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The starter pack to physics and differential geometry
Philosophy and Links
Our point of view is that the texts typically used in physics and especially mathematics degree tracks are window dressing for the real job of being a mathematical physicist or even an engineer. Excellent texts meet a certain standard we set here; Texts should be concise to respect the reader's time and occupations, interdisciplinary, at least relating mathematical tools between areas of mathematics:
- Lang's algebra text contains examples and applications in geometry and number theory throughout
- Vaisman emphasizes the typically algebro-geometric method of sheaves in a differential geometry setting and to develop the theory of multiple sorts of manifolds
- We choose physics texts which connect to heavy mathematical machinery such as curvature and covariant derivatives in general relativity or symplectic/variational geometry in mechanics
- The algebraic topology texts are not "pure" either - focusing on applications to differential or algebraic geometry, and many more.
Thus, the structure of this book list will be centered around core topics in theoretical physics which are already given direct connection to technology and reality, and the mathematics that follows from the theory rather than simply chasing popular formalisms. Future iterations will make an effort to connect with more computational content, such as that seen in representation theory or Olver's text on applications of Lie groups. This pertains to our last criterion that there should be some elementary aspects in a text - showing the translation of the abstract machinery into basic computations to make the relationships with other areas even more transparent.
Also see this list of video lectures, the lectures by Schuller concisely summarize various algebraic and geometric constructions commonly appearing in theoretical physics.
A related set of texts to this one, working with the same basics has been produced leading all the way up and through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory here.
List Structure
Calculus is not in the pictured starter pack because it is found more often in high school curricula, while linear algebra (despite being core to "applied mathematics" topics such as engineering, numerical computing, and statistics) is often missing at the required level of rigor. Thus, we suggest looking at any Basic Mathematics to quickly fill in your gaps and as a source of quick and dirty computational techniques used universally.
The texts by Landau are the list's core. While on that track, you should start dipping into the texts listed under the Landau volumes to enhance your perspective on repeated readings
The General Mathematics section covers the knowledge that would be acquired in standard (but basic) graduate courses on the different areas of mathematics that later develop into modern topics, and should be developed alongside Landau.
The Aspirational section contains some of the big ideas, which may be repeated from earlier but deserve emphasis. These are the triumphs of mathematics, peaks that everyone deserves to climb.
Basic Mathematics
Basic Mathematics
Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
Linear Algebra
Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
Calculus
Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.
Landau
Mechanics
Classical mechanics of particles by Lev Landau.
Symplectic geometry and other mathematical Structures of Classical Mechanics
The Classical Theory of Fields
Classical field theory of electromagnetism and general relativity by Lev Landau.
Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
Quantum Mechanics
Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz
Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics
Quantum Electrodynamics
Quantum Electrodynamics by Landau, written by Berestetskii, Lifshitz, and Pitaevskii.
Statistical Physics
Statistical Physics by Landau and Lifshitz.
Fluid Mechanics
Fluid Mechanics by Landau and Lifshitz.
Theory of Elasticity
Theory of Elasticity by Landau and Lifshitz.
Electrodynamics of Continuous Media
Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.
Statistical Mechanics part 2
Statistical Mechanics part 2 by Landau and Lifshitz.
Physical Kinetics
Physical Kinetics by Landau and Lifshitz.
General Mathematics
Sets for Mathematics
Categorical approach to set theory by F. William Lawvere.
Ordinary Differential Equations
Ordinary differential equations by Vladimir Arnold.
Complex Analysis
Complex analysis by Lars Ahlfors.
Applications of Lie Groups to Differential Equations
Applications of Lie Groups to Differential Equations by Peter Olver.
Topology: A Categorical Approach
Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. Click here for the Open Access version.
Lectures on Differential Geometry
Differential geometry by Shlomo Sternberg.
Generalized Functions: Properties and Operations
Generalized Functions: Properties and Operations by Israel Gel'fand and Georgi Shilov.
Generalized Functions: Spaces of Fundamental and Generalized Functions
Generalized Functions: Spaces of Fundamental and Generalized Functions by Israel Gel'fand and Georgi Shilov.
Algebra
Algebra by Serge Lang. The most direct approach to the subject.
Differential Forms in Algebraic Topology
Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.
Representation Theory
Representation Theory by William Fulton and Joe Harris.
Algebraic Geometry
Algebraic Geometry by Robin Hartshorne.
Cohomology & Differential Forms
Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.
A Concise Course in Algebraic Topology
A Concise Course in Algebraic Topology by Peter May.
Aspirational
Honorable Mentions
The following are some other good books, which are either redundant or otherwise didn't fit into the main collection of texts.(Olver PDEs, Coxeter books to be inserted)