I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, Stokes' theorem, de Rham cohomology, Hodge star operator? Our text is A Comprehensive Introduction to Differential Geometry by Spivak, but I think this book is very difficult for a beginner to learn.  Thanks in advance.
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I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.  I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.  Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.   Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?  Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.  Suggestions about important theorems and concepts to learn, and book references, will be most helpful.
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I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that procede by  following a: "Definition, theorem, proof" style). In addition, the book that I'm looking for should be as much self consistent as possibile. I'm a first year graduate student in nuclear engineering. My prerequisites are a good understanding of (multivariate and vector) calculus,  linear algebra, and a little of functional analysis, Lebesgue integration theory, PDE. I know nothing about differential geometry, but to my (very) poor understanding  differential forms and concept like manifolds and so on are linked to each other.
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I am reading now E. Soper, Classical Theory Of Fields, now and sometimes it is very hard to follow the equations. So I need a side book on  to read it comfortably. Landau &amp; Lifshitz's book is not helping as its content and topics are very much different.
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I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)
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I have finished Carroll's general relativity book, Spacetime and Geometry. I am specifically interested in cosmology, so is there any book which goes more in depth into cosmology? I prefer a mathematically rigorous book, which goes beyond the cosmology chapter in Carroll's book. The Big Bang and inflation theory should be dealt with in depth.
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Hi my course specifically talks about :  Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one and two sheets and their properties.  This is merely indicative, and I would really like if the book talks about these (atleast) and some more. Would appreciate if it contained solved examples and exercise problems as well  Help Much Appreciated, Soham
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I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and eigenvectors and eigenvalues. I am looking for a book that handles every of the aforementioned topics in details. I also want to build a solid basis of the mathematical way of thinking to get ready to an exciting abstract algebra next semester, so my main aim is to work on proofs for somehow hard problems. I got Lang's "Intro. to Linear Algebra" and it is too easy, superficial.  Can you advise me a good book for all of the above? Please take into consideration that it is for self-study, so that it' gotta work on its own. Thanks.
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I'm struggling a bit with functions(limits, squeeze theorem, etc).  I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would appreciate if some experienced people here could advise me which books would be better to master the basics so I can move on with derivatives, etc. Limits is my main problem, so I need a book which explain this in depth so I can master it. Then I'll keep going on with the same book for other chapters.  Below is the list which I found :   Apostol T.M. Calculus Vol. 1 Burn R.P. Numbers and Functions: Steps into Analysis  Courant R. and John F. Introduction to Calculus and Analysis I - One of the better calculus text in print.  Crowell B. Calculus  Dawkins, P Calculus I and Calculus II  Garrett P. Calculus  Ghorpade S.R. and Limaye B.V. A Course in Calculus and Real Analysis  Gill G.S. Calculus Bible  Guichard D. Whitman Calculu.  Hardy G.H. A Course of Pure Mathematics  Hwang A.D. Calculus for Mathematicians, Computer Scientists, and Physicists: An Introduction to Abstract Mathematics  Santos D. Differential Calculus  Shapiro B.E. Calculus and Analysis  Spivak M. Calculus  Strang G. Calculus  Thomas G.B. and Finney R.L. Calculus and Analytic Geometry  Tranter C.J. Advanced Level Pure Mathematics  Tranter C.J. Techniques of Mathematical Analysis  Veeh J.A. Lectures Notes on Calculus  So far I narrowed my list down to Hardy, Spivak and Apostol based on further research, but I haven't got my hands on these books yet to compare them and I'm not sure if these books will be good for me.  Thanks.
