8 ms·

On Leibniz Notation

zero-sharp 3y ago
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 3y ago
  > 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 3y ago
    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 3y ago
      dx,dy,dz are differential 1-forms. It's like i,j,k in a vector field.
      - bsaul 3y ago
        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 3y ago
        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 3y ago
        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.
        nyssos 3y ago
        Skimming through Zorich, it looks like it's roughly at the level of an American honors freshman calculus class, i.e. one aimed at math and physics majors. Engineers are unlikely to see any rigorous analysis in undergrad unless they're personally interested in math.
    - galaxyLogic 3y ago
  - shunyaekam 3y ago
    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 3y ago
      > 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 3y ago
        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 3y ago
        I see my mistake now, sorry for being stupidly blind.
    - bikenaga 3y ago
      > t acts like a mnemonic scribble for an intermediate step. It doesn't have any mathematical meaning. That's about right. When you cover elementary solution methods for differential equations, you start with something like y = dy/dx. You're supposed to separate variables ("get the x's on one side and the y's on the other, then integrate"). So it's tempting to just write "dy/y = dx", even though as you say it doesn't have any mathematical meaning. But it's helpful in keeping track of the algebra. You then forget you wrote that meaningless but helpful step and write "∫ dy/y = ∫ dx" which is okay, and go from there. Looking at one of my old diff eq books I see whole sections where this kind of casual algebra with differentials is the norm. When anyone would ask me about this in class, I'd say something like this. Think of a solution curve for y = dy/dx as a parametrized curve, so x = f(t) and y = g(t). Then interpret the equation as y = (dy/dt)/(dx/dt), write dx/dt = (1/y) (dy/dt), then integrate both sides with respect to t: ∫ 1/y (dy/dt) dt = ∫ (dx/dt) dt Change variables to get "∫ dy/y = ∫ dx". After a while we believe that this will always work, and we just suppress the stuff about parametric equations.
- LudwigNagasena 3y ago
  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 3y ago
    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/books_pres_0/9579/sicm_edition_2.zip/preface001.html https://mitp-content-server.mit.edu/books/content/sectbyfn/b...
movpasd 3y ago
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.
rocqua 3y ago
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 3y ago
  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 3y ago
  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 (λt. f(x(t), t))'(t) = (λa. f(a, t))'(x(t)) x'(t) + (λb. f(x(t), b))'(t), which reuses the ordinary one-variable derivative and has none of these ambiguities.
BruceEel 3y ago
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 3y ago
  > 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 (à la Spivak), it must be recognized that the naming-based notation has its merits, and sometimes is clearer.
  - BruceEel 3y ago
    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 3y ago
    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.
    - archgoon 3y ago
      [dead]
    - enriquto 3y ago
      > 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) https://en.wikipedia.org/wiki/Expression_(mathematics)
LudwigNagasena 3y ago
A somewhat relevant article, it explores an alternative to Leibniz notation: Extending the Algebraic Manipulability of Differentials (https://arxiv.org/pdf/1801.09553.pdf https://arxiv.org/pdf/1801.09553.pdf).
tobinfricke 3y ago
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/books_pres_0/9579/sicm_edition_2.zip/preface001.html 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 3y ago
  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 3y ago
    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 3y ago
Tainnor 3y ago
> 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 3y ago
  When starting multivariable calculus it might be a good place to start with clarifying this, at the latest.
amatic 3y ago
> "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 3y ago
  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 3y ago
    Cf https://en.wikipedia.org/wiki/Parameter_%28computer_programming%29#Parameters_and_arguments https://en.wikipedia.org/wiki/Parameter_%28computer_programm...
fiforpg 3y ago
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 https://archive.org/details/huygensbarrownew0000arno
- Q6T46nT668w6i3m 3y ago
  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 3y ago
  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.
  - bikenaga 3y ago
    The second derivative notation makes sense when you think of d/dx as an operator on functions. (As someone noted below, d/dx is essentially a tangent vector, which is something that operates on functions.) d/dx takes the derivative of what follows with respect to x, so (d/dx)(f) = df/dx is the derivative function of f with respect to x. Now you want to differentiate df/dx with respect to x, so you do "d/dx" to "df/dx": (d/dx) (df/dx) (This looks better if you write it with "real" fractions.) On the top you have "d d f" with is naturally written "d^2f". On the bottom, you have "dx dx", which is naturally written "dx^2". So the result should be written d^2 f/dx^2.
sirsinsalot 3y ago
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 3y ago
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 3y ago
  >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 3y ago
  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 3y ago
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 3y ago
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 https://arxiv.org/pdf/1801.09553.pdf
- bvan 3y ago
  Thanks for that reference. Very useful.
cygx 3y ago
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
- bikenaga 3y ago
  This is a good point. Noting that "d/dx" is a vector (field) is an antidote to the ideas that vectors are little arrows, or "a magnitude and a direction", or an ordered list with numbers in the slots. Vectors can be thought of in lots of ways, and here they're thought of as operators on functions.
seanhunter 3y ago
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 3y ago
Not just Leibniz notation, but for example https://en.wikipedia.org/wiki/Partial_derivative 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)#Definition https://en.wikipedia.org/wiki/Function_(mathematics)#Definit...
- tesdinger 3y ago
  > Functions don't have variables, named variables, that's Python, not math[0]. No, Python named arguments are a good analogy for Leibnitz notation.
  - deleted 3y ago
    [deleted]
  - jnwatson 3y ago
    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 3y ago
  The verb is "to differentiate", not "to derive". You differentiate to find the derivative.
tlb 3y ago
> "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 3y ago
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 https://philpapers.org/archive/ALEFDV.pdf
- pokerface_86 3y ago
  fantastic paper, i really enjoyed reading it. college senior who majored in math and finished the major by my sophomore year. always wished there were more calculus based courses to take but throughout my degree the vast majority of math “electives” ended up being applications of linear algebra or probability.
tesdinger 3y ago
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 3y ago
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 3y ago
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?
- bikenaga 3y ago
  If you mean u-substitution in definite integrals, that is something called the change of variables theorem. I think you mean indefinite integrals or antiderivatives, and I think you're asking for an explanation with using the "du/dx" sort of notation. Here's a quick explanation which may be more than you wanted to know. :-) You have ∫ f(g(x)) g'(x) dx. [So it might be something like ∫ [cos(x^2)] (2 x) dx, where f(x) = cos x and g(x) = x^2.] You do the u-substitution u = g(x), so u'(x) = g'(x): ∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx You want to say that the second integral is ∫ f(u) du -- essentially (to go back to "fractional" derivative notation), you want to do (du/dx) dx = du to get rid of the x's. Why is this justified? Let F(u) = ∫ f(u) du: F is the antiderivative of f. By definition of the antiderivative, F'(u) = f(u). So by the Chain Rule, d/dx F(u) = F'(u) u'(x) = f(u) u'(x). But since F(u) = ∫ f(u) du differentiates to f(u) u'(x), by definition F(u) is the antiderivative of f(u) u'(x). But the antiderivative of f(u) u'(x) is ∫ f(u) u'(x) dx. So putting all this stuff together, ∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx = F(u) = ∫ f(u) du.
captainmuon 3y ago
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 Kalkül, 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 3y ago
  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 3y ago
All notations are wrong. But some of them are useful.

