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On Leibniz Notation

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Re: On Leibniz Notation

#81

Post by bikenaga »

fiforpg wrote: post #20

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) +…

Leibniz and the product/quotient rules: "Although at the outset Leibniz was uncertain about his method and hesitated as to whether or not d(x y) is the same as dx dy and whether d(x/y) is equal to dx/dy, he in the end answered these questions correctly, determining that d(x y) = x dy + y dx and d(x/y) = (y dx - x dy)/y^2. These values he found by allowing x and y to become x + dx and y + dy respectively. Upon subtracting the original value of the function from the new one and observing that dx dy is infinitely small in comparison with the terms x dy and y dx, the results are obtained." [1]

Newton's algebraic vs. geometric approaches to calculus: Boyer [1] notes that in Principia (1687) "Newton presented them [his propositions] in the form of synthetic geometrical demonstrations with an almost complete lack of analytical calculations" - hence the geometrical approach you point out. But he actually wrote up accounts of his calculus in three papers (De analysi ... (1669), Methodus fluxionum ... (1671), De quadratura ... (1676)), all of which were published after 1700. In those earlier papers, his approach is algebraic, often using the binomial theorem. In one incredible result from De analysi, he shows that the area under y = a x^(m/n) is given by z = (m/(m + n)) a x^((m + n)/n). To do this, he increments z and x:

  z + o y = (m/(m + n)) a (x + o)^((m + n)/n)
Expand the right side using the binomial theorem, subtract the z-expression from both sides, divide both sides by o, then drop any terms on the right containing o ... and you get y = a x^(m/n). That's pretty algebraic, right?

But the incredible thing is that he's basically showing that (roughly) "the rate of change of the area under the y-curve is y", which is the Fundamental Theorem of Calculus! He's figured out that derivatives (rates of change) and integrals (areas under curves) are somehow inverses (at least in this case). Newton was pretty amazing!

[1] Carl Boyer, "The History of the Calculus and Its Conceptual Development". Dover Publications, 1949.

Re: On Leibniz Notation

#82

Post by bikenaga »

impossiblefork wrote: post #62
fiforpg wrote: post #20

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) +…

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.

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.

Re: On Leibniz Notation

#83

Post by bikenaga »

BruceEel wrote: post #5

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.

Yeah, I noticed a thing students in math class would get confused is the difference between local and global variables. So I'd ask people if we're discussing functions from the reals to the reals, are these the same function?

  f(x) = x^2   vs.   f(t) = t^2
They are, because they produce the same set of ordered pairs (input, output).

Some of the confusion is the fault of math profs, because we get lazy. We could say "the function f: R -> R defined by f(x) = x^2", but we wind up saying "the function f(x)" (wrong - f is the function, f(x) is the value when x is plugged in) or worse "the function x^2" (really wrong). But if you always say the absolutely correct thing "the function f: R -> R defined by f(x) = x^2" you sound really pedantic, and the extra words make it hard for students to comprehend. Oh well.

What was more confusing to people was if you had f(t) = t^2 and people thought "t" had some sort of independent existence outside the definition of f. Some guy was asking me about it and I was trying various explanations and he wasn't getting it. Then I remembered he was a CS major and I said "t is an argument to f and it's local to the function block" and he got it and nodded his head.

So I wonder if these kinds of distinctions are easier for programmers than for typical math students - in the sense that programmers would know there's no difference between these:

  int f(int x)
  {
     return x * x;
  }

  int f(int t)
  {
     return t * t;
  }

Re: On Leibniz Notation

#84

Post by bikenaga »

cygx wrote: post #36

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 identif…

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.

Re: On Leibniz Notation

#85

Post by bikenaga »

shunyaekam wrote: post #9
volemo wrote: post #3

Earlier quoted context omitted.

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

The answer to this question is answered in the FA. Leibiz notation is confusing except in the most simple cases. For example: > Isn't fractional form useful, for example solving differential equation y = dy/dx => dy/y = dx? What exactly does "dy/y = dx" mean? What is on the LHS and what is on the RHS? It acts like a mnemonic scribble for an intermediate step. It doesn't have any mathematical meaning.

> 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.

Re: On Leibniz Notation

#86

Post by nyssos »

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

Earlier quoted context omitted.

Thanks for the term, but it still doesn't help me understand what i'm allowed to do with it. Vectors is a great example: as soon as you're introduced to vectors, you immediately starts to be given definitions on how to multiply / add them together and with regular numbers. dx remained a mystery even during my first 2 years of calculus in university. I used them purely as a notation tool, but really didn't understand…

Well, I don't know much about the way calculus is taught in the West, but I remember that Zorich's Mathematical Analysis (ch. 5 Differential Calculus and ch. 8 The Differential Calculus of Functions of Several Variables) was pretty clear about everything.

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.

Re: On Leibniz Notation

#87

Post by pokerface_86 »

xamuel wrote: post #50

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 i…

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.

Re: On Leibniz Notation

#88

Post by BruceEel »

bikenaga wrote: post #83
BruceEel wrote: post #5

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.

Yeah, I noticed a thing students in math class would get confused is the difference between local and global variables. So I'd ask people if we're discussing functions from the reals to the reals, are these the same function? f(x) = x^2 vs. f(t) = t^2 They are, because they produce the same set of ordered pairs (input, output). Some of the confusion is the fault of math profs, because we get lazy. We could say "the f…

Brilliant. Love the programming analogy. I like that the above works particularly well in C as:

   int f(int x);
   int f(int t);
will compile with no errors, while changing t to, f.i., "float" obviously won't. I wonder whether this could be pushed farther to come up with programmer-friendly descriptions of things like the chain rule starting from something like:

   int h(int x) {
     return f(g(x));
   }
etc.

Re: On Leibniz Notation

#89

Post by impossiblefork »

bikenaga wrote: post #82
impossiblefork wrote: post #62

Earlier quoted context omitted.

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.

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…

The problem is that d/dx(dy/dx) is not d^2 y/(dx)^2.

d/dx (dy/dx) is really d(dy/dx)/dx, which is ((d^2 y)/dx + dy d(1/dx))/dx = d^2 y/dx - dy/dx (d^2 x)/(dx)^2, where df is the actual differential.

Re: On Leibniz Notation

#90

Post by [deleted] »

laingc wrote: post #69
zero-sharp wrote: post #32

Earlier quoted context omitted.

>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+...=-1/1…

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.

[deleted]
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