Great if this works for you but honestly I don't find this "easy" at all. The writing style is annoyingly formal and already on page 2 it jumps into "draw the rest of the fucking owl" territory:
>> A very simple example will serve as illustration.
>> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.
I'm of the opinion that there's a reason why a subset (myself included) of people who when initially exposed to infinitesimals, and specifically the part where you start just disregarding terms, reject them (it's one of the oldest arguments related to calculus! [0]). Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor hurts those arguments until you have a rigorous justification for them (that didn't appear until the 1960's if my memory is right).
I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.)
> Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor (...)
Yes it's equivalent to limits, but limits are a very cumbersome machinery, specially if you use the epsilon delta definition (there exists .. such that all ..).
But note that I just linked you a PDF that does fully 100% rigorous calculus using only infinitesimals with no limits. Yhey aren't disregarding small terms willy nilly (like it was done in the early history of calculus)
The only catch about SIA is that it requires you to use intuitionistic logic rather than classical logic in your mathematical arguments (which I admit is a barrier, but it also buys you some things). And what it offers is much simpler proofs that support intuitive reasoning.
There is also this book, "A Primer of Infinitesimal Analysis" [0], which develops a big chunk of calculus and classical mechanics using only infinitesimals, and is fully rigorous.
I wonder if it's working with floating point numbers that made me less uncomfortable when first discovering infinitesimals. The idea that something just falls out of our current representable scope under certain operations seemed fine to me. I've always had a soft spot for infinitesimals and a slight dislike for epsilon-delta limits.
Looks like something went wrong with your cut-and-paste:
>Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x²+2x⋅dx+(dx)². The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x².
I always enjoyed calculus, I thought it was kind of magical and didn't worry too much about where the magic came from. I've known the derivative of x^2 is 2x for nearly 50 years but just found out why due to your formatting correction here. Thanks! Also gratitude for the poster who explained that the fundamental theorem of calculus (i.e. reverse differentiation is integration) is essentially just the calculation involved in going from an odometer reading to a speedometer reading then back again!
IMO "made easy" would involve connecting everything single concept in calculus immediately to the whole reason it exists - physics.
I made the mistake of taking algebra-based physics, then calculus, and only after the calculus course did I realize how much harder I made my life by not starting with calculus (and learning it as the mathematical language of physics).
I get the feeling that if someone understands how to square x + dx and can also follow the similar triangle argument on the next page, they'll be totally fine studying calculus starting with limits instead of this.
I find the geometric proof of the product rule using differentials much more intuitive than the difference quotient proof. The difference quotient proof is a clever algebraic trick, but it doesn't (at least for me) give any deep insight about why the product rule works.
You make a very reasonable point but I'm sure you would have been rather more accommodating if you'd known (via an acknowledgement) that the original author wrote exactly these words in 1910 in a style which was undoubtedly a lot more readable and welcoming for kids learning calculus than other texts available at the time.
>> A very simple example will serve as illustration.
>> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.