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I'm a mathematics undergrad student who finished his first university year succesfully. For this post, it should be interesting to note that I already took a course in group theory.  I also had a course in linear algebra where we treated the following topics:  1) Vector spaces; basis; span; linear dependency; direct sum   2) Linear transformations, matrices, rank  3) Linear varieties, system of linear equations  4) Determinants  5) Eigenvalues and eigenvectors, diagonalisation, triangalisation, cayley-hamilton,  6) Euclidean spaces: inner products, norm, hermitian transformations, orthogonal basis, Gramm-Schmidt, orthogonal transformations  7) Prehilbert spaces  8) Isometries  9) Bilinear forms and quadrics in $2$ and $3$ dimensions  Now, I want to revise these topics, but it would be too boring to reread these books again, so now I look for a book that has all these topics, but also some additional requirements:   Focus on intuition (e.g. connections between linear algebra and geometry). This is a hard requirement for me, since there was no attention given to this in the course I took) Additional topics such as Jordan form of matrices, quotient vector spaces, ... Good exercises (preferred with solutions somewhere) The book must be rigorous   In the course I had, we mainly discussed finite-dimensional vector spaces, but I am also interested in a theory of infinite dimensional vector spaces.  Can someone hint me towards a good book that would suit me? If I have to add any information, please leave a comment and I will edit my post.  Thank you for your time.
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During the next quarter at uni, I'll be taking a course in real analysis and since I prefer studying with an additional text I thought I'd come here to look for some book recommendations.  My background: I'm comfortable with linear algebra and single variable calculus, but shakey on multivariable calculus. I did have a slight introduction in to geometry.  For linear algebra I found the common recommendation of Hoffman &amp; Kunze to be very helpful (although the abstractness came as a bit of a kicker at first).   For Single variable calculus I used the book Calculus by Adams and Essex, which I found frustrating to work with, because it often left me guessing at what they were trying to do or why they were doing it. I also felt that they exercised a certain lack of rigour. (Which may be due to the subject which would make it hard to derive from axioms.)  Details: Though it would be a complementary text, I'd prefer one which will still hold value as a work of reference at a later point in my studies. I've heard that Rudin's Principles of mathematical analysis is one of the better textst out there, but also read several comments discouraging Rudin's book as a first introduction to real analysis, hence my quest to gather more information and maybe get some more personalised recommendations.  The course: The mandatory course literature will consist of lecture notes, but 4 texts are suggested as recommended reading.   T.M. Apostol, Mathematical analysis. Addison-Wesley (1974) J. Dieudonné, Foundations of Modern Analysis. Academic Press (1960) A. van Rooij, Analyse voor Beginners.Epsilon Uitgaven, no. 6 (2003)   R.S. Strichartz, The way of analysis (1995)   Thanks in advance!
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I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra.  My motivation is that I (eventually) want to understand the theory underpinning    .  The problem is, I am at the  stage where I don't know what I don' t know. The  makes it clear that I lack several prerequisites in modern geometry.  So, what would be a proposed study path to get to this stage?  My background is electronic engineering and am comfortable with associated topics (signal processing, estimation theory, Kalman Filtering, etc.)
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What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
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Can someone give me a recommendation on homological algebra textbooks? I would like something that are accessible to a beginner (i.e., someone who have studied abstract algebra) and that have  1) an account on preadditive, additive, monoidal, abelian, triangulated categories, respectively,  2) motivations from algebraic topology exploited, (an elementary example that comes into my mind is the mapping cylinder/cone construction explained in contrast with Puppe sequence,)   3) an explanation on module theoretic topics like injective / projective resolutions,  4) a coverage of sheaf theory, cohomology of groups, and Galois cohomology.
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I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).  I have , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.  Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing.
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I'm planning on getting into research in mathematical physics and was wondering about interesting topics I can get into and possibly make some progress on.   I'm particularity fond of abstract algebra and topology and if possible any topics that involve abstract algebra would/topology/calculus of variations would be especially appreciated
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I am looking for a multivariable calculus book that is really physics oriented.  Anyone know of any?  EDIT: My wife is looking to brush up on multivariable at the same time she needs to brush up on the related physics.
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I'm a bachelor student majoring in math, and pretty interested in physics. I would like a book to study for classical mechanics, that will prepare me to work through . What books would be good for a beginner?