I think you need to have a look at Kolmogorov's Mathematics: Its Content, Methods and Meaning 
You can also get it in print 
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:
This is a classic and exactly what you are seeking for. I think it was originally published in 1962.
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.
Just from a search, there are some great results:
1. Mathematics for Computer Science - HN Submissions: https://news.ycombinator.com/item?id=9311752, https://news.ycombinator.com/item?id=3694448
2. How to Read Mathematics - HN Submissions: https://news.ycombinator.com/item?id=4030812, https://news.ycombinator.com/item?id=1576969
Here are some other excellent mathematics books:
1. Mathematics: Its Content, Methods and Meaning
Containing the thoughts and direction of numerous mathematicians including Kolmogorov, this is a great survey of the field of mathematics. It touches upon Analysis, Analytic Geometry, Probability, Linear Algebra, Topology, and more. [1.]
2. Concrete Mathematics: A Foundation for Computer Science
Containing the thoughts and direction of mathematician and computer scientists such as Donald Knuth, this is a great reference for computer science related mathematical concepts focusing on continuous and discrete concepts. [2.]
One of the masterpieces that has gone the opposite direction is:
Mathematics: Its Content, Methods and Meaning (three volumes bound as one) by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev (18 authors total)
This book is really good companion for autodidacts. It's basically overview of mathematics.
Covers something like three years of an undergraduate degree in mathematics. Lots of words - but that text is used to develop an understanding of the concepts and images. Considered a masterpiece. An enjoyable read.
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.
Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come even close to this. Reading this book together with the The Princeton Companion to Mathematics was real treat.
The Russian school put out some pretty massive volumes, like
but I found those to be a little too caught up in the proposition/proof cycle to be useful as a guide to the uninitiated.
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