When it comes to mathematical concepts- to understand everything broadly, connected to its context, and to understant "what's the point?", nothing tops this book- Mathematics: Its Content, Methods and Meaning (3 Volumes in One) by Aleksandrov, Kolmogorov, and Lavrentev [0].

[0]: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

I haven't quite reached the "mathematical maturity" but what you describe seems like something I have be after too.

I think you need to have a look at Kolmogorov's Mathematics: Its Content, Methods and Meaning [0]

You can also get it in print [1]

[0]: https://archive.org/details/MathematicsItsContentsMethodsAnd...

[1]: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

Book is a collaborative effort of many mathematicians from different soviet regions

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Grad...

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: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

I studied physics in undergrad many years ago, and it's been a long time since I used higher level math on a regular basis. I just picked up Mathematics: Its Content, Methods, and Meaning, on the recommendation of someone here. It's over 1000 pages, so it's going to be a lifetime reading project for me, but it's been wonderful to start reading. The first part of the book traces the earliest origins of math, and everything was grounded in real-world physical problems.

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.

https://www.amazon.com/gp/product/0486409163/

"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov ,M. A. Lavrent'ev. (3 Volumes)

This is a classic and exactly what you are seeking for. I think it was originally published in 1962.

https://www.goodreads.com/book/show/405880.Mathematics

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

This book is fantastic and pretty much takes you through an entire undergrad mathematics course: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...