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 3y ago
> 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 3y ago
  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 3y ago
    dx,dy,dz are differential 1-forms. It's like i,j,k in a vector field.
    - bsaul 3y ago
      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 3y ago
        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 3y ago
        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.
        nyssos 3y ago
        Skimming through Zorich, it looks like it's roughly at the level of an American honors freshman calculus class, i.e. one aimed at math and physics majors. Engineers are unlikely to see any rigorous analysis in undergrad unless they're personally interested in math.
  - galaxyLogic 3y ago
- shunyaekam 3y ago
  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 3y ago
    > 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 3y ago
      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 3y ago
        I see my mistake now, sorry for being stupidly blind.
  - bikenaga 3y ago
    > t acts like a mnemonic scribble for an intermediate step. It doesn't have any mathematical meaning. That's about right. When you cover elementary solution methods for differential equations, you start with something like y = dy/dx. You're supposed to separate variables ("get the x's on one side and the y's on the other, then integrate"). So it's tempting to just write "dy/y = dx", even though as you say it doesn't have any mathematical meaning. But it's helpful in keeping track of the algebra. You then forget you wrote that meaningless but helpful step and write "∫ dy/y = ∫ dx" which is okay, and go from there. Looking at one of my old diff eq books I see whole sections where this kind of casual algebra with differentials is the norm. When anyone would ask me about this in class, I'd say something like this. Think of a solution curve for y = dy/dx as a parametrized curve, so x = f(t) and y = g(t). Then interpret the equation as y = (dy/dt)/(dx/dt), write dx/dt = (1/y) (dy/dt), then integrate both sides with respect to t: ∫ 1/y (dy/dt) dt = ∫ (dx/dt) dt Change variables to get "∫ dy/y = ∫ dx". After a while we believe that this will always work, and we just suppress the stuff about parametric equations.
LudwigNagasena 3y ago
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 3y ago
  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/books_pres_0/9579/sicm_edition_2.zip/preface001.html https://mitp-content-server.mit.edu/books/content/sectbyfn/b...

movpasd 3y ago

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.

rocqua 3y ago

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 3y ago
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 3y ago
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 (λt. f(x(t), t))'(t) = (λa. f(a, t))'(x(t)) x'(t) + (λb. f(x(t), b))'(t), which reuses the ordinary one-variable derivative and has none of these ambiguities.

