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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.
Applications
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.
Applications
Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
Quantum Mechanics
Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz
Applications
Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics