Are you ready to test my knowledge of tensors?

[CosmicKitten]

I finally got my library fine paid off last week and I Picked up Schaum's Outlines Tensor Calculus by David C. Kay. I figured I really need to learn about tensors because every time I read a book or paper about certain subjects such as relativity, nonlinear optics, aerodynamics, etc., I see stuff about tensors and so i will need to eat sleep and breathe them for a little while before my physics studies can progress.

I have been studying the book for around half to most of the day for the past three days, I had been studying it at the library for a while as well before I got my fine paid off, and I am kind of on chapter 7 now, which is about Riemannian geometry of curves. It all seems really easy to me, I got the hang of Einstein summation really quick, the metric is intuitive, I think I get how to test for tensor character, Christoffel symbols are easy to figure out, although I will have to pick up a more advanced book later to figure out the theory behind them, as well as the theory behind the differences between covariance and contravariance. But I think it's good to do some problems and see some problems being done before I read the theory, that way I have some material in my head to work on so I can concentrate on the more theoretical books better.

I have mostly been reading through the problems, working some out and just absorbing the language of tensor calculus, even addictively at moments, the meanings and ideas and answers to things I have been wondering about just jumping out at me; and after I read through all of the chapters I intend to go back and do a bunch more of the problems.

Does anybody want to test me?

 

[Jorriss]

Does anybody want to test me?

What's a tensor?

 

[CosmicKitten]

That's a very good question...

Tensors seem to evade any form of straightforward definition. These objects seem to be defined more by their operations and how they transform than by what they... well, are.

Tensors seem to be a generalization of vectors and scalars into higher dimensions. A vector is just a type of tensor, of rank one, which is expressed as a single row or column of elements. A rank zero tensor is a scalar, or a single element. A rank two tensor is a matrix, which can be used to map a vector from a space with the matrix's number of columns to a space with the matrix's number of rows. A rank three tensor I imagine could be expressed as a three dimensional box with elements along all three dimensions, but since that's rather difficult to draw a matrix with vectors as elements (as taking the outer product of a matrix and a vector) will have to suffice. Or a (rank four) matrix of matrices, if the outer product of two matrices is taken. The inner product subtracts two from the sum of the ranks of the tensors, like multiplying a matrix with a matrix is still a matrix, and can only done if the columns on the first is equal to the rows on the second, or equivalently in shorthand, the indicated indexes are the same and therefore summable. Or like multiplying a row vector with a column vector makes a scalar. But multiplying a column with a row creates a matrix, the outer product.

A covariant tensor seems to be defined as a tensor that, when transformed into another coordinate system, equals the tensor in the original coordinate system multiplied by the derivative of the coordinates of the original coordinate system with respect to the coordinates of the new system. A contravariant is defined as one that transforms such that the one in the new coordinate system equals the original multiplied by the derivative of the coordinates in the new system with respect to the corresponding ones of the old system. Checking to see if a possibly tensorial object can be put in this form seems to be the main method to test tensor character, and apparently the Christoffel symbols had to be introduced because the derivative of a tensor that is not linear (that is, its elements are not all constants) doesn't pass this test otherwise.

When finding the length along a curve, the form of the covariant and contravariant versions of the vector multiplied with the metric and integrated over the endpoints of the curve reminds me of multiplying the wave function and its conjugate, with the position or momentum operator and integrated over space to find the probability. But when I compute the Christoffel symbols, turning the second kind into the first kind multiplied with the contravariant metric element, it seems it equals the reciprocal of the covariant version. I suppose the distinction and the meaning of all this eventually clears up, or will dawn on me sooner or later...

And, well, that's my interpretation.

 

[Jorriss]

That is mostly fine (if not a bit long winded) but that perspective on tensors is out of date. You should find a more modern treatment.

 

[CosmicKitten]

The book is dated 1988. Has tensor theory really changed that much since then? What new developments have they made?

 

[1MileCrash]

The book is dated 1988. Has tensor theory really changed that much since then? What new developments have they made?

I learned it the same way as you (k^n tuples where k is space dimension and n is tensor rank that transform according to a transformation law) about a year and a half ago.

It is because I was a physics major then, but yes, it is outdated. It is still taught because learning it that way apparently allows you to circumvent a lot of concepts in linear algebra, which a lot of physics degree programs don't include.

Though someone else may be able to correct me on that.

 

[CosmicKitten]

Wow how is linear algebra not required for a physics degree, I thought it was?

Exactly what math is required for a physics degree anyway? Not that it matters to me, since I intend to study mathematics as well. I'm surprised that they don't list any requirements to learn tensor calculus for undergraduate classes in general relativity, or do they teach that with the class? Or is tensor calculus usually taught in graduate school?

