Not quite. d(1) *is* zero. The differential of a constant is zero, basically by definition. If e is an infinitesimal, 0/e is still zero. However, d^2x/dx^2 != d(dx/dx)/dx. d(dx/dx)/dx, using the new notation, is "d^2x/dx^2 - (dx/dx)(d^2x/dx^2)", which is obviously zero by inspection.

The problem with e-book math books is trying to make it look right on a small screen. If you just want a PDF of it, send me an email and I'll send you one, especially if you consider telling other people how great it is. Unfortunately, you can't just tell Amazon to take your PDF and make it an e-book :(

I've actually got a second paper on partial derivatives just about ready to go. It was originally part of this paper, but it got a little long, and I wanted to rethink and clarify a few concepts. Anyway, partial differentials have the same notational problem *plus* one more. The problem is that there are several partial differentials which all go by the same name. Once you name them properly (i.e., give them each a distinct name) the problems go away.

My coauthor has been doing this to good effect. His book "Controllability of Dynamic Systems: The Green's Function Approach" utilizes it. My role in mathematics is primarily in teaching high schoolers, so I don't spend a lot of time with differential equations. That's also the reason I *have* a co-author. I needed someone to tell me I wasn't crazy :)

Except that, in the first derivative, it *is* used as a fraction. Otherwise you couldn't reformulate your equation for integration (i.e., you have to multiply both sides by dx, which is treating it as a fraction). So, to say that in one case, it is a fraction, but this next case it isn't, but still written as a fraction, even though it *could* be written as a fraction, but we just decided not to, seems strange, at least to me.

You never did a second derivative test to determine whether you are at a local minima or maxima?

Most intro calculus books at least show the notation for the second derivative. However, it is true that they rarely take it far enough to hit any problems with the notation.

I actually figured this out while trying to find a good way to explain the notation to my students, which is a homeschool co-op class (I have a range of 9-12 graders - the 9th grader is an exception, but she is ridiculously smart). I read through numerous calculus textbooks trying to find the justification for the notation, and none of them even attempted it. So, I decided to try it out myself, and found out that the standard notation was wrong.

This is my thought as well. Interestingly, I developed this while writing a book (Calculus from the Ground Up) to use for my homeschool co-op calculus classes. I was trying to find a good way to explain the notation, and I literally had 20 calculus books that I read through trying to find a good explanation for the standard notation in any of them. None of them even attempted an explanation, just "this is the way it is, but don't treat it as a fraction." So, I tried to deduce the notation myself. That's when I realized that it was not just limited, it was actually wrong. So I wrote the paper and finished the book (it's Appendix B in the book).

It's a bit of both. Some of the facts of the matter were known, but it was assumed that this was just "the way it was". That is, no one considered it an open problem. For instance, we view the inability to divide by zero just a fact of mathematics, not a flaw. Likewise, this was not known to be a flaw, it was just assumed that this was the way things worked.

If you need to point to a definitive flaw, it was in our understanding of how it was supposed to work - the relationship between our understanding and the notation. Once *that* flaw was discovered, the actual notation just spilled right out. That is, the flaw was that people were *not* treating dy/dx *sufficiently* as a fraction, due to 19th century preferences against infinitesimals. Once you realize that dy/dx really is a fraction, and has to be treated accordingly, everything automatically works.

It's almost humorous because there was no real advanced work to do. Literally everything needed is available in intro calculus. The problem was (a) the mathematics community had a habit of *not* treating dy/dx as a fraction, and (b) new students who didn't know better were simply taught *what* to do, not *why* to do it, and continued to repeat the mistake for over a century.

Thanks! I appreciate it. Given that this was my first peer-reviewed mathematics paper, I had no idea how long the process was. I submitted the paper over a *year* ago. The necessary changes were minor. But the actual time it took to go through the process was excruciating. I'm happy to finally be on the other side :)

That's the Undertaker's Music!!!!

Yes - drinking on a flight to Vegas, then landing and continuing to drink, has led to some of the most black-out drunk periods of my life. Now I have a technical explanation for why!

In California, businesses (like restaurants) can refuse to accept money from anyone for any reason. They are not required to do business just because you have dollars or legal tender. If they don't like the fact that you are trying to pay in pennies, they don't have to accept it. Therefore, some business could legally refuse dollars but accept bitcoin if they wanted to.

It's not zero/one. The greatest electricity demand is on hot days, during the daytime. You can equate solar energy with air conditioners.