but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.
The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that
Integral over the boundary = Integral over the interior of the divergence, or more poetically
Int_(dA)A = Int_A dA
Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.
So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus
f(b) = Int(df) + f(a).
The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).
I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.
I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov:
I've been a high school math teacher for most of my life, and I have deep frustrations with how removed from meaning math is presented to most students. Just because the teacher knows and states the possible relevance doesn't mean students should be expected to take the relevance at face value.
I was mostly focused on teaching algebra 1 classes, which is why I didn't use higher math all that often. But my understanding of higher math grounded my teaching of lower level concepts all the time, and I often spoke of higher level concepts with my students to help demystify math. My 8yo son loves math for now, and the moment school makes math meaningless to him I am planning to find some way to intervene.
a) You don’t do this full time.
b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.
c) You are skilled with programming/software in general.
In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.
Another thing you have going for yourself is - you can basically skip high school math and jump
right in for the good stuff.
I’d recommend getting great and cheap russian recap of mathematics up to 60s  and a modern coverage of the field in relatively light essay form .
Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.
Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.
For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.
I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.