As far as my math level, although I haven't taken an actual math class since I dropped out of calc 2, I have studied such that I feel confident in everything up to ordinary differential equations. I intend to brush up on partial differential equations as well, although I know a bit about those, and I studied a bit of group theory and abstract algebra and so far it seemed pretty easy, but I feel I need firmer grounding in analysis and proofs, as in I need to become familiar enough with the required concepts such that knowing how to prove something comes easily. Computation of everything from simple arithmetic to derivatives, most integrals and basic ordinary and partial differential equations, anything that I have practiced and am familiar with the techniques used for, comes like breathing to me, although I make mistakes easily especially if I do half the steps in my head which is why it helps to have a book with answers to the problems, to let me know what kinds of mistakes I make; if I have the answers, I can usually spot my computational errors quite quickly.

Can I be asked to prove something tensor related, such as say if some expression has tensor character?

 

[Jorriss]

Exactly what math is required for a physics degree anyway?

It depends on the program. Most decent programs require calculus, differential equations, linear algebra and a mathematical methods sequence at minimum.

Can I be asked to prove something tensor related, such as say if some expression has tensor character?

Yes, though depending on the type of question (such as just asking if some quantity is a tensor) the homework section might be more appropriate.

 

[dextercioby]

If you know calculus (multivariate one) and got a decent grip on linear algebra, then Schaum's book is not what you want. I'd try: Isham, C.J. - Modern differential geometry for physicists (2ed., WS, 1999)(306s)

 

[CosmicKitten]

I would have thought so, analysis/proofs wouldn't be necessary unless one wanted to go into theoretical physics - which is actually what I prefer, since I hate lab activities. I would probably enjoy designing an experiment though, but I don't see much point in setting up a computer and a bunch of toys to verify something that can be easily computed or found in a textbook.

I hope that biology and chemistry programs have similar requirements. I had noticed that a lot of people going into biology were people who wanted to go into science but weren't good enough at math to do chemistry or physics, who only had to take one or two semesters of calculus for a math requirement and wound up flunking calc I two or three times until they were told to change majors. I have some interest in biophysics, which I imagine has the same math requirements at least.

What exactly does 'mathematical methods' entail? Is it just a repeat of math you've already taken, showing you how to use it to solve or model physics problems? Or does it also feature how to use software like MATLAB?

 

[CosmicKitten]

If you know calculus (multivariate one) and got a decent grip on linear algebra, then Schaum's book is not what you want. I'd try: Isham, C.J. - Modern differential geometry for physicists (2ed., WS, 1999)(306s)

I found a book similar to that, if not the same. I need to read something with more problems and answers first so that I have a good enough idea of what that is in my long term memory such that the theory and proofs don't weigh too heavily on my working memory.

 

[Jorriss]

I would probably enjoy designing an experiment though, but I don't see much point in setting up a computer and a bunch of toys to verify something that can be easily computed or found in a textbook.

People are not setting up a computer simulation or 'a bunch of toys' to verify things that can be easily computed or found in a textbook. They are running experiments so theorists and computational scientists have an actual measurement to compare their numbers to and computational scientists do simulations on systems the analytic theorists have (usually) no hope of finding the behavior of.

I hope that biology and chemistry programs have similar requirements. I had noticed that a lot of people going into biology were people who wanted to go into science but weren't good enough at math to do chemistry or physics, who only had to take one or two semesters of calculus for a math requirement and wound up flunking calc I two or three times until they were told to change majors. I have some interest in biophysics, which I imagine has the same math requirements at least.

Biology and chemistry majors have fewer math requirements in general. My experience in academia is very different than yours. People who go into biology are either premed so there interests lie in medicine or are interested in biological sciences. It's not a field for people who can't do physics or chemistry.

What exactly does 'mathematical methods' entail? Is it just a repeat of math you've already taken, showing you how to use it to solve or model physics problems? Or does it also feature how to use software like MATLAB?

Depends on the university. Broadly speaking it's a cookbook of mathematical techniques needed to solve physics problems. One might be cover vector calculus, Fourier analysis, greens functions, etc. Some programs, such as UCSD, use it as an opportunity to teach computational software such as Mathematica though I do not know if that is standard.

 

[CosmicKitten]

I didn't mean to say that actual experimentation is computers and toys used to solve problems that can easily be computed or found in a textbook; rather, that's what fake freshman lab activities are about. I think they would get much more out of laboratory exercises if they had to design their own experiments and problems and maybe even make a competition out of it. My physics I lab class bored me so much I got a C, despite getting an A in the lecture section, and my performance was also mediocre in my chemistry lab - chemistry labs are somewhat more interesting, since you get to work with things and observe reactions that one doesn't get to see every day, but it irritated me that they were making me do experiments to see things change color and stuff without teaching the quantum mechanics behind it. I would rather learn all of the theory before I even began experimentation, since no professional experiments can be done without lots of help if you say don't know dynamic similarity for a vehicle flight test, or how potassium makes a flame turn purple for a solid state chemistry experiment.

Well after the first year or so of course all of the weak math students would be weeded out. Those that somehow get by would find themselves working in fast food after graduation. But then I imagine UCSD has a far better crop of students to start than Pitt-Johnstown, which is the one four year college I went to for one semester before moving and transferring to the school I flopped at.

I have considered saving up to take a graduate level class at UCSD by extension, as the only way to circumvent the transfer requirement, after I feel I have studied enough, but given the cost and my exorbitant food requirements it is rather impossible. My other option might be to outline an honors contract at the community college where I ask to do graduate level honors work but I know not all instructors are willing to do that.
I have a book that covers those topics, but it mostly just gives a long list of proofs for all of the theorems that are developed, and then a few (mostly proofs) exercises with no solutions. Tensors are covered only very briefly in the first chapter. I can follow the proofs and the developments but I don't know where to start for the exercises, I think more worked out examples are necessary ro feed it to my ADHD mind, which can somehow see the logic for every step of a proof and be unable to replicate it. Well actually sometimes as I run through and verify the steps in my head I run into one I don't immediately see the logic for, and so my mind goes on a tangent, which is why I never finish any of the math books I check out, or sometimes they pull some trick out of the blue such that I don't even know how they thought to come up with that, as I saw in some stuff about functional analysis I was reading. Now the ideas were completely intuitive to me if I was in a focused state while reading it, as in my mind would jump ahead and see the big idea of everything while reading the definitions, but I clearly have not found the right approach to learn proofs properly. Do you know a good source to learn proofs, like with a list of techniques used, tips on how to know what technique to use and a long long list of problems?

 

[Jorriss]

I think they would get much more out of laboratory exercises if they had to design their own experiments and problems and maybe even make a competition out of it.

More advanced lab classes often do give students considerable more flexibility in the labs and much more independence. With that being said, students with insufficient past guidance will not be able to design their own labs in any sensible way and teaching intro labs that way has potential for disaster.

but it irritated me that they were making me do experiments to see things change color and stuff without teaching the quantum mechanics behind it.

You can also view it another way - you are doing these experiments to appreciate why the theory you are learning needed to be developed. Certainly the founders of quantum mechanics, who were well aware of atomic spectral lines, or chemists, who were familiar with the spectra of various metals in fire, did not have access to quantum mechanics...

But then I imagine UCSD has a far better crop of students to start than Pitt-Johnstown, which is the one four year college I went to for one semester before moving and transferring to the school I flopped at.

I went to community college and UCSD and the students at UCSD were better on average (though quite below UChicago).

I have considered saving up to take a graduate level class at UCSD by extension, as the only way to circumvent the transfer requirement,

This still seems like a mistake. I still recommend finishing community college the standard way and transferring the standard way. Rushing to graduate course work with incomplete preparation is not a good idea.

I have a book that covers those topics, but it mostly just gives a long list of proofs for all of the theorems that are developed, and then a few (mostly proofs) exercises with no solutions.

Personally, I spend a lot of time researching which textbooks are good and appropriate to my skill level, knowledge and interests. I'd recommend if you need a book on subject asking for advice in the textbook section of this forum or checking out the reviews we have here.

Do you know a good source to learn proofs, like with a list of techniques used, tips on how to know what technique to use and a long long list of problems?

Many of these types of books exist though I am only familiar with two. Velleman, How to prove it and Eckles, Introduction to Mathematical Reasoning. I would just ask in the textbook section.

 

[1MileCrash]

Wow how is linear algebra not required for a physics degree, I thought it was?

A lot of linear algebra concepts.

There is "linear algebra," and there is an actual proof based study of linear algebra.

