I want to lodge an official complaint (but Trump says warming doesn't exist): I left Yosemite at 8 AM and it was 34º, I arrived in Santa Clarita about 1 PM and it was 94º. @robjohn: That is obscene.
@TedShifrin why does multiplication of vectors not make any sense? i.e. $(1,2)\cdot (3,4)$. Or otherly asked: does there exist something like that, which is not completly different from normal multiplication?
$\int \int \frac{ln(1+2x^{2}+y^{2})}{1+x^{2}+y^{2}}dxdy$
over D : $R^2$
using $ln(1+2x^{2}+y^{2})\leq \sqrt{(1+x^{2}+y^{2}}$ and showed convergence by the comparison test ,but according to my friends it diverges .
Who is right ? Thanks in advance.
@TedShifrin Regarding warming, I think it is folly to ignore the climatic trends, but I think that since we are still recovering from the last ice age that the climate would be warming anyway. However, I don't deny that we are helping it along. In a few thousand years, we will be back up to our skyscrapers in ice again.
@Kasmir: We never finished talking. But if you switch to polar coordinates, you have an integral that looks like $\displaystyle\int_0^\infty \frac{\ln u}u\,du$ and that you can settle immediately.
@TedShifrin out of whack from what? we don't have a good baseline for a long enough period. The earth is warming, but it has been since the last ice age. It will peak and then descend into the next ice age in several thousand years.
@Maks: Oh, some books use the word differently for lines. But that's why I asked if we were still in the plane or in 3-space. Because equations of planes will only make sense in 3-space.
Take an N by N 2D space with integer X, Y coordinates, for example if N were 3, then there are 9 possible positions/coordinates in this space.
Where M points in this space can be marked.
Q: What is the most unique vectors that can exist that connect pairs of marked points in this space?
Assume...
I already said you need to learn to recognize certain standard curves (like from high school) — lines, circles, parabolas, ellipses, hyperbolas, ....
I have 112 multivariable calculus/linear algebra lectures on YouTube, but the level is a bit more on the theoretical side, although all the computational stuff is in there.
@TedShifrin but as we are studying computer science, nobody cares much about math, so everyone just memorizes everything and dont learn the logic of it
I haven't watched stuff like multivariable calc on KA to see how much I'd hate it.
I am not in general a big fan of videos which tell you how to operationally do something without understanding what's going on. (Kudos to @Maks for wanting to understand.)
is "For every $x\in\mathbb{R}$ there exists at least one $n\in\mathbb{Z}$ with $n\leq x<n+1$." equivalent to: "between any integer and it's successor exists at least one real number"?
@Null: It says that every real number is between some integer and its successor. Actually, that integer is unique (so the original statement should be stronger).
@Fargle: Don't get me wrong. I think computation is extremely important!! All my former students will tell you that. I just think being given things to memorize and regurgitate without understanding is worthless.
Well, @Fargle, there's tons of research on learning and pedagogy (that's what keeps schools of education in business), but I'm not sure how much of it I truly value, personally. I've read some of it and been on a bunch of Math Education doctoral committees.
That is, research pedagogical methods and come out with an understanding of how to teach math better. Incoming totally-uncontroversial-on-MSE statement: it's a beautiful subject and I wish it were taught better, from the very outset.
@TedShifrin how do you tell this to you random kiddo that has to "pass just this one test"? in some sense "stupid memorizing" is a necessary evil we have to cope with. no?
@TedShifrin professors on our faculty know a lot, and they to researches and everything, but they dont know how to teach to others, they are missing that pedagogical ability
@TedShifrin That's definitely a downside. I hope I can take some time to educate myself in that vein, or at least listen to the wisdom of my immediate family--half of them are educators or former educators and one of them is a college professor.
@TedShifrin well, i passed my france exams (not on university, in school) purely by memorizing. it made no fun, it wasnt a PITA, just saying i memorized without context.
I would not have gotten promoted in today's academic world. (I wrote some excellent research papers that were in good journals, but I didn't write nearly enough to make today's administrators happy.)
