Calculus Made Easy (1910)

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One thing that seems to be missing (and wasn't given a lot of attention when I was in school) was the notion that the reals are continuous. Calculus made a lot more sense to me once I internalized how that one simple idea basically serves as the rug that really ties the room together.

I've often thought that an interesting treatment would start with differences and sums of integers as approximations, demonstrate their errors and then introduce reals and limits as a tool for making better theoretical models using the infinite "zoom button" continuity property of the reals.

Hey, there's more!

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

Why Σ? That will have to wait a little.

Sums and differences cancel out:

ΣΔy_i = (y_1 - y_0) + (y_2 - y_1) + (y_3 - y_2) + ... + (y_n - y_{n-1}) = y_n - y_0

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].

[1]https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_calcu...

Thank you for taking the time to do this. It's one of my favorite math books, and it's always been hard to read as scanned PDFs.