>The preliminary terror [..] can be abolished once for all by simply stating what is the meaning–in common-sense terms–of the two principal symbols:
(1) d, which merely means “a little bit of.” Thus dx means a little bit of x; or du means a little bit of u.
(2) ∫, which is merely a long S, and may be called (if you like) “the sum of.”
As someone who taught Calculus, how I wish every book on the subject started like that!
If I ever have to do it again, I will use this book. Wish I had known about it earlier.
The book includes a very important chapter on compound interest, which is too often glossed over in texts used today. I wrote notes to remedy that (as an extra-credit reading project for the students), and was glad to find that the book has a similar approach:
Indeed. They never explained this at school and we were just memorizing and applying formulae having no idea of the meaning. So many years have passed an I've only realized what does the d actually mean some weeks ago and now I see this book explaining it this easy at the very beginning written in 1910!
I've seen a similar effect in various other topics too, and propose the saying "the closer a subject is to its infancy, the clearer it will be taught". Two examples of this come to mind. The first is in the automotive industry, with early videos such as https://news.ycombinator.com/item?id=15122031 (and older discussion at https://news.ycombinator.com/item?id=8513209 ) as well as the detailed yet straightforward explanations in the service manuals of the time. The second example is with early computers; I remember an engineering text from the late 50s/early 60s that managed to include a surprisingly lucid chapter on assembly language programming, essentially showing how to write programs to solve numerical problems. Later on, microprocessor and home computer user manuals would also contain such information.
First, why "d"? Well, "d" is for "difference". As in: as x changes from x_1 to x_2, the difference (x_2 - x_1) -- when it's very small.
But wait, there's more!
The commonly used symbols for finite difference like that is the Greek letter Delta: Δ
For a list of values x_1, x_2, x_3, x_4,.. we write Δx_i = (x_i - x_{i-1}). That is, Δx_i is the i'th change. (Side note: an airline had a marketing slogan Change is Delta, which some nerd must have been immensely proud of).
Ok, bear with me for a bit more!
The symbol we use for finite sums is Σ: we write
Σy_i = y_1 + y_2 + ... + y_n
That is, summing up small succesive changes gives you total change. Simple?
Now apply this to the situation where the small changes in the quantity you are looking at are proportional to changes in another:
Δy_i = Δx_i * f(x_i)
Say, y is position, x is time; then f(x_i) is the speed at time x_i: as time increases a little, so does your position; the ratio of the changes is the speed. Δy_i is how much you moved from time x_{i-1} to time x_i, which is proportional to change in time
Δx_i.
Note that f(x_i) = Δy_i/Δx_i here (speed = change in position div. by change in time).
Now write:
y_n - y_0 = ΣΔy_i = Σf(x_i)Δx_i
Again, just summing up small changes to get the net chnage.
NOW, what you've been waiting for!
Imagine you took infinitely many measurements. The changes become infinitely small, and the sum becomes of infinitely many things.
We need new notation for this.
But let's keep it similar. Instead of using Greek letters, let's use the same letters... in Latin.
Δx becomes dx
ΣΔy becomes S dy
And, with some sloppy handwriting of the letter S, the net change equation becomes:
y_final - y_initial = ∫ f(x)dx
where
f(x) = dy/dx.
You now see that Σ and the sloppy S -- ∫ -- stand for Sum.
And that, my friend, is pretty much all there is to Calculus and its symbols, fundamentally[1].
Do you realize this comment of yours is worth more than years of studying (high school + Bc.)? Perhaps schools are better at your location but the ones I've attended never explained this, just write that down, this equals that, memorize and f-ck you. Thanks G-d we have the Internet and people like you nowadays...
I think that Knuth's concrete mathematics made this connection at some point - the difference between the infinitely small abstraction of an integral and a big-sigma sum as being discreet quantities that can be measured by actual numbers that measure a thing.
I remember when I had calculus in school and the textbook was a huge book weighing like five pounds...and my mom's calculus textbook from the 1950s was this tiny little thin book. But they hadn't added anything to the subject of calculus since then!
Too true! I took Calculus in College, but I read this 2 months ago and loved it! Even better is it thought me something new - that is infinitesimal calculus.
How I wish there were books like this written for more topics!
>I To deliver you from the Preliminary Terrors
>The preliminary terror [..] can be abolished once for all by simply stating what is the meaning–in common-sense terms–of the two principal symbols:
(1) d, which merely means “a little bit of.” Thus dx means a little bit of x; or du means a little bit of u.
(2) ∫, which is merely a long S, and may be called (if you like) “the sum of.”
As someone who taught Calculus, how I wish every book on the subject started like that!
If I ever have to do it again, I will use this book. Wish I had known about it earlier.
The book includes a very important chapter on compound interest, which is too often glossed over in texts used today. I wrote notes to remedy that (as an extra-credit reading project for the students), and was glad to find that the book has a similar approach:
http://romankogan.net/math/A_paper_of_interest/A_Paper_of_In...