If we have an integral, say $\int_{-\infty}^{\infty} f(k,k') dk'$, is it possible to interpret this integral as an integral along a straight line in a $k$, $k'$ plane. We might then make a change of variables $k = e^{\rho} \cos{\theta}$, $k' = e^{\rho}\sin{\theta}$ where we are now integrating a function $g(\rho, \theta)$ along some other contour in the $\rho$, $\theta$ plane that can be parameterized in terms of whichever variable we chose. Does this all sound reasonable?
So interpreting the integral as being along a straight line contour in the $k$, $k'$ plane isn't problematic, is it? Its just the change of variables which is unreasonable
@TedShifrin I don't understand your analog/digital analogy. It seems like the problematic idea though is that the original integral is defined for independent parameters $k$ and $k'$. The idea of a plane with contours, though, suggests that along most paths they are in fact not independent.
when I said a straight line, I actually meant a horizontal/vertical line such that $k$ is a constant and $k'$ varies, if that was unclear
@TedShifrin Well, if the interpretation as a horizontal/vertical line in a plane makes sense, it just confirms that my weird idea had at least solid foundation. If the change of variables is unreasonable, and your point has convinced me that it is (as I sort of always expected), then there is no point to it
@TedShifrin Oh yes, of course. I should have written $\int_{0}^{\infty}$. But if I understand your point correctly, the specific transformation isn't the problem. Its the idea of doing a transformation
Because education is going down the tubes and I believe in making people think and learn, not just throwing high grades at them the way most want (along with their parents).
@Pedro: I have been trying to make sure you can actually do concrete, interesting computational stuff. I think you should take my advice and take much more advanced classes.
That's terrible @Ted. Luckily I'm not far enough up the chain to get calls from parents. I feel like graduate programs in medicine and law are at least partly to blame. For physics, your GPA matters but it isn't a deciding factor. And we put a reasonable weight on where you went as an undergrad. From what I've seen though for med and law school it doesn't matter if you went to Yale or Southwestern Kentucky State, you GPA is treated the same
First, I knew we'd have grade inflation and all the things I'm complaining about. Second. It's mostly poor people who play the lottery, and they're subsidizing mostly rich families.
True, @Kevin, so when Harvard and Stanford give everyone As and we don't ...
@Pedro: As long as you make sure you learn basics, go for challenging stuff, because you're unhappy in standard courses. At MIT and Berkeley and here, the exceptional students don't take the same classes that the weaker students take.
@Ted Indeed. Grade inflation is certainly a big problem. And what's worse is that its a Nash equilibrium. If you try and unilaterally deviate from inflation, you potentially hurt you students or your career
Well, I am still tough and give real grades, but lots of students shun me. My career has never been in jeopardy for being a great but demanding teacher. :) If I had worries, it was about the quantity of my research, not the difficulty of my classes. But most people sell out.
@Ted I can attest that the same thing happens at Duke. There is a whole Math-1??X series in additon to the usual Math-1?? series for 1st and 2nd year students who are exceptional
No, not a prodigy at all, but you don't need to be a prodigy to be on the fast track. Talent + hard work! If it turns out overwhelming, then drop back a step.
@Pedro If you're anything like me then taking classes where you know most of the material already is a recipe for disaster. Like low grades. I was required to take multivariable calculus a second time by Duke, even though I had taken it in high school and gotten an A. I ended up with a C in that class because I never did the homework.
@Pedro Sadly, yes. It was actually a HUGE shock to me when I went to Duke that they wanted to grade my homework. I sent the majority of my last 2 years of high school at a technical college where the only grades were midterms/finals/papers/projects
@Ted Probably. I also should've pushed to take Math 103X instead. I only asked once and got some kind of "Well...... most students aren't ready and we prefer them to take....." speech
@Pedro In that class there were 3 tests and a Final. Homework was worth 1.5 tests, I believe
@KevinDriscoll Well, our university as a particular characteristic: one really has to look after oneself. They publish the problem sets, we have a theoretical class (2hs) and then a practice class (3hs, but never really that long) where we can ask questions and where problems are solved for everyone. But there is no such thing as homework, we have two midterms which we have to pass (not graded) and the only grade is the final.
If you don't pass a midterm you have a makeup exam which is usually tougher, and if you don't pass them, you have to take the course again.
@Pedro That would be a little harsh for me. I'd be lazy enough to just pass the midterms and then try and catch up for the final. At least that's what I would've done as a freshman. But I would've preferred it to having to hand in homework twice a week
@TedShifrin Because it is not what the OP is asking. "My question is whether my proof is correct, because I have a little bit doubt. If it's not correct, what is the best I can do? "
So, wikipedia, and my book, both define functionals as elements of B(E,F) (spaces of bounded linear mappings from a normed space E to a scalar field F). Is this correct? I.e. it would not be right to call the elements of a family of bounded linear mappings from a Banach space X into a normed space Y for functionals"?
Hmm. Let me retract part of that. The book certainly doesn't say anything else can't be called functionals as well, though they do make a point of the scalar field, so I figured that was central.
@KevinDriscoll Thank you. It more or less confirms the interpretation I have gotten as well. I was trying to make sense of what a classmate told me earlier, but I think he was just wrong to call them functionals in that case.
I'm working with FEM and continuum mechanics, and in those cases, we almost always end up with the simplest, finite spaces, so I'm not sure I'll ever get much pratical use out of all this fancy theory
amsldoc = documentation. A pdf you'll find on the net. Download it and keep it open. Tons and tons of examples in it for everything you might want to do
