Almost thirty years ago, I switched from the dream of aerospace engineering to lawyering, because integral calculus broke my brain.  I failed it twice.

Oddly, enough I still geek out over math news that hit my feed; New algorithms that speed up calculations, centuries old math puzzles that get solved, etc.  Unfortunately, I don't always grok everything that I'm reading.

My question is could you recommend a book or books for self-teaching or a better understanding of "advanced" mathematics?  Something akin to Hawking's "A Brief History of Time," or a general survey of advanced mathematics?

Hawking's "God Invented The Integers" is pretty good for what it is.

Thank you.

All righty, that was worth rummaging through my library.

The World of Mathematics, Newman.
The Mathematical Experience, Davis, Reuben.
Mathematics:  The Loss of Certainty, Kline.

I've guessed at ranking them for you.

Here's a few more.  The first is a text book but should still be accessible.  The harder parts of second gave me headaches.  The rest are lighter reads but still very nerdy.

Galois Theory, Ian Stewart
Gamma, Julian Havil
The Equation That Couldn't Be Solved, Mario Livio
An Imaginary Tale: The Story of sqrt(-1), Paul Nahin
E: The Story of a Number, Eli Maor

And then there are the 'tweaks' that engineers use to produce useful answers.

How do you establish a failure rate when you cannot create a failure?

Dividing by zero makes strictly formal statistics fail.

We assign a failure to the test and the next time increment.

We than note that this allows the calculation of a bound on the failure rate.

"The MTBF (Mean Time Between Failures) will be better than (a huge number of hours) based on testing millions of units for millions of hours.

One of the toughest problems in engineering reliability is accelerating low activation energy failures.

There are simply some failure mechanism,s that cannot be 'increased' in a reliable manner.

Some simple things like the failure rate of solder joints from repeated temperature cycles.

Colder or hotter (wider temp swing) does not reliably increase the rate.
Unless it is so large that it 'works' the solder joint excessively.
But that is NOT the failure mechanism being investigated.

It is the more 'normal' range we are interested in.

There are numerous satellites that orbit the earth once every 24 hours.
Often in polar orbits so they can pass over every spot on the earth eventually.

Heating and cooling them normally takes pretty much one hour in a test chamber.
Just like they undergo every orbit.

So in 24 hours I can test 24 cycles of temperature stress.
And that is what the satellite is subjected to every day.
One cycle every hour.

If textbooks are what you seek here is a free one I use for calc 2. Same website has a calc 1 book as well.

https://openstax.org/details/books/calculus-volume-2