books

Multivariable Calculus, Stewart
(Multivariable calculus.) The emphasis is not on theory, but rather on methods---read an example, solve ten similar problems, ad nauseam. Proofs are uncommon and often abstruse when present; little intuition is provided. On the bright side, there are a large number of examples and problems to work through. The chapters on vector calculus are particularly weak, and Schey's div, grad, curl, and all that may be a helpful companion. A much more concise resource is Appendix A of Griffiths' Introduction to Electrodynamics, which is very handy.

div, grad, curl, and all that, Schey
(Multivariable calculus.) A short book on the fundamental theorems of vector calculus. Schey conveys quite clearly the intuition and rationale behind these theorems, but I feel like the book could have been (even) more succinct.

Advanced Calculus: A Geometric View, Callahan (★)
(Multivariable calculus.) As its title would suggest, a geometric viewpoint is adopted throughout, and helpful figures abound in the text. The material is considerably more advanced than in Stewart; linear algebra is used liberally in the text, and general mathematical maturity is assumed. Callahan also defines area (with respect to double integrals) in terms of Jordan content, which is an interesting approach I have not seen elsewhere. Nevertheless, the geometric approach is highly visual and intuitive, and the book is useful even for beginners.

Understanding Analysis, Abbott (★)
(Real analysis.) The presentation of material in other real analysis books appear quite dull and uninteresting; I have not experienced the same feeling with Abbott's book. The material is introductory and the breadth of topics allegedly quite limited, though my limited knowledge of real analysis does not allow me to comment. I found the third chapter, on topology, quite difficult. (Some disjoint notes: 1, 2.)

Linear Algebra and its Applications, Lay
(Linear algebra.) Quite good. The presentation is clear, but the coverage of material is somewhat limited. Advanced topics like dual bases, Jordan normal form, generalized eigenvectors, and the spectral theorem are not covered. There are a large number of questions, which is helpful.

An Introduction to Ordinary Differential Equations, Coddington
(Ordinary differential equations.) Standard introduction. The scope of the book is limited, but the presentation is done well. Answers to most exercises are at the end of the book, which is nice.

Fundamentals of Differential Equations, Nagle, Saff and Snider
(Ordinary differential equations.) The emphasis is not on theory, but rather on methods. The material is an amalgam of the standard theory of ordinary differential equations and basic concepts in dynamical systems, integral transforms, and partial differential equations. Chapters 10 and 11, on partial differential equations, are dense and quite difficult to read.

Differential Equations with Applications and Historical Notes, Simmons (★)
(Ordinary differential equations.) Standard textbook with a similar scope as the directly preceding book. The star is for the word problems at the end of chapter 1, which are particularly good, and the introduction to the calculus of variations in chapter 13.

Symmetry Methods for Differential Equations, Hydon (★)
(Ordinary differential equations.) Applies group theory to solve differential equations. The standard tricks---scalings, Möbius transformations, integrating factors, inspired substitutions, etc.---used for solving differential equations are collected under and understood from a group-theoretic viewpoint, and this book is also a good, if short, introduction to Lie groups. An introductory differential equations background is sufficient; the book presents a self-contained treatment of groups and the like, though at one point it does assume knowledge of the method of characteristics for partial differential equations. The presentation is fairly geometric. (Some notes: 1.)

Chaos and Fractals, Feldman (★)
(Dynamical systems.) The level of math required is pitched very low---all that is necessary is an elementary understanding of functions; the book provides a self-contained introduction to complex numbers and differential equations. Feldman is able to explain and elucidate an amazing amount of complicated phenomena simply using iterative maps, and because so much of the material can be investigated using the same principles, the book is very easy to read.

Nonlinear Dynamics and Chaos, Strogatz (★)
(Dynamical systems.) The coverage is similar to the directly preceding book, though a higher level of mathematical sophistication is assumed---single-variable calculus, basic physics, and a sprinkling of multivariable calculus and linear algebra. Strogatz delves deeply into the mathematical concepts in dynamics and draws many fascinating examples from physics.

Complex Variables and Applications, Brown and Churchill (★)
(Complex analysis.) Standard introduction to complex analysis. The material is partitioned into many small sections, and Brown and Churchill do a good job motivating and presenting the material. The coverage is relatively standard but done very well, and the proofs are relatively simple to comprehend. I was able to go through this book without a background in any real analysis, and, though against conventional wisdom, I think studying complex analysis from this book would be a good way to bridge calculus and real analysis.

