual physical underlying
| quantities. And there can be a lot of them. The answer evokes
| thermodynamics, and I couldn't think of a better example. Are we
| really to introduce a distinct symbol for _every_ possible
| functional relation between _every_ state variable? If I have T =
| f(P, V) and P = g(T, S), do I need to remember in which position
| exactly each function has been ordered before I can write down
| "how P varies with T when S is fixed"?
|
| Leibniz notation, although formally tricksy, is just the best
| tool for communicating the _intent_ of a physical relationship
| without getting bogged down in the mathematically detail. The
| purpose of an equation is to express that relationship to the
| reader. Think if it like code -- is it so bad if my code does a
| bit of magic behind the scenes to allow for a clearer reading,
| even if the semantics aren't immediately obvious? Well,
| ultimately, it depends on the situation, and a balance must be
| reached. I don't believe that expressing everything the way TFA
| suggests is striking the right balance.
|
| Notational abuse happens all the time in physics, and this is
| certainly not the most egregious example. Just compare it to the
| path integral. It's easy to assume that this is because of a lack
| of sophistication or rigour by physicists. (Certainly I did
| throughout my physics education, being more mathematically or
| pedantically inclined.) But it's a simplistic view.
|
| Now, while I'll defend the usage of these sort of unrigorous
| conventions even if they are strictly speaking meaningless
| mathematically, what I won't defend is the slapdash approach that
| is often used to _teach_ partial derivatives to physicists. Some
| exposure to concepts like distinguishing real
| variables/quantities from functions is needed, or, as TFA does
| mention at the end, the student won't be able to unpack the
| notational convenience into clear semantics, which can lead to
| unclear reasoning. I used to share the views of the author for a
| period when first introduced to Leibniz partial derivative
| notation in my first thermodynamics course, and, probably like
| them, found it to be totally incomprehensible symbol soup. But
| for myself, I see now that it was mostly a failure of teaching
| rather than a failure of the notation itself.
|
| I'll add one last thought. There is a degree of "primitive
| obsession" at work here, trying to fit everything into positional
| functions and real numbers. I have thought that a formalism that
| better reflects the structure of "physical quantity" (as opposed
| to thinking of them as plain real numbers) may help bridge the
| gap between rigour and conceptual convenience. The tools are
| really already there. We need two key concepts: first, borrow
| from programming the idea of keyword arguments (there is a book
| which sadly I can't remember the name of which does as much to
| formalize Einstein notation for tensors in a coordinate-free
| way); second, to model quantities not as real numbers but as
| differentiable homomorphisms from a state space (modeled as a
| manifold in a coordinate-free way) to the reals. This is how
| physicists already think about it, it needs only be formalised.
| 082349872349872 wrote:
| 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?
| lupire wrote:
| No mathematician has ever accused a physicist of being
| rigorous.
| amatic wrote:
| > "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 argument
|
| For partial derivatives, it may be ok to use 1 and 2 to show they
| are the first and second parameter, but maybe they could also be
| named by some convention, like x and y or alpha and beta or
| whatever
| tesdinger wrote:
| 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.
| cygx wrote:
| Cf https://en.wikipedia.org/wiki/Parameter_%28computer_progra
| mm...
| fiforpg wrote:
| Such discussions show that teaching of calculus often tends to be
| overly algebraic, for no good reason.
|
| Clearly, Leibniz notation does not _intrinsically_ contain any
| deep insights since at one point Leibniz himself was erroneously
| induced by it to think that d(xy)= d(x)d(y), which is false.
| Newton on the other hand thought in terms of simple geometric
| concepts (areas), which make it crystal clear that d(xy) = x d(y)
| + y d(x).
|
| On the topic of history of early analysis I cannot recommend
| enough this collection of anecdotes from Arnold:
|
| https://archive.org/details/huygensbarrownew0000arno
| Q6T46nT668w6i3m wrote:
| Ultimately, algebra _is_ important and calculus is a useful
| setting to practice algebraic manipulation. Nevertheless, the
| trend in modern mathematics education in the United States is
| increasingly towards a more balanced understanding (e.g.,
| greater emphasis on geometric interpretations).
| impossiblefork wrote:
| But you do have 1/(dy/dx) = dx/dy and you can get the insight
| deep if you follow on with actually using differentials
| properly, as in the paper linked here in this very thread by
| LudwigNagasena.
|
| But then of course, you see that d^2 f/dx^2 is a bad expression
| for the second derivative, and that you should actually use
| differentials properly and write it as (d^2 f)/(dx)^2 - df/dx
| (d^2 f)/(df)^2.
| sirsinsalot wrote:
| I'm currently reading "Calculus Reordered: A history of the big
| ideas" by David M Bressoud, which really helped me contextualise
| all the peculiarities of calculus.
|
| It really does feel like a hodgepodge of poorly fitting parts
| that could use some revision in terms of notation and teaching,
| but I feel like that's part of the charm.
|
| I like that it encodes so much history and culture "up front". It
| reminds me of PHP. There is no way a good language designer would
| create PHP intentionally but as ugly as it was, it was brilliant
| in 1999. Index.php and you're done.
| robinhouston wrote:
| 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 here in the land of the Y Combinator
| should have less trouble with, is the distinction between free
| and bound variables.
|
| In lambda calculus, variable binding is explicit and denoted by a
| lambda. Leibniz notation binds variables in just the same way:
| the operator (d/dx) binds the variable x in the expression to
| which it is applied.
| zero-sharp wrote:
| >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.
| lupire wrote:
| The answer author rambled on without well structured
| paragraphs, but the author seems to have a correct intuition
| for it, since they wrote that "f(x)" and "df/dx" are not well-
| formed function identifiers (free variables), but "(df/dx)(x)"
| is OK (bound variables.
| BlackFly wrote:
| The choice of denominator letters are not irrelevant in Leibniz
| notation. They are chosen in conjunction with parameterization in
| order to suppress arguments to avoid being lengthy. Similar to
| how Einstein summation convention is used to suppress summation
| symbols. All notational conveniences lead to confusion in certain
| scenarios.
|
| Given a field W, you will want to talk about various derivatives
| of the field. However, you might not be interested in the
| derivative of the fields with respect to coordinate fields like
| `\partial W / \partial x^i`, you might be interested in
| derivatives of that field with respect to some other field like
| `\partial W / \partial \lambda_i`. Then you end up introducing
| all kinds of auxiliary notation for the functional
| representations of the field. You end up with
| `\tilde{W}(\lambda_i)` and `\bar{W}(I_i)` next to `W(x_i)`.
|
| Or just consider changes of variable, one generally doesn't
| distinguish the electric field `E` from cartesian, to
| axisymmetric, to spherical to whatever other coordinate system
| you dream up. Instead, the presence of the chosen coordinate
| symbols denotes the coordinate system in use and you don't need
| to distinguish the field from its functional representation. In
| order to understand `\partial_2 E` you would need to understand
| what coordinates I am working with while `\partial E / \partial
| y` is a bit more self explanatory in classical electrodynamics.
| Thermodynamics picks up confusion because of the constant changes
| in variable when the variables aren't coordinates.
| Buttons840 wrote:
| 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
| cygx wrote:
| 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 ph a chart, eg
| cartesian coordinates ph: p -(x(p), y(p)) where x,y: M - R. Then,
| [?]/[?]x denotes the holonomic vector field tangent to the
| coordinate lines t - ph-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|[?] f = lim_{h - 0} ( f(ph-1(x(p) + h,
| y(p))) - f(p) ) / h
| seanhunter wrote:
| 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
| "<something> times d by dx of <some other expression>" 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.
| bmacho wrote:
| 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...
| tesdinger wrote:
| > Functions don't have variables, named variables, that's
| Python, not math[0].
|
| No, Python named arguments are a good analogy for Leibnitz
| notation.
| jnwatson wrote:
| 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.
| bowsamic wrote:
| The verb is "to differentiate", not "to derive". You
| differentiate to find the derivative.
| tlb wrote:
| > "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.
| xamuel wrote:
| 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
| tesdinger wrote:
| What if your physics problem involves the same physical quantity
| twice, like speeds and masses of two bodies? Then single letter
| variable names are no longer unique nor meaningful.
|
| f(v_1, m_1, v_2, m_2)
|
| For the derivatives I would prefer writing
|
| d_v_2 f
|
| instead of
|
| d_3 f
|
| when counting arguments from 1
| OscarCunningham wrote:
| The way to think about Leibniz notation is that there's a
| manifold M, and all the quantities f,g,x,y... are functions from
| M to the reals. Then you can talk about the derivative of any of
| these quantities with respect to any of the others, without
| thinking of any of them as the arguments of any of the others.
| tolmasky wrote:
| Is there a good explanation of u substitution (or any other
| integral stuff that makes use of the denominator) using non-
| Liebnitz notation? Is it just that [?]1 is actually [?]/[?]1, and
| thus you can begin using [?]1 the way you would have used [?]x?
| captainmuon wrote:
| 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 luck. How can you
| distinguish between f(x) and f(t) which are very different
| functions? And 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.
|
| 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. I don't recall seeing an explicit proof _why_
| it works (except for some specific cases), or a list of exact
| rules, but I 'm sure that exists and it would have been neat to
| have seen that in my studies.
|
| [1] calculus here in the sense of German Kalkul, a notational
| system and a method of mechanically manipulating symbols, and not
| neccessarily meaning "differential and integral calculation",
| although "the" calculus is the prime example of "a calculus" of
| course.
| pinkmuffinere wrote:
| 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
| amai wrote:
| All notations are wrong. But some of them are useful.
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