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> Those are differential forms ... Yup. You gave the modern treatment. What I wrote was: > Physicists keep using old versions of differentials, and the math

> Those are differential forms ... Yup. You gave the modern treatment. What I wrote was: > Physicists keep using old versions of differentials, and the math books got rid of those 50+ years ago; clean ups are not too difficult but need to be done by either physics or math. So, I was talking about the "old" stuff. That is relevant because a LOT of the physics literature elementary enough to be in courses in universities just asks the student to accept that dA is a "little unit of area" and dV is "a little unit of volume" and that that is enough to define an integral. The people who write such physics are rarely or never actually thinking the modern treatment of differential forms, tangent spaces, etc. So, often a physics student is stuck with the old versions of dA and dV and with a prof who doesn't know the modern treatment either. So, that is an example of my theme: The physics people are not so good at the math.

selimthegrim 5y ago
The reality is we cannot start freshmen on Courant-Hilbert. Even Apostol may be too much. If you have solutions I’m all ears. And for all its flaws FLP is still one of the best intros to complex numbers for laymen out there.
- graycat 5y ago
  > All ears Start with at least a good course in calculus. Then, for the divergence theorem, in one dimension that is essentially just the fundamental theorem of calculus on a closed interval on the real line. Then with multiple integrals can get a proof on any box, rectangular parallelepeped with edges parallel to the three orthogonal coordinate axes. If draw a picture and use the Jacobian for change of variable, can get a somewhat more general proof. That's enough for most courses in E&M, Maxwell's equations, and fluid flow. An okay source in PDF from Taiwan is in https://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/divergence%20theorem.pdf https://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/dive... Apostol does Jordan curves, and that's a bit much; it's a strange physics course that needs Jordan curves. The last edition of Rudin, Principles of Mathematical Analysis is a good source. With some good lectures to explain to the students just what is going on with compact sets, Rudin does not have to be too difficult or severe. Fleming in Functions of Several Variables does both the Lebesgue integral and exterior algebra -- at times it has been taught to ugrads. For the Lebesgue theory, I prefer either of Rudin, Real and Complex Analysis (just the first, real, half) and Royden, Real Analysis to Fleming. The Lebesgue integral does not have to be more difficult to teach or learn than the Riemann integral -- maybe the Lebesgue approach is easier than the Riemann approach where usually dig into compact sets. Here is the fishy part of improper integrals: Rudin's Principles discusses conditionally convergent series. Then it's just an exercise that a rearrangement can be made to converge to negative infinity, positive infinity, or anything between. Well, can generalize this to improper integrals: Partition the real line into intervals of length 1, (0, 1], (1, 2], .... Then start with a function defined from a conditionally convergent infinite series, say, each interval with integral the same as one of the terms of the series. Then in the limit in the improper integral, integrate over the intervals in whatever order want, not necessarily from -a to a and, thus, converge to anything might want. So, the value of an improper integral depends all on just what order in which add up the pieces -- so that's not a very good definition and stands to resist proofs of good properties. The Lebesgue theory gets around this issue. First, that theory just takes the given function and takes it apart into the positive part and the negative part. Second, it is easy, then, to define the integral of each part separately. If both parts have integral with absolute value infinity, then the integral of the function is not defined because it would require subtracting infinity from infinity which is not defined, even in the extended real number system. And that approach right away rules out the conditionally convergent issue. Otherwise the Lebesgue integral is defined. It also has good convergence properties, e.g., can prove a good version of Leibniz's rule. In grad school, the department suggested Courant and Hilbert. So, I got a copy: The content didn't seem to have much to do with the math I was studying or the physics I was interested in (I lost my copy in a move). For the Feynman Lectures, yes, I like most of it. I didn't like his treatment of the bra-ket notation, but then I don't like that notation anyway. The whole Lectures are available in PDF at https://feynmanlectures.caltech.edu/ https://feynmanlectures.caltech.edu/ The move also cost me my copy of Feynman's Lectures. There is something, maybe, US STEM field students should understand: In the US, there was The Bomb, the atomic bomb. That made a lot of people in Congress awake and afraid, and they decided that the US research universities should be the world leaders in research in the STEM fields. So, Congress appropriated a lot of money (NSF, NIH, DARPA, etc.) for US research universities and STEM field researchers. The universities like the money, e.g., commonly take 60% for "overhead" and, thus, fund the theater group, the string quartet, the English department, the art department, the new fountain in the quadrangle, the limo for the president, etc. But the STEM field profs and students also get funded. Since the members of Congress tend to be old, they also like funding medical research. For something like the Webb telescope, who's to say, that might lead to some fantastic discovery crucial for US national security (it's happened before). Well, the Congress is smart enough to want some evaluation, grades, measures, of the work funded. In simple terms, the evaluation is to count prizes and papers in peer-reviewed journals. Net, the big bucks say that the emphasis is on the research. It's not on the teaching. It's not on writing books that clean up rough edges in old research. E.g., no one questions the basic correctness of Maxwell's equations, Schrödinger's equation, the divergence theorem, and polishing that material will NOT get published in the best peer-reviewed journals and is NOT the results desired from the funding. If some student wants to polish the material, the library is just off the quadrangle, and they can go for it -- or now just use the Internet. But the emphasis is on the research. What Congress is paying for is the research that might be crucial for US national security (or medical research that might make them live another 10 years). What's wanted is the research, to be the best in the world in research, and polishing and teaching are not much of a part of that. Actually, Congress has zero curiosity about the research and couldn't care less about the research. All Congress wants is to have the US the world leader in anything that might yield another surprise for US national security such as The Bomb. If there were no competition from Russia, China, etc., the US Congress would cut back the research funding to pennies or less.

