Are you ready to test my knowledge of tensors?

[CosmicKitten]

Well THAT'S simple, but how would they know how to formulate problems that got a little more involved? Are they taught about Hilbert spaces and the Legendre/Laguerre/Hermite/other orthogonal sets of polynomials in the vector space, and how they can be used to solve for the wave function spherical harmonics? What about solving the heat equation, the wave equation? Those are simple, but what about finding the heat equation in three dimensions for a material that conducts heat anisotropically (Is there such a thing?) or with an active heat source that would make it inhomogenous... if students don't know this stuff, then what they are learning in class amounts to pure memorization. I don't know how they even remember the parts that aren't used in their jobs.

What exactly do they teach in an undergrad quantum mechanics class? Is more than one semester required? I wouldn't doubt that some professors don't cover everything, either they don't reach the end or they skip to it, often because they took pity on the students and went more slowly... My physics 1 teacher, I know it's a basic freshman class but I'm pretty sure he didn't cover everything he was supposed to. Is it normal or ok that I could get As on the tests even though the other students were doing so poorly he had to cut them a steep curve, while I wasn't even doing the homework?

 

[Jorriss]

Well THAT'S simple, but how would they know how to formulate problems that got a little more involved? Are they taught about Hilbert spaces and the Legendre/Laguerre/Hermite/other orthogonal sets of polynomials in the vector space, and how they can be used to solve for the wave function spherical harmonics? What about solving the heat equation, the wave equation? Those are simple, but what about finding the heat equation in three dimensions for a material that conducts heat anisotropically (Is there such a thing?) or with an active heat source that would make it inhomogenous...

They are probably taught methods to know how to approach such problems.

What exactly do they teach in an undergrad quantum mechanics class? Is more than one semester required? I wouldn't doubt that some professors don't cover everything, either they don't reach the end or they skip to it, often because they took pity on the students and went more slowly... My physics 1 teacher, I know it's a basic freshman class but I'm pretty sure he didn't cover everything he was supposed to.

A typical undergraduate course (1 year) will cover the entirety of Griffiths plus perhaps some basic nonrelativistic QM or another topic or two.

Is it normal or ok that I could get As on the tests even though the other students were doing so poorly he had to cut them a steep curve, while I wasn't even doing the homework?

You have this thing about needing to point out how easy everything is for you and it is quite off putting.

All I got out that sentence is that you didn't do your homework so you probably didn't learn the material that well.

 

[CosmicKitten]

Don't I point out the things that aren't easy for me in equal measure?
Paying attention, understanding what people say, multitasking, remembering short-term, you know the stuff that really matters.

And no I didn't learn anything in that class. Which makes it all the more of a travesty that I got the highest or second highest grade in the class. I studied what should have been taught later on my own, of course... I honestly tried to do the homework, but I was too tired in the head every day from having been at school all day to concentrate, and hesitant at my friends' suggestions that I get on ADHD meds (the shrink that I did see back then just tried to put me on an antipsychotic!) I was surprised that I got good grades, I always had the irrational thought that I was going to fail but... It was mostly stuff I remembered from high school anyway, the few points I missed were for not drawing a diagram the way the teacher wanted, the complicated solutions I came up without of not having studied or remembered the simple way had actually been right and I didn't make too many stupid errors.

Are they supposed to cover another semester of classical mechanics and or electromagnetism in the undergrad curriculum? I don't really know much the point of even teaching it before the students know enough math such that they can teach more advanced concepts. I would have liked to learn some calculus of variations, even a very simple do-it-with-the-teacher version, to solve classical mechanics problems, if that isn't too hard for a calculus 1 class. I still need to learn about Lagrangian and Hamiltonian mechanics, all this talk of invariants and such should become clearer if I know about tensors.

I just read about Riemann tensors. They are kind of like the commutator operation in quantum mechanics, that is, you take the derivative with respect to two different variables in both orders and take the difference; it doesn't commute thanks to the Christoffel symbols. The quantum mechanics momentum and position operators don't commute because taking the derivative and multiplying by x comes out different than multiplying by x and then taking the derivative WITH RESPECT TO x. I think this is the difference in the outcomes based on whether you measure one variable or the other first, or am I mistaken?

Anyway, do they teach about Christoffel symbols in an undergraduate relativity class? Those are easy, just use matrix elements no need to learn about tensors.

 

[Jorriss]

Don't I point out the things that aren't easy for me in equal measure?
Paying attention, understanding what people say, multitasking, remembering short-term, you know the stuff that really matters.

'humility is usually received better than arrogance' - the greatest man to ever live.

I don't really know much the point of even teaching it before the students know enough math such that they can teach more advanced concepts.

Yes, you do not know the point, that does not mean there is not one (or many). One would be doing themselves a large disservice by not spending the time to go through Newton's laws and all their CM I difficulties and face those challenging problems.

 

[jgens]

Does anybody want to test me?

With all this talk of Christoffel symbols you should be familiar with (affine) connections. A foundational theorem in differential geometry states that every Riemannian metric uniquely defines a torsion-free connection compatible with this metric. If you want (a very basic) test of your understanding of this material, then I propose that you prove this result in two different ways.

Edit: I am not personally big on nitty-gritty details. They are important but tend to obscure the ideas, and since understanding is what you are testing here, try to come up with proofs that are no longer than a paragraph. If you also want to work out the proofs including a verification of all the finer points, then by all means do so.

 

[CosmicKitten]

But if the students did well in a basic physics class in high school, then they already got all of that, didn't they?

The point of making you take it again in the first year of college I thought was to show you how it's based in calculus? But using an integral, force differential x to find the work done is so simple, it reduces to an easy equation, F*x unless the force is a function of displacement. I spotted momentum as the derivative of kinetic energy with respect to velocity very easily. But really, I think someone who had not taken calculus but had good grounding in algebra and trigonometry would easily pass it. Is that a problem for most freshmen, remembering their algebra and trigonometry?

If I have a problem with something, it's usually because of an equation I don't know. Being too hardheaded to listen to how others approach the problem, I just try to figure out my own overcomplicated way. Is this how people are supposed to learn it? By having to figure out the equations all on their own? When it would be easier to just read a lengthy paper on how the equations were originally derived, a paper so lengthy you will never forget the equation that it is about? Should all children thus be kept away from advanced educational materials in case they spoil themselves for the mental exercise of knowledge deficiency? Come to think of it, that may be why it's so much easier for me than for other people, because I was knowledge deprived in the home for a good while. No internet, no books, I wrote down equations if I was lucky enough to see them on Science Channel. I borrowed a noncalculus book on physics and read it and played with equations in it just for fun. All of a sudden give me library books and internet, what happens?

 

[Jorriss]

But if the students did well in a basic physics class in high school, then they already got all of that, didn't they?

This is all irrelevant, let's get back to the topic of tensors.

What do you think of jgens questions above?

 

[CosmicKitten]

This is all irrelevant, let's get back to the topic of tensors.

What do you think of jgens questions above?

The connection is Levi-Civita, torsion-free means a symmetric matrix which is why the two indices at the bottom of the Christoffel symbol commute, and proof of this concerns the development of the Christoffel symbols and why they are necessary, that much I know, but I'm too tired to think of a proof right now. I should be sleeping, and tomorrow I'm obligated to do stuff all day that does not allow me to study until... Thursday afternoon. I'll keep it on my mind until then, perhaps even come up with it sooner, although I'm a very different creature without my meds, as is the case on the days when I don't study...

 

[Vanadium 50]

Closed pending moderation.