 

[CosmicKitten]

If they can't design a laboratory experiment, then I don't see much point in even having a lab section, except to teach usage of dangerous equipment, which community colleges can't afford anyway. The skills taught in basic classes are covered in high school anyway, but then again it is rightfully assumed that most of the students are rusty in whatever they learned in high school, even if some of them have a frightfully clear recollection of it. a

I would rather bypass community college in any way possible, since by now I know most if not all of the mathematics required for a physics degree program, and almost enough physics to score well on the GRE, and any treatment of electromagnetism or optics that doesn't even require knowledge of tensors is trifling. I asked the teacher once a question about how light predicts which way to travel through glass is the way that would get it out of the glass in the same time it would take to get out of the same length of air if it just continued traveling in its initial direction, and he seemed to think the answer to that was unknown. But it's not; it's explained by some intuition developed after pondering Fermat's principle. The treatment of everything was high school level, and I was being dumbed down into a state of depression. The only part where I think I would suffer would be with anything that requires social involvement, unless it's made interesting, and I would have to know how to do research and perhaps harder experiments with lab equipment that cannot be bought at Big Lots. But it sounds like many undergrad programs don't even train you in that?

I am surprised that tensors are not a requirement for every degree program? Or partial differential equations? Is general relativity not always a physics requirement? Even quantum mechanics requires it at some level. Not to mention tensors for stress and strain, or permittivity/permeability of dielectrics, well some of that might only be required for engineering programs. Some physics majors that don't take extra classes probably won't get accepted into a lot of degree programs, I figure some schools' physics programs are geared toward training high school level physics teachers? I would hate teaching.

 

[1MileCrash]

I am surprised that tensors are not a requirement for every degree program? Or partial differential equations?

I've never heard of a physics program that does not cover these things (though I'm sure they would exist), where did you find this program?

 

[CosmicKitten]

I've never heard of a physics program that does not cover these things (though I'm sure they would exist), where did you find this program?

Jorriss just listed the usual math requirements, though perhaps tensors or PDEs are covered under them somewhere. You can bet that an associate's program will let you get away without taking a few of those. Honestly, I don't know what an associates degree is good for that an honors high school diploma shouldn't be for.

 

[jgens]

I've never heard of a physics program that does not cover these things (though I'm sure they would exist), where did you find this program?

The undergraduate physics program at Chicago does not. Most graduating physics majors have seen this material since they take classes not required by the degree program, but it is definitely still possible to graduate without it.

 

[CosmicKitten]

The undergraduate physics program at Chicago does not. Most graduating physics majors have seen this material since they take classes not required by the degree program, but it is definitely still possible to graduate without it.

That I presume would be the difference between a B.S. in physics and a B.A. in physics? I don't even know why the latter exists...

 

[jgens]

That I presume would be the difference between a B.S. in physics and a B.A. in physics? I don't even know why the latter exists...

Chicago only offers a BA in physics. It is not like the program is bad either (ranked 7th in the nation on USNews). Most stuff at the undergraduate physics level has no real need for tensors or a general theory of PDEs so the department does not require it for graduation.

 

[Jorriss]

I am surprised that tensors are not a requirement for every degree program? Or partial differential equations? Is general relativity not always a physics requirement? Even quantum mechanics requires it at some level. Not to mention tensors for stress and strain, or permittivity/permeability of dielectrics, well some of that might only be required for engineering programs. Some physics majors that don't take extra classes probably won't get accepted into a lot of degree programs, I figure some schools' physics programs are geared toward training high school level physics teachers? I would hate teaching.

Partial differential equations does not need to be required. The bare necessary techniques are learned in either math methods or E&M.

Most tensors that show up in an undergraduate curriculum can just be called a matrix and you move on with the physics.

Further, general relativity is not required in most programs I have come across.

 

[CosmicKitten]

Chicago only offers a BA in physics. It is not like the program is bad either (ranked 7th in the nation on USNews). Most stuff at the undergraduate physics level has no real need for tensors or a general theory of PDEs so the department does not require it for graduation.

No, just for getting into most graduate programs. I don't know what's the real point of getting just a bachelors in physics since it's pretty useless if you don't get a graduate degree afterwards. Unless you just want to teach it in which case you have to take a load of teaching classes as well, or use it to get more tutoring clients on Craigslist and as an excuse to charge higher prices.

I don't know what good the (I'm assuming required) undergraduate relativity class would be without tensors. I learned basic algebra based equations for basic BASIC relativity when I was in high school. Maybe if there's a brief crash course in tensors.

I'd rather go for a B.S., since a B.A. means more humanities classes, which I'm sure are really good humanities classes, but I find humanities classes to be too easy since you can't really make mistakes and you don't need to learn any new skills that you didn't learn in high school, except perhaps in language classes and music theory classes (in my case), or art classes in the case of people who are less adept at art. I consider my writing and art abilities to be far more developed than my mathematical and science abilities, since I had been drawing pictures since I was a toddler and writing stories since kindergarten, by contrast I had not recognized a talent at math until I was almost in high school.

 

[jgens]

No, just for getting into most graduate programs.