BruceEel 3y ago

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 3y ago
> 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 (à la Spivak), it must be recognized that the naming-based notation has its merits, and sometimes is clearer.
- BruceEel 3y ago
  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 3y ago
  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.
  - archgoon 3y ago
    [dead]
  - enriquto 3y ago
    > 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) https://en.wikipedia.org/wiki/Expression_(mathematics)

LudwigNagasena 3y ago

A somewhat relevant article, it explores an alternative to Leibniz notation: Extending the Algebraic Manipulability of Differentials (https://arxiv.org/pdf/1801.09553.pdf https://arxiv.org/pdf/1801.09553.pdf).

tobinfricke 3y ago

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/books_pres_0/9579/sicm_edition_2.zip/preface001.html 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 3y ago
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 3y ago
  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 3y ago

Tainnor 3y ago

> 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 3y ago
When starting multivariable calculus it might be a good place to start with clarifying this, at the latest.

amatic 3y ago

> "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 3y ago
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 3y ago
  Cf https://en.wikipedia.org/wiki/Parameter_%28computer_programming%29#Parameters_and_arguments https://en.wikipedia.org/wiki/Parameter_%28computer_programm...

fiforpg 3y ago

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 https://archive.org/details/huygensbarrownew0000arno

Q6T46nT668w6i3m 3y ago
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 3y ago
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.
- bikenaga 3y ago
  The second derivative notation makes sense when you think of d/dx as an operator on functions. (As someone noted below, d/dx is essentially a tangent vector, which is something that operates on functions.) d/dx takes the derivative of what follows with respect to x, so (d/dx)(f) = df/dx is the derivative function of f with respect to x. Now you want to differentiate df/dx with respect to x, so you do "d/dx" to "df/dx": (d/dx) (df/dx) (This looks better if you write it with "real" fractions.) On the top you have "d d f" with is naturally written "d^2f". On the bottom, you have "dx dx", which is naturally written "dx^2". So the result should be written d^2 f/dx^2.

sirsinsalot 3y ago

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 3y ago

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 3y ago
>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 3y ago
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 3y ago

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 3y ago

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 https://arxiv.org/pdf/1801.09553.pdf

bvan 3y ago
Thanks for that reference. Very useful.

cygx 3y ago

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

bikenaga 3y ago
This is a good point. Noting that "d/dx" is a vector (field) is an antidote to the ideas that vectors are little arrows, or "a magnitude and a direction", or an ordered list with numbers in the slots. Vectors can be thought of in lots of ways, and here they're thought of as operators on functions.

seanhunter 3y ago

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 3y ago

Not just Leibniz notation, but for example https://en.wikipedia.org/wiki/Partial_derivative 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)#Definition https://en.wikipedia.org/wiki/Function_(mathematics)#Definit...

tesdinger 3y ago
> Functions don't have variables, named variables, that's Python, not math[0]. No, Python named arguments are a good analogy for Leibnitz notation.
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- jnwatson 3y ago
  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 3y ago
The verb is "to differentiate", not "to derive". You differentiate to find the derivative.

tlb 3y ago

> "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 3y ago

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 https://philpapers.org/archive/ALEFDV.pdf

pokerface_86 3y ago
fantastic paper, i really enjoyed reading it. college senior who majored in math and finished the major by my sophomore year. always wished there were more calculus based courses to take but throughout my degree the vast majority of math “electives” ended up being applications of linear algebra or probability.

tesdinger 3y ago

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 3y ago

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 3y ago

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?

bikenaga 3y ago
If you mean u-substitution in definite integrals, that is something called the change of variables theorem. I think you mean indefinite integrals or antiderivatives, and I think you're asking for an explanation with using the "du/dx" sort of notation. Here's a quick explanation which may be more than you wanted to know. :-) You have ∫ f(g(x)) g'(x) dx. [So it might be something like ∫ [cos(x^2)] (2 x) dx, where f(x) = cos x and g(x) = x^2.] You do the u-substitution u = g(x), so u'(x) = g'(x): ∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx You want to say that the second integral is ∫ f(u) du -- essentially (to go back to "fractional" derivative notation), you want to do (du/dx) dx = du to get rid of the x's. Why is this justified? Let F(u) = ∫ f(u) du: F is the antiderivative of f. By definition of the antiderivative, F'(u) = f(u). So by the Chain Rule, d/dx F(u) = F'(u) u'(x) = f(u) u'(x). But since F(u) = ∫ f(u) du differentiates to f(u) u'(x), by definition F(u) is the antiderivative of f(u) u'(x). But the antiderivative of f(u) u'(x) is ∫ f(u) u'(x) dx. So putting all this stuff together, ∫ f(g(x)) g'(x) dx = ∫ f(u) u'(x) dx = F(u) = ∫ f(u) du.

captainmuon 3y ago

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 Kalkül, 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 3y ago
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 3y ago

All notations are wrong. But some of them are useful.