Visual Complex Analysis, Needham (★)
(Complex analysis.) Complex analysis from a geometric viewpoint. Complex-analytic functions are very "nice" mathematically, with rich geometric structure. Needham conveys this geometric insight excellently: two examples are the representation of complex differentiation as an "amplitwist"---a scaling combined with a rotation---and the geometric integration of $z^n$ over the unit circle for integral $n$, which gives insight as to why the integral should vanish for all but $n = -1$. Goes well with standard complex analysis books like that above; this book could work as a first introduction to the subject, but is better used once

Concrete Mathematics, Graham, Knuth, and Patashnik (★)
(Discrete mathematics.) A rather unique book with topics from continuous and discrete mathematics: sums, number theory, and generating functions. I found the first chapter a much harder read than the next few chapters. There is a fascinating discussion on discrete calculus, the generalization of calculus to discrete domains. Full solutions to problems (hard!) are provided at the end of the book.

Stochastic Processes and Models, Stirzaker (★)
(Probability and stochastic processes.) Simplified introduction to probability and stochastic processes that doesn't require a background in real analysis or measure theory. A lot of books purport to be introductory; I chose this one because it doesn't discuss $\sigma$-algebras in the context of probability spaces or filtrations in the context of martingales. This necessitates a loss of rigor, of course, and a considerable number of results are stated without proof or proven in weaker forms, but my goal was simply to survey the field and not be subject to a rigorous introduction. This book is appropriate for self-study, with solutions provided to a significant number of the harder problems and exercises. I found the end-of-chapter exercises considerably harder than the end-of-section problems. A more mathematically inclined reader might look at Grimmett and Stirzaker's Probability and Random Processes and its companion exercise book One Thousand Exercises in Probability. (Notes: 1. I spent some time elaborating on convergence of different types, which I felt was covered quite superficially, then realized that it was probably because that material was relatively unimportant.)
Generatingfunctionology, Wilf (★)
(Discrete mathematics.) Cool book on generating functions. Wilf's passion for generating functions is infectious, and the manipulations by which generating functions can be derived and put to use are quite fascinating. The first three chapters are of moderate difficulty, and then the remaining two get quite hard. Full solutions to problems are provided at the end of the book.

Introduction to Tensor Analysis, Grinfeld (★)
(Differential geometry and tensor analysis.) A gentle introduction to tensor analysis. Most other books I've tried reading swamp the reader in an orgy of indices right away; Grinfeld's book is more relaxed. The tensor notation is introduced gradually, and the various conventions neatly fall into place. Intuition is emphasized. As a bonus, Grinfeld's so-called Voss-Weyl formula allows for simple evaluation of the divergence in arbitrary coordinate systems, which dramatically cuts down the time it takes to derive, say, the divergence and Laplacian in spherical coordinates. A considerable part of the book is devoted to the calculus of moving surfaces, which is quite interesting. The book does have a number of errors, however, and the contravariant surface basis in chapter 12 does not seem to have been explicitly defined. (Some notes: 1. I derive representations for the covariant derivative for tensors of arbitrary order by introducing the tensor product and proving that the product rule of calculus holds for it. As far as I know, the proof is original (read: may contain errors).)

Differential Geometry, Kreyszig (★)
(Differential geometry and tensor analysis.) An introduction to differential geometry proper. Kreyszig writes very lucidly, and the book proceeds naturally from curves to surfaces to tensor analysis, though I was still uncomfortable with tensors and index manipulations before reading Grinfeld's book. The material is partitioned into small sections, which is convenient. Full solutions to problems are provided at the end of the book. Most are doable with results and information from each section, though there are some more unusual problems that require outside knowledge. (Some notes: 1.)

A Short Course in Differential Geometry and Topology
(Differential geometry and tensor analysis.) I haven't progressed far enough with this book to comment upon it, but I would like to highlight Figure 1.6, which shows a smooth mapping turning a giraffe into a hippopotamus.

Inside Interesting Integrals, Nahin (★)
(Integrals.) Fascinating book of tricks for solving integrals: symmetry, clever $u$-substitution, power-series solutions, reduction through recurrence relations, special functions, differentiation under the integral sign, contour integration. The author's interest in the subject is evident, and the book is a relatively easy read. A lot of the book is dedicated to working out quite complicated integrals, and it is worth the time to try solving a presented integral before reading about how the book does it.

Street-Fighting Mathematics, Mahajan (★)
(Mathematical reasoning in physics.) An analysis of problems through approximations, pictorial approaches, dimensional analysis, and analogies. The problems covered are not particularly hard, but are made much more obvious from a geometric viewpoint, and often succumb to simple approximations. This book describes less a bag of tricks and more a useful mindset for tackling problems, and is particularly helpful for beginning students.

The Mathematical Mechanic, Levi (★)
(Mathematical reasoning in physics.) Standard problems in mathematics are solved using physical principles, rather than the other way around, which is quite refreshing. The book is simple to pick up, and the physical analogues of mathematical problems quite didactic. For example, a physical proof of the $n^\text{th}$ roots of unity summing to zero for any integer $n$ can be visualized as an object in the center of a circle being pulled in $n$ different directions equally spaced about the circle; geometrically, we see that the forces are evenly distributed and that the object will not move, so the sum of the forces---the sum of the roots---must be zero.

