Nano-Passion said:
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I find that to be very intriguing... Throughout my life I've been introduced to things that are largely solvable. The fact that higher mathematics such as complex analysis can help you with finding an integral sounds interesting in itself.
Yeah, complex analysis is pretty cool, and very useful to physics. (I will say, though, with the power contained in Mathematica, you won't use the methods to evaluate integrals as much as the 60-year-old professor who taught my course.)
Also, if you ever take some analysis (it was in our second course in analysis) we discussed other ways of defining integrals, which sounds a little boring, but it allows you to integrate some functions that you couldn't even dream of with your run-of-the-mill calculus II methods.
By the way, I ended up wasting a lot of time in my calculus test (which I like to believe that I completely aced but I didn't get the results back yet). I found myself erasing everything I've done for the problem a few times because it was the very long way. Luckily I finished the exam in time. But that took out a huge chunk of my time! That scares me because in calculus II integration can often get much trickier. I can't afford the time to approach the problems in the long way.
Well, being proficient at figuring out ways to solve problems quickly can be the name of the game in math and physics (even in some of the more advanced classes, though certainly not as much.) The best thing to do is to solve a lot of problems, sure, but you should also understand
when certain methods work, and when they don't. I've found that the more practice I have, the less time I spend writing out all the menial steps explicitly. (I.e., I don't recall the last time I wrote out what my u, du, etc., etc. were in a u-substitution problem.) You see a problem, and you just have a good idea of what will work and what won't.
He is a very inspiring professor himself. Its different than other professors, you feel like you get something out of most conversations. I don't even know why he didn't pursue being a mathematician, I have to clarify that with him. But one thing that struck me as inspiring (and scary) is his story of studying 18 hours a day for a whole summer. He shared some funny stories as a side effect.
That's awesome. He sounds like a good professor to keep in contact with, as well. If you want to go to graduate school, you really should always keep your eyes open for good opportunities to build relationships with your professors.
homeomorphic said:
Well, I'm not judgmental about teaching, actually. I don't care if they are a crappy lecturer as long as they have mathematical insight. They can have the most boring monotone for all I care, as long as their math is interesting. It's mathematics that I am judgmental about, not teaching. I've done some teaching myself and I sympathize with people who have trouble with it.
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I totally understand this sentiment. I could be pretty critical of some of my professors as an undergraduate, but now that I am a TA, I understood more what it's like - and I really sympathize. (Also, with my brother who teaches high school in an under-privileged district; I tell him I don't know how he does it.)
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There was a certain vampiress who did irreparable damage, along with the vampires who wrote the textbook (Thornton and Marion--the book is despised with a passion by countless physics students, not just me, as you will see if you look at the reviews on amazon).
I'm not sure if rigor had too much to do with it, except there was this attitude that we get to prove everything from scratch. There were typically no theorems that were just quoted, and we had to use them. We proved it all.
You should check out John Taylor's "Classical Mechanics." When I was doing a lab report on chaos theory as an undergraduate, one of my friends recommended that I read his chapter on chaos; I was so blown away. I am going through the rest of his book in preparing for my qualifiers, and I am just as pleased with his treatment of other topics. It is not the most rigorous, but very conversational. When I read it, I feel like I am getting insight into how
he thinks about physics. (I also echo another poster's recommendation of the Feynman Lectures.)
Along the same lines of what you said, I think the most valuable thing you can get out of a professor is some understanding about how they, ideally a researcher, think about problems in physics (or math.) I had a professor who could present some absolutely dreadful lectures, but sometimes I would just be blown away at how differently (and insightfully) he approached some of the problems in our text than the author did.
victor.raum said:
I know exactly what you mean. I'm not personally in school, but I've sat in on a few of my girlfriend's lectures for her first year physics course. It was totally terrible, and very much like that "just giving sample problem & heartlessly listing rules" method that you described in your post.
I'm very into calc and physics. And though I'm not in school, I'm still reading through one of the very stock and standard intro physics textbooks of the day, namely "University Physics" by Young and Freedman. To give an example of how I go about studying from said book, I just read the chapter on work and kinetic energy the other week. However, I didn't do any of the problems at the end of the chapter, and I didn't even bother to read their sample problems which were provided inline in the chapter. Being good at calculus though, I did understand the math very clearly, and being a software programmer I wrote myself a little simulation which had a point-mass dropping or being thrown upward in a field of a=9.8m/s^2 Earth gravity. I had the simulation displaying the point's KE and PE values at any given moment, and I suspect that watching my simulation's KE and PE counters as it ran gave me a much better feel for how KE and PE work in practice than doing any of the book's problems ever would have.
Then I went on and just had a ton of fun fiddling with the equation, like for example I evaluated {{\rm d} \over {\rm d}t}\left({1 \over 2}mv^2\right) in the context of v being the proper length of a vector, defined as v=({\vec v} \cdot {\vec v})^{1 \over 2} The resulting derivative involved a simple chain rule application and gave a very interesting result, though I won't list it here. I did several hours of some more such playing around with the equations in the chapter, and then I moved on. All the while I still never touched a single one of the numerical engineering-style problems or examples in the book (though from the one or two I did glance at, I can't imagine they're very difficult, or for that matter interesting).
I think I'm moving through the book a lot more enjoyably using that method, and at about the same pace any college course would. Plus it's a lot more fun to boot, and what I might be "missing" by not doing the problems I think I'm making up for in other ways.
I always sort of assumed that higher level physics classes used a method more like the one I just described above, but obviously I'm not certain. I'd be curious to know myself.
To be fair, if it is a survey course aimed at a varied audience (other scientists and engineers, etc.), solving problems using physics really is the goal of the course. Many of those students would be bored with more theoretical discussions.
I do think that more advanced physics courses are similar to what you are discussing, but that obviously depends on the lecturer. In my mechanics lectures, the professor would give motivation and go about some basic derivations; in our problem sets, we would be expected to show some of the more important results and relationships ourselves. A particular problem set on non-linear physics and chaos theory was basically "have some fun exploring different properties of chaos." It has been my experience that, in the upper-level courses, there is certainly more discussion of what the deeper physics is.
