Most of us who know it learned it in college, and have tried hard to forget it It's the understanding of it that's important, not so much doing actual integrals and remembering math rules.
I got an A in my honors differential equations course in college, but that was ten years ago and I hardly use it as an electronics design engineer.
I'd argue that it's far more important to know how to use the tools you have than to understand how they work (from the point of view of getting something done).
I don't agree, if you understand how the tools work (and if your mind is sharp ), then you can think through possible sequences and determine what would be the best way to solve your problem. If you just know how to use the tools but don't know why, you'll only go so far and then be stuck. Then you'll have to go all the way back again and try and understand it. Say for example, Ohms Law. If I didn't understand the relationships between voltage, current, and resistance, I could be in big trouble. What If I wanted to power an LED and I didn't know about current? I could say well R=E/I and I'll give this LED 1A of current (because I don't understand what is too much) and the LED will self destruct. Or, what If I would like to give someone a little zap of current and I give them 2A of current? Since, I didn't understand that could kill the other person!
Yes, the vast majority of it isn't ever needed in real life, it's just something they make you learn at college.
That clearly falls under knowing how to use the tool, not understanding how it works (aka where it comes from). That would involve all the conduction and valence band stuff. Similar to knowing how to use matrix reduction rules, but not knowing where they come from.
With many mathematical formulas we have today, you have to understand the concepts so that you apply the correct one. You have to understand why your doing that and how you will benefit from that. I like to know why I'm applying a formula, wouldn't you?
Yes, in theory. But then again you (and me) would both probably like to know everything.
There's a difference between knowing the math, and knowing the theory behind the math.
Well, I would say that knowing the theory is part of math.
In the calculus class we were trying to figuring out how to derive everything, but not necessarily how to use it. Once we derived the result, we stopped.