> While we're at it, it's probably not the best idea to represent derivatives as fractions either (for the sake of notational consistency).
Do you mean dy/dx? Why isn't that a good idea? Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx? What are the alternative you'd prefer: prime notation, D-notation, or else?
I don't know about the person you're responding to, but those operations over what usually appears at the denominator of the derivative purely for notation purpose has always looked to me as complete magical garbage. What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ?
I think this notation is single handly the reason why i've never been comfortable with calculus.
PS: i've stumbled a few years ago on a math book that described the original concept of "infinitesimals" and how a whole different way of doing calculus exists. And it seems to me those kinds of computations over "dx" come from there. But the end result of mixing concepts really looks like trash.
> 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.
>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?
For teaching purposes you are definitely allowed to lie, as long as that lie can be resolved eventually (not necessarily in this semester ;-). That's how we have been generally taught about integer divisions and negative square roots. But behind the scene, the `dx` notation can be fully generalized and made rigorous with differential forms, or that was what I have been told.
This is definitely not an apples to apples comparison. Integer division is something everybody is expected to learn. Also we don't teach imaginary numbers to middle schoolers as soon as they learn about square roots.
To some extent we have to speak to our audience. I consider that part of effective communication. I don't think "assume the person you're speaking to is/will be a mathematician" is an effective way to interact.
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.
dx can be a differential form, but in (elementary) calculus books it's exposited this way: Suppose you have a function y = f(x) taking real numbers to real numbers, so x is an independent variable, y is the dependent variable. You define an independent variable Δx and define dx = Δx. Also define dependent variables dy and Δy by
Δy = f(x + Δx) - f(x) and dy = f'(x) dx.
Then f'(x) = dy/dx. This may look like a stupid hack to make the last formula work, but actually it's a little more. If you use nonstandard analysis, you define the derivative of a function f from reals to reals by
f'(a) = st( (f(a + Δx) - f(a)) / Δx )
where st takes the standard part of a hyperreal number and Δx is a nonzero infinitesimal. This is like the usual limit definition, without limits. Then you can use the formulas above and "dy" and "dx" are numbers, albeit hyperreal numbers.
(The "dx as a differential form" vs. "dx as a number" is probably coming from the fact that the tangent space to the reals at a real number is isomorphic to the reals, so the dual space [where dx lives] is too.)
(Calculus via infinitesimals is pretty cool; a good resource for this is H. Jerome Keisler's "Elementary Calculus" and "Foundations of Infinitesimal Calculus", both available for free: https://people.math.wisc.edu/~hkeisler/)
I second the recommendation for Barrett O'Neill's book - I used it in my differential geometry class at MIT.
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.
> 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.
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.
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.
Well, I don't know much about the way calculus is taught in the West, but I remember that Zorich's Mathematical Analysis (ch. 5 Differential Calculus and ch. 8 The Differential Calculus of Functions of Several Variables) was pretty clear about everything.
I’ve found the book online and will definitrly look into it. It’s going to be my first time with russian math teaching style, i’m really looking forward to it. From what i’ve seen browsing the first chapter it seems very down to earth and straightforward, i really like it. Also, it covers exactly the scope of calculus i want to get better at, so thanks again.
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.
> 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.
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.
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?