Knowledge of tensors and PDEs is not even required for that! I know plenty of people in the physics program at Chicago who have no idea what a tensor is, but who got into great graduate schools for physics.

I don't know what good the (I'm assuming required) undergraduate relativity class would be without tensors.

The undergrad relativity class at Chicago is not required for graduation and also does not cover tensors. The graduate course does cover that material, but many majors do not take it.

I'd rather go for a B.S., since a B.A. means more humanities classes

Not necessarily. The difference in a BA/BS math major at Chicago is essentially the number of chem/physics courses you take. So someone who spends all their time taking math and no physics would only qualify for a BA. I have no doubt other majors (at other schools) are like this too.

 

[Jorriss]

No, just for getting into most graduate programs. I don't know what's the real point of getting just a bachelors in physics since it's pretty useless if you don't get a graduate degree afterwards. Unless you just want to teach it in which case you have to take a load of teaching classes as well, or use it to get more tutoring clients on Craigslist and as an excuse to charge higher prices.

Graduate school isn't for everyone, some people get their BS and just go to industry, go to teaching as you mentioned, etc. Also, a pure course in PDEs is not necessary for graduate school either - though I can't imagine it not helping!

 

[CosmicKitten]

Graduate school isn't for everyone, some people get their BS and just go to industry, go to teaching as you mentioned, etc.

Industry? You mean just assembling things without knowing how they really work? Sounds boring.

 

[Jorriss]

Industry? You mean just assembling things without knowing how they really work? Sounds boring.

Why do you assume they don't know how things really work?

In any event, industry means a lot of things. Not that I am familiar with any of it.

 

[CosmicKitten]

Knowledge of tensors and PDEs is not even required for that! I know plenty of people in the physics program at Chicago who have no idea what a tensor is, but who got into great graduate schools for physics.



The undergrad relativity class at Chicago is not required for graduation and also does not cover tensors. The graduate course does cover that material, but many majors do not take it.



Not necessarily. The difference in a BA/BS math major at Chicago is essentially the number of chem/physics courses you take. So someone who spends all their time taking math and no physics would only qualify for a BA. I have no doubt other majors (at other schools) are like this too.

But they take classes in grad school to teach them what it is right? At least if they're in grad school for theoretical physics? Or condensed matter/solid state physics, such as research and development in stuff involving nonlinear optics? Or continuum mechanics? Who could possibly be passionate enough to go to grad school and only be satisfied taking the bare minimum?

 

[CosmicKitten]

Why do you assume they don't know how things really work?

In any event, industry means a lot of things. Not that I am familiar with any of it.

Well they might if they researched on their own. But if it's like me in that chemistry class, well... I would expect myself to know why any given substance is whatever color it is, and, given atomic/molecular structure and other conditions, how to predict what color it is without looking at it. I don't see anything in the curriculum that could help with that - I've seen material for undergrad quantum mechanics classes (are those even required?), and it barely gets through showing you how to calculate the spectrum of hydrogen. The equations to calculate that are barely comprehensible if you don't know partial differential equations, I hope they teach that in mathematical methods, and also how to approximate for larger problems. Or is the boss with the graduate degree expected to do all that? For that matter, would an associates degree even qualify for any industry job? I don't even... If requirements are that low, I want them to just let me take the finals for all the lecture classes and then just get all the lab classes over with and then just give me the paper.

The spectrum of potassium in a flame test would look very different if applied a much hotter flame, am I correct? Higher temperature means harder collisions and so some of the more tightly bound electrons will be excited. How would one calculate that?

 

[Jorriss]

I don't see anything in the curriculum that could help with that - I've seen material for undergrad quantum mechanics classes (are those even required?), and it barely gets through showing you how to calculate the spectrum of hydrogen.

An undergraduate QM course will go well past how to calculate the spectrum of hydrogen.

The equations to calculate that are barely comprehensible if you don't know partial differential equations, I hope they teach that in mathematical methods, and also how to approximate for larger problems. Or is the boss with the graduate degree expected to do all that?

What do you mean by know partial differential equations? Solving the hydrogen atom just requires separation of variables and then ODE techniques. It's not that complicated. A PDE course will talk about existence and uniqueness of solutions, sobolev spaces, issues of convergence, etc - not that much about solving a PDE (necessarily).

For that matter, would an associates degree even qualify for any industry job? I don't even... If requirements are that low, I want them to just let me take the finals for all the lecture classes and then just get all the lab classes over with and then just give me the paper.

Some industry jobs require a bachelors, others a masters, PhD, some particular certificates. I'm sure there exists some industry job where an associatives degree qualifies - though I have no idea which.

 

[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.