Introduction to Classical Mechanics, Morin (★★)
(Analytical mechanics.) An amazing introductory textbook. Morin spends just enough time on expositions and the rest on problems, solutions, and exercises. The worked problems and solutions in each chapter are really very good, and there are also a large number of exercises for each chapter. Other introductory books lack the same emphasis on problem-solving and the quality and number of examples and problems presented. Tricky points are highlighted and discussed extensively---Morin has a section in the appendix for subtle issues with problems with varying mass, as well as a detailed discussion of physical models for falling chains. There is also a full solutions manual for the exercises floating around the Internet with instructive solutions for the exercises.

Classical Mechanics, Kibble and Berkshire (★)
(Analytical mechanics.) Standard development of classical mechanics, somewhat beyond an introductory book but not too advanced. The book is quite concise and interesting to read. Points of subtlety are often discussed clearly but not emphasized, so many passages have to be read multiple times. I did not lose interest in this book, as I did with many other analytical mechanics textbooks. There are both numerical and theoretical problems that span a good range of difficulty, and numerical answers (but not solutions) are provided at the end of the book. I especially liked chapter 6, on potential theory, and chapter 14's discussion of action-angle variables.

Mechanics, Landau and Lifshitz (★)
(Analytical mechanics.) This book is surprisingly thin for its remarkable insight. The development is quite concise and not particularly beginner-friendly---the first few pages present a variational approach to mechanics starting with Hamilton's principle of extremal action, and the lack of instruction regarding the mathematical manipulations of variations was quite frustrating the first few times I read it. Nevertheless, with a good foundation in classical mechanics, this is a wonderful reference book with much to present. I found section 10, on mechanical similarity, quite astounding, and the geometric development of two-body collisions in section 17 simplifies the topic considerably.

An Introduction to Thermal Physics, Schroeder (★)
(Thermodynamics and statistical physics.) An intuitive introduction from a physical perspective. The coverage is limited, but whatever is covered is done so well.

An Introduction to Statistical Mechanics, Hill (★)
(Thermodynamics and statistical physics.) A comprehensive introduction from a chemical perspective. After introducing the theory, the rest of the book essentially studies a wide range of chemical examples and applications. I found the first few chapters on parallels and connections between statistical mechanics and thermodynamics quite revealing.

Concepts in Thermal Physics, Blundell and Blundell (★★)
(Thermodynamics and statistical physics.) My favorite textbook. The material is divided into concise chapters with interesting end-of-chapter problems, and briefly covers a very wide range of supplementary material. I have not found their argument for deriving the Gibbs entropy formula presented elsewhere, and I was quite surprised to find non-equilibrium topics like Brownian motion and Onsager's reciprocal relations covered. Some of the problems were quite difficult my first time through, though careful perusal of the chapter provides sufficient information for solving them.

Introduction to Modern Statistical Mechanics, Chandler (★)
(Thermodynamics and statistical physics.) As stated, a modern perspective on statistical mechanics. The development is formal and centered around the second law, introduced as a postulate of maximum entropy. The topics covered are somewhat more advanced than those in the previous books, and the discussion of phase transitions was particularly insightful. On the other hand, this book is quite terse and important points are often left for the reader to discover.

Principles of Condensed Matter Physics, Chaikin and Lubensky (★)
(Thermodynamics and statistical physics.) This book discusses advanced topics in soft condensed matter physics. The mathematical maturity required is quite high---there are frequently places where the reader is expected to fill in a number of steps in a derivation. The book presents a summary of Fourier transforms and functional derivatives, but I feel that the treatment is too limited for a reader seeing these concepts for the first time. I've only read a small portion of the book, but examples from the literature abound, and the treatment is relatively comprehensive.

Introduction to Electrodynamics, Griffiths (★)
(Electricity and magnetism.) A standard introduction to electrostatics and electrodynamics. A background in multivariable calculus is recommended, though Griffiths does provide a concise self-contained treatment of the required mathematics. Griffiths' tone is rather conversational, which can be off-putting to some readers---I found his quantum mechanics book very annoying to read, but his electrodynamics book is a lot better. There is a strong emphasis on intuition and physical understanding, and Griffiths does a very good job of communicating important but subtle concepts to the reader. A complementary book is Purcell and Morin's Electricity and Magnetism, which is supposedly very good also, but I haven't read enough of it to comment. In addition, I really like Appendix A, which provides a simple way of obtaining the gradient, curl, and divergence in spherical and cylindrical coordinate systems. (Some notes: 1.)