I've read that Pillai's conjecture is still unproven. it states that for any given positive integer k there are only a finite number of pairs of perfect powers whose difference is k
@Maks: Since so many foreign students want to come here to study (or at least wanted to before the recent election), I have to assume it's considered above average. I think professors in the US care a little bit more (but not hugely more) about students than professors in Europe and Asia.
@TedShifrin my mother travelled to Los Angeles to give some advices and ideas to high schools on how to teach to new generations and the new ways of learning, not just memorizing everything but understanding it
@TedShifrin I'm from Argentina, I study on the UNC (National University of Cordoba) which is a public university, one of the first here in Latin America and the most important one, as it found the pillars of a new way of teaching.
@Null That I can't say. I'm still a student, not a teacher--I can only see the problems in the system, and I'm not educated enough on education itself to judge any solutions.
@TedShifrin Agreed. It's almost hard to say which comes first. Are people racist, etc., because they're anti-intellectual, or are they against current research because it doesn't confirm their prior beliefs?
@TedShifrin the problem with the education system at the moment is: you just have to memorize certain things, otherwise you can't even understand anything.
@Null: I won't argue entirely against memorization. In math one must memorize definitions and understand them, or it's pointless. In foreign languages, one must memorize vocabulary and declension/conjugation patterns. Now if one has an "ear" for language and a "brain" for math, this memorization is not just rote ...
I think that's not too practical, @Maks, and it would cost you a fortune to fly me there and teach me Spanish (which I am trying to learn) in a hurry.
Yeah--there's a definite difference between memorization and memorization with understanding, and the second one is far more likely to stick for longer.
@Fargle: As I used to tell my students, often the understanding comes from using the concepts. That's definitely the case with things like differential forms (or the definition of linear independence).
@TedShifrin It's the same even with basic things like proofs. When I first was confronted with them I was very intimidated--now that I've done enough, they don't intimidate me nearly as much, because I've used the basic methods of proof a good number of times. But that's more a meta-example.
@Null I'm just saying that proofs in general are easier the more of them you do, similar to how quadratic expressions are easier to factor the more of them you factor.
@TedShifrin Yeah, analysis is still a bit hard to work with for me, but that's because I'm a bit unused to the proof techniques. It was fine in the metric topology examples in Munkres, but past that I struggle.
OK. It is, although there are hard things in algebra.
But some people are more naturally good at estimates and analysis, and other people are more naturally good at algebraic stuff. So it depends on the person.
@TedShifrin as you can read some german, maybe it will clear it up, altho one has to say, the chapter of II is not complete yet, because the semester has not ended.
Yes, I can read German (and even speak it). OK, I was expecting multivariable analysis, both derivatives and integrals. All that's there is derivatives and differential equations. I wonder where they teach differential forms and Stokes's Theorem.
Maybe you need to take a graduate course in manifolds to get to those.
oh, you originally said Linear Algebra II, not Algebra II.
One of my favorite linear algebra/geometry exercises (section 2 of either book). A triangle inscribed in a circle with one side a diameter is a right triangle.
The exam is about series and sequences, taylor, maclaurin ,etc, vectors, functions with multiple variables , multiple integrals and maximum and minimum of functions with multiple variables
It has 5 exercises, 3 of them, are of maximum and minimum, only 2 of them cover the other stuff
$\int \int \frac{ln(1+2x^{2}+y^{2})}{1+x^{2}+y^{2}}dxdy$
over D : $R^2$
using $ln(1+2x^{2}+y^{2})\leq \sqrt{(1+x^{2}+y^{2}}$ and showed convergence by the comparison test ,but according to my friends it diverges .
Who is right ? Thanks in advance.
Take an N by N 2D space with integer X, Y coordinates, for example if N were 3, then there are 9 possible positions/coordinates in this space.
Where M points in this space can be marked.
Q: What is the most unique vectors that can exist that connect pairs of marked points in this space?
Assume...
