The linked document below contains an integration 'trick' that's very interesting to me, but it arguably shouldn't be. First, the interesting bit. The problem shown is from a derivation of the Fourier transform of the ground state of the hydrogen atom from position space into momentum space The trick here shows how to easily integrate theta dependency using simple 'u substitution'. There' a second possibly much more interesting bit that I have to check out later, we wind up with sin(kr)/kr which is a prototype for a delta function, which is related to the whole Fourier transform process, but I digress.
While I'm very proud of my u substitution, I arguably shouldn't be. I'll out myself. I'm in grad school and I'm not an expert at integrals! The shame! After all, I learned the technique of u substitution in freshman calculus.
Here's a little bit of history followed by a suggestion I'd appreciate your thoughts on. When I took freshman calc, it was a less than stunning experience, (except for the quarter I had Dr. Davis, thanks for all the pizza Dr. Davis!). Even worse, we didn't really use hard core integrals in physics. Consequently, two sleep deprived years later when I needed the hard core integral skills, they were gone. Now I'm in grad school with (hopefully) the full understanding of the horrible, horrible thing I did by not memorizing every integral technique, which brings us to physics home rooms.
Would our undergrad experience be more cohesive and useful if we had daily updates on what's actually important and why it's important? Would the equivalent of what I always imagined, (we didn't have them in Ruidoso), a high school home room to be work? A simple half hour meeting every morning where the importance of the previous days classes was revealed. My first impression was that the class shouldn't be graded, but perhaps grades are exactly what is needed. In addition to emphasizing that integration by parts would be the favorite trick of each professor in three years, or that the silly top problem would ultimately lead to the precession of an electron in its orbit what if there were quizzes to make sure the material was sticking?
Those are all the ideas I have for now, what do you think? Does this affliction of 'lost knowledge' hit enough students to warrant an effort to fix it? Would a home room kind of activity/environment help? How could it be done better?
Picture of the Day: