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i am a freshman physics student and naturally my curriculum includes math-classes. The thing is, that -at least for the time being- teachers cover only the surface of topics so as to have only a practical knowledge of math. However i understand the philosophy behind that strategy, i find mathematics too beautiful to confront them as only a box of tools for my physics. More over i believe that the deeper in understand mathematics the better theoretical physicist i will become.

So, to be more explicit this semester i studied linear algebra. Since my book covered as i said only the basics i naturally researched the internet. However, the deeper i got the more unknown topics and definitions poped up like tensors, permutation group,metric spaces etc some of them i later got an idea of .

So what i would really like to ask is, if there is a right sequence of topics to start learning deep mathematics that will eventually help me with physics as well.

For example: I know that calculus is like the bible to a physicist...so i would like to study for example the principia at some point because i would like to know the very basics of calculus as well.

Another Example:I really liked linear algebra and i know that a physicists makes use of it a lot as well. Should i have studied something else before linear algebra to make more sense to me? for example group theory etc?

Hope i am making some sense. Thanks in advance.

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  • $\begingroup$ At present, how well do you know the theoretical aspects of calculus and linear algebra? $\endgroup$ – David Feb 10 '16 at 5:58
  • $\begingroup$ Well, although i am good at them according to my grades i would still say that i find some parts of them pretty confusing! $\endgroup$ – George Smyridis Feb 10 '16 at 6:02
  • $\begingroup$ What do you find confusing? $\endgroup$ – David Feb 10 '16 at 6:05
  • $\begingroup$ For example determinants. $\endgroup$ – George Smyridis Feb 10 '16 at 6:06
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    $\begingroup$ I would like to express disagreement with thedude's comment, since Russell's book may not be the Principia you're talking about (maybe you meant Newton's), and at any rate it is a book about symbolic logic with deeply antiquated notation and I'm not sure symbolic logic should be high on your priority list. Also, I do not offhand see any reason why you would not learn the standard topics of mathematics from books directed at mathematics undergraduates. For calculus Spivak is said to be good and for linear algebra I like Lang. Others may disagree... $\endgroup$ – ForgotALot Feb 10 '16 at 6:07
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Depends what are your aims. Indeed I think that the best thing you could study as freshman in physics is basic group theory.

In Italy we do that in first year of Mathematics taking a two semester course called "Algebra", while often is not in a first year of Physics. Anyway modern Physics relies deeply on Algebra and so knowing a deeper notion of groups, field and some algebraic tecniques will turn out to be even more useful in nowdays Physics then plain calculus.

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In response to your question about calculus for physicists, I would suggest you study the book by Keisler entitled Elementary Calculus and freely available online here.

The book uses an intuitive approach exploiting infinitesimals and is particularly well-suited to a future physicist who is less concerned with epsilon-delta techniques that many math majors find confusing but which are considered a sine-qua-non for a well-rounded mathematics education. Since you seem to plan to study physics this may be ideally suited for you.

For more advanced applications of Robinson's framework to physics, you can see the book

Albeverio, Sergio; Høegh-Krohn, Raphael; Fenstad, Jens Erik; Lindstrøm, Tom. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986. xii+514 pp.

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  • $\begingroup$ I disagree with this idea. Physicists should study basic math in the same form mathematicians do, with epsilons. The only difference might be in terms of the applications that are emphasized. $\endgroup$ – David Feb 10 '16 at 17:58
  • $\begingroup$ @David, based on my acquaintance with the literature it is my impression that physicists use infinitesimals more often than mathematicians, and in fact have never stopped doing so since Leibniz, Euler, and Cauchy. $\endgroup$ – Mikhail Katz Feb 10 '16 at 18:12
  • $\begingroup$ Of course, that's correct, but the best of them do it with a rigorous understanding of the underlying math, when that is feasible. And epsilons are ultimately more widely applicable in rigorous math than are infinitesimals. Mathematical physicists even use concepts from algebraic topology, representation theory and algebraic geometry. Just try opening a book on these subjects if you don't have an understanding of the usual approach to analysis. $\endgroup$ – David Feb 10 '16 at 18:38
  • $\begingroup$ @David, you are mistaken if you think that Robinson's framework is limited to undergraduate mathematics. Thus, the ultrapower construction can be applied to any construct in algebraic topology, representation theory, and algebraic geometry. For examples from physics, just open Albeverio's book mentioned in my answer. If this is too technical for you, you can see some of my answers in nonstandard-analysis. $\endgroup$ – Mikhail Katz Feb 13 '16 at 21:12
  • $\begingroup$ See for example this answer: math.stackexchange.com/questions/405492/… $\endgroup$ – Mikhail Katz Feb 13 '16 at 21:30
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As one who started in physics major and ended up being in math, my experience is that math is much more difficult than physics at undergraduate level. That said, basic calculus and linear algebra education should be the same in either division. At this level, any textbook would pretty much serve well enough. As long as you get decent grades, you in a good shape. It may be worth mentioning that mathematics is not a subject learned via the old classics, unless you are interested in its history. It is more important to adopt the modern viewpoints.

The fundamentals of undergraduate math are analysis and algebra. Both will study abstract objects axiomatically. For a taste of these, my suggestion would be to read an advanced book on linear algebra, like the one by Hoffman and Kunze. In particular, you should study vector spaces very carefully. It is profound in nature, crucial for both math and physics, but relatively easy to learn.

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