selimthegrim 5y ago

The reality is we cannot start freshmen on Courant-Hilbert. Even Apostol may be too much. If you have solutions I’m all ears. And for all its flaws FLP is still one of the best intros to complex numbers for laymen out there.

graycat 5y ago
> All ears Start with at least a good course in calculus. Then, for the divergence theorem, in one dimension that is essentially just the fundamental theorem of calculus on a closed interval on the real line. Then with multiple integrals can get a proof on any box, rectangular parallelepeped with edges parallel to the three orthogonal coordinate axes. If draw a picture and use the Jacobian for change of variable, can get a somewhat more general proof. That's enough for most courses in E&M, Maxwell's equations, and fluid flow. An okay source in PDF from Taiwan is in https://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/divergence%20theorem.pdf https://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/dive... Apostol does Jordan curves, and that's a bit much; it's a strange physics course that needs Jordan curves. The last edition of Rudin, Principles of Mathematical Analysis is a good source. With some good lectures to explain to the students just what is going on with compact sets, Rudin does not have to be too difficult or severe. Fleming in Functions of Several Variables does both the Lebesgue integral and exterior algebra -- at times it has been taught to ugrads. For the Lebesgue theory, I prefer either of Rudin, Real and Complex Analysis (just the first, real, half) and Royden, Real Analysis to Fleming. The Lebesgue integral does not have to be more difficult to teach or learn than the Riemann integral -- maybe the Lebesgue approach is easier than the Riemann approach where usually dig into compact sets. Here is the fishy part of improper integrals: Rudin's Principles discusses conditionally convergent series. Then it's just an exercise that a rearrangement can be made to converge to negative infinity, positive infinity, or anything between. Well, can generalize this to improper integrals: Partition the real line into intervals of length 1, (0, 1], (1, 2], .... Then start with a function defined from a conditionally convergent infinite series, say, each interval with integral the same as one of the terms of the series. Then in the limit in the improper integral, integrate over the intervals in whatever order want, not necessarily from -a to a and, thus, converge to anything might want. So, the value of an improper integral depends all on just what order in which add up the pieces -- so that's not a very good definition and stands to resist proofs of good properties. The Lebesgue theory gets around this issue. First, that theory just takes the given function and takes it apart into the positive part and the negative part. Second, it is easy, then, to define the integral of each part separately. If both parts have integral with absolute value infinity, then the integral of the function is not defined because it would require subtracting infinity from infinity which is not defined, even in the extended real number system. And that approach right away rules out the conditionally convergent issue. Otherwise the Lebesgue integral is defined. It also has good convergence properties, e.g., can prove a good version of Leibniz's rule. In grad school, the department suggested Courant and Hilbert. So, I got a copy: The content didn't seem to have much to do with the math I was studying or the physics I was interested in (I lost my copy in a move). For the Feynman Lectures, yes, I like most of it. I didn't like his treatment of the bra-ket notation, but then I don't like that notation anyway. The whole Lectures are available in PDF at https://feynmanlectures.caltech.edu/ https://feynmanlectures.caltech.edu/ The move also cost me my copy of Feynman's Lectures. There is something, maybe, US STEM field students should understand: In the US, there was The Bomb, the atomic bomb. That made a lot of people in Congress awake and afraid, and they decided that the US research universities should be the world leaders in research in the STEM fields. So, Congress appropriated a lot of money (NSF, NIH, DARPA, etc.) for US research universities and STEM field researchers. The universities like the money, e.g., commonly take 60% for "overhead" and, thus, fund the theater group, the string quartet, the English department, the art department, the new fountain in the quadrangle, the limo for the president, etc. But the STEM field profs and students also get funded. Since the members of Congress tend to be old, they also like funding medical research. For something like the Webb telescope, who's to say, that might lead to some fantastic discovery crucial for US national security (it's happened before). Well, the Congress is smart enough to want some evaluation, grades, measures, of the work funded. In simple terms, the evaluation is to count prizes and papers in peer-reviewed journals. Net, the big bucks say that the emphasis is on the research. It's not on the teaching. It's not on writing books that clean up rough edges in old research. E.g., no one questions the basic correctness of Maxwell's equations, Schrödinger's equation, the divergence theorem, and polishing that material will NOT get published in the best peer-reviewed journals and is NOT the results desired from the funding. If some student wants to polish the material, the library is just off the quadrangle, and they can go for it -- or now just use the Internet. But the emphasis is on the research. What Congress is paying for is the research that might be crucial for US national security (or medical research that might make them live another 10 years). What's wanted is the research, to be the best in the world in research, and polishing and teaching are not much of a part of that. Actually, Congress has zero curiosity about the research and couldn't care less about the research. All Congress wants is to have the US the world leader in anything that might yield another surprise for US national security such as The Bomb. If there were no competition from Russia, China, etc., the US Congress would cut back the research funding to pennies or less.