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?
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.
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.
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.
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?
It depends on the program. Most decent programs require calculus, differential equations, linear algebra and a mathematical methods sequence at minimum. 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.
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 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?
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.
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. Biology and chemistry majors have fewer math requirements in general. My experience in academia is very different than yours. People who go in to 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. 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.
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?
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. 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... I went to community college and UCSD and the students at UCSD were better on average (though quite below UChicago). 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. 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. 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.
A lot of linear algebra concepts. There is "linear algebra," and there is an actual proof based study of linear algebra.
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.
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.
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...