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21

There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether ...


7

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics". As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult ...


5

This is what I recommend to students learning analysis as a good companion: http://minds.wisconsin.edu/handle/1793/67009


5

Just to state my opinion: when you get to study functional analysis it is almost certainly that you are going into research, one way or another. At such level, "rich of complete, step by step, rigorous, and enlightening solutions" is probably not a good idea. When doing research, you won't find "complete, step by step, rigorous, and enlightening solutions" ...


3

This quickly got longer than I thought it would be, and I'm sure I inadvertently skipped something. If you have any questions, feel free to ask. Definition: (Lie Algebra) A Lie algebra is a real vector space $V$ equipped with an operation $$[\cdot,\cdot]:V\times V\to V,$$ called the bracket of the Lie algebra, such that $[\cdot,\cdot]$ is ...


3

You can check these books: http://www.amazon.com/Problems-Analysis-Second-Charalambos-Aliprantis/dp/0120502534/ref=sr_1_1?s=books&ie=UTF8&qid=1411894207&sr=1-1&keywords=problems+in+real+analysis And ...


3

Here is another bunch of texts. Like the ones suggested by Mathemagician1234, they are not general texts. The level of formality is variable. Classical mechanics F. Scheck, Mechanics, Springer, 2010. Although not specifically geared toward mathematicians, it makes use of mathematically advanced tools. I consider it the best book on classical mechanics ...


2

At Oxford we used http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710 for the first year of the Physics Course.


2

Try Principles of Physics by Resnick, Halliday and Walker


2

There is no conventional symbol for the set of radical expressions. If you find yourself needing one, just define one and be explicit about it.


2

Maybe you'll find the Princeton Companion to Mathematics helpful. It has some nice and accessible surveys of mathematics, both past and present.


2

Check these books: http://www.amazon.com/Problems-Variables-Lebesgue-Integration-Applications/dp/0070602212 http://www.amazon.com/Problems-Solutions-Undergraduate-Analysis-Mathematics/dp/0387982353/ref=sr_1_sc_1?s=books&ie=UTF8&qid=1411897062&sr=1-1-spell&keywords=Problems+and+Solutions+for+Undergraduate+Analysis+Rami+Shakarch ...


2

In terms of actual formal prerequisites for Hatcher's book, basic algebra and those notes on topology should mostly cover what you need. If you find that the notes aren't enough, it's most likely because you don't have enough practice using the theory that's presented there. The answer for that doesn't necessarily have to be another book on topology; if ...


2

The book Algèbre linéaire et Géométrie élémentaire by Jean Dieudonné is a highly formal presentation of elementary geometry using linear algebra. It's the way the Bourbaki school might have taught geometry in high school. Two books called Linear Algebra and Geometry, one by Kostrikin and Manin, the other by Shafarevich and Remizov, apply linear algebra to ...


2

You have the right ideas. Presumably, $F$ should be any subset of $\Bbb R^n$; what you're describing there is Carathéodory's criterion. Also, I think you have some extra words in you definition of Lebesgue measure; I would write $$ \lambda^{*}(E) = \inf\limits \{ ~\sum_{k=0}^{\infty} \text{vol}(Q_{k}):~\{Q_{k}\}_{k} \text{ are rectangles such that } E ...


2

Well, as far as your actual example is concerned, your proposed definition seems quite sensible to me. You are trying to translate the following definition into natural language (more or less): given groups $(G,*)$ and $(H,\diamond)$, $$ \mathbf{Grp}((G,*),\ (H,\diamond)):=\{((G,*),\ f,\ (H,\diamond)):\ (G,\ f,\ H)\in \mathbf{Set}(G,\ H)\ \wedge$$$$\wedge\ ...


2

Tikz is a popular drawing tool for $\LaTeX$. It is fairly dificult to get going, but the manual is very valuable, and there is also a tex SE site. With a lot of valuable information. What you can do with it is endless, but it will take time. Note that it is also possible to make simple figures/diagrams in programs such as maple, mathematica and the like and ...


1

I prefer sticking to the classical context for the first round of dealing with the spectral theorem; in particular, I would use Riemann-Stieltjes integrals instead of Borel measures. Once you have the Riemann-Stieltjes version, it is a fairly trivial matter to extend to the measure theoretic, when you want it. I highly recommend this text for self-study at ...


1

I think you'll also enjoy A Hilbert Space Problem Book by Paul Halmos. The Preface of this book is interesting, and the book is written in the Halmos style. Not much else needs to be said.


1

For the first part of your question, Volume 2 of Apostol's Calculus contains a full treatment of basic linear algebra immediately followed by chapters on differential equations. Hoffman and Kunze covers both topics you mentioned; eigenvalues are discussed in Chapter 6 and are called "characteristic values." I think the only problem will be if you find the ...


1

As Youngsu mentioned above, there is in general no relation between the primary components of $I$ and $J$, or the numbers $r$ and $s$ (and Youngsu has given the example of $I = m$ in a local ring for a case where $r < s$). For an example with $r > s$: take $R = k[x,y]$, $I = (x) \cap (x,y)^2 = (x^2,xy) \supseteq J = (x^2)$. On a positive note, there ...


1

I presume you must have already taken a basic course into calculus and analysis, so the book Analysis In Vector Spaces-A course in advanced Calculus can be a good place for you to start with. The chapters $3$ to $5$ which I have read were very well-explained ...


1

The following two papers might be useful, but note that their focus is not on finite sets. However, you might be able to find something more appropriate by searching for papers that cite one of these two papers. Edward O. Marczewski [Szpilrajn], Concerning the symmetric difference in the theory of sets and in Boolean algebras, Colloquium Mathematicum 1 #3 ...


1

From my side Thomas calculus is best book for calculus which will give rigorous explanations of differentiation using geometry. It starts from the basics of differentiation and goes on to advanced level like vector calculus. Not only geometrical approach but it also provides writing exercises. A hallmark of this book has been the application of calculus to ...


1

There is something to be said for Calculus in Context by Callahan et al. It emphasizes applications in the sciences. Students who will not use calculus in science or engineering courses that they take later should learn why calculus is important and considered a great achievement, rather than just learning to chant "n x to the n minus one", as in the ...


1

Looking through the slides you linked, it looks like they can all be done without working in coordinates. Do you perhaps have specific identities in mind for which you mean ' possible to argue without coordinates.' You mentioned Cartan's magic formula can be proved without working in local coordinates, and so can $\mathcal L_{[X,Y]}\alpha=\mathcal ...


1

Lemma 1.4.a in: J. M. F. Castillo and M. Gonzalez, Three-space problems in Banach space theory, Springer Lecture Notes in Math. 1667, 1997. Proof. Let $E$ be a Banach space and let $\{x_i\colon i\in I\}$ be a dense subset of the unit ball of $E$. Define a map $Q\colon \ell_1(I)\to E$ by $$Q ( (a_i)_{i\in I} ) = \sum_{i\in I} a_i x_i\quad (a_i)_{i\in ...


1

The group of unitary matrices is compact, and for any $\epsilon>0$, there are distinct integers $m$ and $n$ such that $\| U^m- U^n\| < \epsilon$. So $\|U^{m-n}-I\| <\epsilon$. This seems so simple that a reference would not be required.


1

For a different take on linear algebra, try Practical Linear Algebra, A Geometry Toolbox by Farin and Hansford. (reviews) For geometry, try Geometry: Euclid and Beyond by Hartshorne. (MAA review)


1

Try The Theoretical Minimum: What You Need to Know to Start Doing Physics and Quantum Mechanics: The Theoretical Minimum.



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