Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed.
Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and those who had memorized them were more likely to solve the problem than someone like me who happened to forget the theorem existed in some book and is trying to rederive everything from scratch.
Of course, those who mostly or only memorize perform the worst.
Basically, most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform. So some level of "take it on faith and memorize a few items" are needed. There are other examples of this.
Also, having studied under many top class physicists, I can tell you that a significant number of them cannot derive much of the mathematics they use, despite being wizards in applying the techniques.
1. none of the exams at Caltech required memorization. They were open book and open note. Memorizing simply wouldn't have helped. For example, one physics exam question was: "Assume magnetic monopoles exist. Derive how Maxwell's Equations would then look." If you didn't understand the ME derivation, you'd be completely lost. The same for FFTs, where the exam question was derive the hyberbolic transforms.
2. one winds up inadvertently memorizing things used often, like I knew all the trig identities from excessive use. I never attempted to memorize any of them. Just like I know a lot of the hex opcodes for the x86 :-)
3. we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.
4. I've forgotten an awful lot in the 40 years since. But I took an online MIT course and was pleasantly surprised that it was still there, it just needed a bit of oiling.
> most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform
Which is why some engineering schools are better than others. I applied to
the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.
I'm sorry you were left behind because of your insistence on learning it thoroughly. You were doing it right. You just were in a school that didn't value doing it right.
BTW, I did not mean deriving a formula from scratch every time you used it. That would be silly. Just that at some point you did, and thereby understood where it came from, hence understood its limitations, and knew how to adapt it to a situation not in the book.
Much of your experience was Caltech specific, and will simply not translate to other schools. Caltech is famous for this - more so than schools like MIT, etc.
1. I learned the hard way, as did others, that one should still memorize with open book exams (or at the least make a 1-2 page cheat sheet). Why? Because there was a time limit and most professors would not alot enough time for people to even look up everything they needed in the textbook. Sure, if they increased the time by 50-100%, you'd do just fine with memorizing.
I'm not even hypothesizing there. After one open book exam I went and asked everyone who got a good score - most had incorporated some level of memorizing. And clearly most who did not memorize at all got a poor score (which is less surprising than it should be, because most students will do poorly regardless ;-) Still, I was the clear exemplar of one who improved from "below average" to "one of the top students" within the duration of one semester when I finally embraced that some level of memorization would be needed.
2. True, and I can relate to trig identities - most of which I remember 25 years later - and even after over a decade of not needing them. However, when I got to more advanced topics in math, the frequency with which I would need to use them dropped significantly, and the approach of "Just do lots of problem sets and you'll passively memorize" failed me in grad school.
> we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.
Did they teach you measure theory before those transforms? Did they teach you measure theory before probability? Did they teach you the theory of distributions before the Dirac-Delta function?
> Which is why some engineering schools are better than others. I applied to the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.
I went to a top 5 engineering school. Can assure you Caltech's approach is not the norm.
I am sadly well aware of that. Caltech was known as unique at the time, I wonder what it is like 40 years later.
1. It was rarely necessary to look anything up. What was on the exam was reliably in the notes or in the assigned textbook. I don't recall ever memorizing things, and managed an A- average. My innate abilities were completely average there, and we all knew who the really smart ones were, like Hal Finney. What I did do to prepare, however, was ensure I attended every lecture and took comprehensive notes, make sure I could solve every homework problem, and every midterm problem (in trolling for the final). I did not look at prior year's stuff.
2. I didn't attend grad school, so can't comment there. But I did match wits with Masters engineers at Boeing, and would wind up fixing their work, too, though far more rarely. That group eventually offered me a position, though they had a Masters as a requirement.
Measure theory wasn't taught, at least in the undergrad courses I took. Neither was the theory of distributions.
> Can assure you Caltech's approach is not the norm.
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