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Other Maths textbooks for self-study

  1. Mar 28, 2015 #1
    I am a 1st-year physics student and over the summer am planning on self-studying some extra maths courses. I was considering geometry, statistics, and algebra as these seem to not require too many prerequisites and are useful in physics. Could anyone recommend some good textbooks/online resources for self study in these areas? I have taken intro courses in algebra and probability this year, and I have very little background in geometry. Thank you for any suggestions.
     
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  3. Mar 28, 2015 #2
    Last edited: May 7, 2017
  4. Mar 28, 2015 #3
    Thank you very much- several of these look really useful. In addition to these pure-maths textbooks, could anyone recommend any books on geometry and algebra written specifically for physics students, i.e. written in a less formal/rigourous style and directed especially towards the areas with physical applications?
     
  5. Mar 28, 2015 #4

    micromass

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    That is extremely vague. Geometry, statistics and algebra are huge fields. If you're somewhat more specific, then I can give recommendations.
     
  6. Mar 28, 2015 #5
    Sorry- I would like to study some 2nd-year courses that I have the prerequisites for but not the time/credits to study next year, so I am looking for material at a fairly introductory level. There are no recommended books and the lecture notes are not made public, but these are some of the main topics included according to the course webpages:

    Algebra:
    Abstract vector spaces, linear transformations, multilinear algebra of determinants, eigenvectors and eigenvalues, fields, rings and modules, quotients, isomorphism theorems, Sylow theorems, Cayley-Hamilton theorem, inner product spaces, spectral theorem, Jordan normal form, Galois groups.

    Geometry:
    Curves in Euclidean space, Frenet-Serret frame, curvature and torsion, vector fields, differential forms, Poincare’s lemma, connection forms, structure equations, surfaces, isometries, geodesics on surfaces, integration of forms, Stoke’s theorem, Gauss-Bonnet theorem, Euler characteristic

    Statistics:
    Random walks, stirling’s approximation, moment generating functions, fourier transform of probability distribution, central limit theorem, error function, least squares fitting, residuals, error analysis, Kolmogorov-Smirnov test
     
  7. Mar 28, 2015 #6

    micromass

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    There are no (beginner) books which cover all of these. You'll have to focus on either linear algebra which covers:
    Abstract vector spaces, linear transformations, multilinear algebra of determinants, eigenvectors and eigenvalues, Cayley-Hamilton theorem, inner product spaces, spectral theorem, Jordan normal form

    or abstract algebra which covers:
    fields, rings and modules, quotients, isomorphism theorems, Sylow theorems, Galois groups.

    You can do either of those. For linear algebra, I recommend highly the (free) book by Treil: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
    For abstract algebra, I recommend Anderson and Feil: https://www.amazon.com/First-Course-Abstract-Algebra-Edition/dp/1584885157

    What you want is (elementary) differential geometry. A good book is: https://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X

    It's a bit difficult to recommend a book for all of these topics. I think the best probability books require measure theory, which you likely don't know yet. Nevertheless, I think you'll find the classical books by Feller very good: https://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087
    Another good option would be the following online course: http://www.math.uah.edu/stat/ It is extremely good, but it doesn't quite cover many topics you listed.
     
  8. Mar 31, 2015 #7
    Thank you- that's really helpful!
     
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