JohnnyB and I have been discussion Liebnitz notation in calculus. JohnnyB teaches calculus and is writing a textbook. When I learned elementary calculus, the treatment of the topic was awful, and it haunted me even in physics grad school. Amazingly advance calculus and real analysis classes didn't use Liebnitz notation at all (at least I don't recall them doing so), except maybe "dx" to indicate the variable of integration. The issue is not just the formalism, but the justification for the symbols, the conventions for manipulating them, and most important to us, the teaching of the ideas.
One would think a topic that has been around for over a hundred years would not need re-examination, but I found out the hard way, that isn't so. Just look at the mess of how Entropy is defined and taught. See this discussion for example:
In Slight Defense of Granville Sewell: A. Lehninger, Larry Moran, L. Boltzmann (http://theskepticalzone.com/wp/in-slight-defense-of-granville-sewell-a-lehninger-larry-moran-l-boltzmann/)
In the first comment (http://theskepticalzone.com/wp/in-slight-defense-of-granville-sewell-a-lehninger-larry-moran-l-boltzmann/comment-page-1/#comment-144684), textbook author Larry Moran admits to giving an imprecise definition of Entropy!!!! :o
October 4, 2016 at 9:45 pm
Textbook authors have been discussing this issue for decades. Most of us understand that there's a statistical mechanistic definition of entropy and we understand that the simple metaphor of disorder/randomness is not perfect.
However, we have all decided that it is not our place to teach thermodynamics in a biochemistry textbook. We have also decided that the simple metaphor will suffice for getting across the basic concepts we want to teach.
I've talked about this with Cox and Nelson (the authors of the Lehninger textbook) and with Don and Judy Voet (the authors of another popular textbook). Sal may not like our decision but it was a very deliberate decision to dumb down a complex subject that doesn't belong in biochemistry textbooks. It's not because we don't know any better.
There are a dozen such issues in my textbook ... cases where difficult terms are deliberately over-simplified. On the other hand, there are some terms in my book that I refused to simplify even if other authors did so.
Sooo, as far as calculus goes and the teach of it, I think it's fair game to revisit the traditional teaching methods. As someone who has learning disabilities, I had to teach myself ways of learning hard material more easily. I hope to share that in some way, hence the motivation for this discussion. Whether that bear fruit on the topic of Liebnitz notation remains to be seen.
So first off, something from the net. They call F(x) the indefinite integral, but I think a better term is Anti-Derivative. In any case, let me post it as I'll build on it. One can see already, a freshman calculus student could get intimidated by the symbology. I'll start with this, and then do something irreverent to drive home a point.
Building on that previous graph, let me point out an that an integrand = 1 can become implicit and not even explicitly shown.
So far so good. Now the irreverent part. When I was taking probably the 2nd most brutal class of my life in graduate school, namely mathematical physics, when the professor, whom we affectionately referred to as "Dr. Phil" (Phillips was his first name), took us all a step back for a minute to try to show the goal of solving one particular class of differential equations. Dr. Phil said, every graduate student should be glad when he can get one side of a differential equation to look a certain way because (and he really did use smiley faces on the black board!):
What Dr. Phil was doing was trying to remedy typical conceptual errors of some of his grad students made that are (like so many math mistakes) rather trivial. Another professor pointed out, to do the math of physics with 100% accuracy. What helps is when you can solve trivial problems or decompose complex problems into something trivial. So then with Dr. Phil's little irreverent display of a basic idea in calculus, I began to understand liebnitz notation (mind you, I had already a math degree and advance calculus, but Dr. Phil's little illustration helped me understand the motivation for the notation). Thus one could hypothetically relate abstract derivatives in liebnitz notation like:
I mean, hypothetically, suppose the above equality was an intermediate result to solving a differential equation. One could find at least one solution right in one variable? So let me do the first part:
What Dr. Phil was helping us do is to compartmentalize and properly abstract away certain details. So the irreverent exercise helped me grasp Liebnitz notation better. Something that I really never got in all the formal training I had to that point. So I make this irreverent digression before going into a more formal treatment.
JohnnyB the following is along the lines of our e-mails and the problems we were discussing. Consider.
Just to make the example easier let f(x,y) = 1. Then
Now by way of extension, suppose, y was just a dummy variable for x.
Just to make the example easier let f(x,x) = 1. Then
Where all this was going is addressing the Liebnitz notation. First the operator, not the derivate itself:
now dx^2 is a bit sloppy because what is squared is "dx" not "x", but oh well, that's the convention!
Now I will apply the operator to the varialble y = f(x):
The obvious notational confusion is that the operator on y and the 2nd derivative look practically the same! So what does d^2y mean? It is an infinitesimal change of change:
Now where all this is going is to justify the notational convention (this is not really a proof, just a justification of convention):
this is obviously the sticking point:
But, since this is really an argument about convention, I'm only going to try to justify the convention, but not make a formal proof by going back to that trivial "differential" equation:
multiplying both sides by dx^2:
and then expanding
Now one might complain about the meaning of "d (dy)", but if I may be irreverent to interpret the meaning:
Thus double integrating both sides of:
Now, I'm only looking for one solution, not a general solution, so when I do the integrations, I'm going to let the constant of integration C = 0 to make things simpler.
So in light of the idea "(dy)" = "Smiley face":
which I'll carry out without the constants of integration C = 0 just to find a simple solution:
which reduces further to:
double checking that this is one of an infinite number of possible solutions of the "differential" equation:
So going back to the sticking point, is this true, or is the notation at least justified?
In light of the fact we can excute the double integration in any order, it seems to me, we can execulte the differentiation in any order. To make the point a little clearer (I hope anyway):
thus it seems reasonable (though I haven't worked it out more formally) as illustrated in the double integral example where the order integration is interchangeable:
again this would be (if true) an idealization of:
recall I showed one can double integrate this:
which is an approximation of:
There is probably a formal way to prove this relation as all the delta's go to zero. The problem is that in elementary calculus books, this proof is omitted! For all the proof-based calculus book out there, this was one case where the proof would have been helpful, if only in the appendix or somewhere, rather than just letting the students remain in a state of confusion:
So trying to deal with:
The right hand side of the equation relates to this. One can see the h^2 correspondes to dx^2 (I guess):
The left hand side (I think) relates to:
I think coverting the "h" to "delta x", plus some clean up, will get the desired result to justify the Liebnitz notation.
It would have been helpful if they showed the connection in calculus text rather than just throwing Liebnitz notation in with not a lot of justification.
I realized I went about things the hard way. Cleaning up a bit, and taking out the "z" and replacing with "x".
this relation has to just be recast to delta-X's and Y's, I think.
Let me recast:
If I define:
which justifies the notation
Yes, it is a terrible tragedy that this never gets explained in textbooks! What is even more of a tragedy is that, I believe, the notation itself is incorrect. This is detectable if one actually takes
seriously as a differential.
To perform this operation, you would actually need to use the quotient rule. Doing so leads to a more expanded form of the second derivative:
Now, this reduces to
in the case of x being an independent variable. But, when x is not an independent variable, then the full expansion is needed.
If you have the full expansion, you can easily convert the second derivative of y with respect to x into the second derivative of x with respect to y using algebraic manipulations only, which is not possible with the traditional notation.
I don't know if there is a LaTeX addon for Elkarte. I'll have a look. ETA seems not yet.
Just to jump back in the topic - my paper on an expanded second derivative notation got accepted into the arXiv! http://arxiv.org/abs/1801.09553