From The Portal Wiki
Linear algebra, Mechanics, Relativity and Fields, Differential Geometry
The starter pack to physics and differential geometry.

The starter pack to physics and differential geometry.


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:

  • Later volumes of Landau on emergent/less fundamental physics refer to more engineering applications and numerical analysis (vols. 5-10)
  • The numerical analysis texts make an effort to discuss the geometry of tensor spaces, or preserve geometric structures in numerical integration
  • 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 additions and pages will make an effort to connect with more project-based and experimental content, as our goal of demonstrating computational aspects (PDEs, Representation Theory, Numerical Analysis) have been fully satisfied and the supplementary material nearly completed. 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. We make no claim to complete coverage, especially since one needs to actively engage in experiment and new theoretical calculations to get further into a topic and this is a life's work. We do try to present essential mathematics and promising new directions.

Related Lists

Fredric Schuller's video lectures concisely summarize various algebraic and geometric constructions that commonly appear in theoretical physics.

A related set of texts works with the same basics to lay a path through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory.

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

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.

Shilov Linear Algebra Cover.jpg

Linear Algebra

Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.

Apostol Calculus V1 Cover.jpg


Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.


Landau Course in Theoretical Physics V1 Cover.jpg


Classical mechanics of particles by Lev Landau.


Symplectic geometry and other mathematical Structures of Classical Mechanics

Landau Course in Theoretical Physics V2 Cover.jpg

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

Landau Quantum Mechanics.jpg

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

Landau 4 Quantum Electrodynamics cover.jpg

Quantum Electrodynamics

Quantum Electrodynamics by Landau, written by Berestetskii, Lifshitz, and Pitaevskii.


Axiomatic and Geometric Structures in Quantum Field Theory

Landau statistical physics.jpg

Statistical Physics

Statistical Physics by Landau and Lifshitz.


Symplectic statistical mechanics, lattice, and stochastic quantization methods

Landau 6 fluid mechanics cover.jpg

Fluid Mechanics

Fluid Mechanics by Landau and Lifshitz.


Topological hydrodynamics, statistical methods of turbulence, cloud physics, geometric and tensor numerical analysis

Landau 7 elasticity cover.jpg

Theory of Elasticity

Theory of Elasticity by Landau and Lifshitz.

Landau 8 electrodynamics of continuous media cover.jpg

Electrodynamics of Continuous Media

Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.

Landau 9 statistical physics part 2 cover.jpg

Statistical Physics part 2

Statistical Physics part 2 by Landau and Lifshitz.


High Tc superconductivity, holography and string theory in condensed matter, emergent gauge fields/Chern-Simons, random matrix theory, the Quantum Hall Effect

Landau 10 physical kinetics cover.jpg

Physical Kinetics

Physical Kinetics by Landau and Lifshitz.

General Mathematics

Lawvere Sets for Mathematics Cover.jpg

Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.

Arnold Ordinary Differential Equations Cover.jpg

Ordinary Differential Equations

Ordinary differential equations by Vladimir Arnold.

Ahlfors Complex Analysis Cover.jpg

Complex Analysis

Complex analysis by Lars Ahlfors.

Olver Applications of Lie Groups to Differential Equations Cover.jpg

Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations by Peter Olver.

Brown groupoids cover.jpg

Topology and Groupoids

Topology and Groupoids by Ronald Brown. Click here for the Open Access version.

Sternberg Differential Geometry Cover.jpg

Lectures on Differential Geometry

Differential geometry by Shlomo Sternberg.

Gelfand Generalized Functions vol 1 cover.png

Generalized Functions: Properties and Operations

Generalized Functions: Properties and Operations by Israel Gel'fand and Georgi Shilov.

Gelfand Generalized Functions vol 2 cover.png

Generalized Functions: Spaces of Fundamental and Generalized Functions

Generalized Functions: Spaces of Fundamental and Generalized Functions by Israel Gel'fand and Georgi Shilov.

Lang Algebra Cover.jpg


Algebra by Serge Lang. The most direct approach to the subject.

Bott and Tu Differential Forms in Algebraic Topology.jpg

Differential Forms in Algebraic Topology

Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.

Fulton-Harris Representation Theory cover.jpg

Representation Theory

Representation Theory by William Fulton and Joe Harris.

Hartshorne Algebraic Geometry cover.jpg

Algebraic Geometry

Algebraic Geometry by Robin Hartshorne.

Vaisman Cohomology and Differential Forms Cover.jpg

Cohomology & Differential Forms

Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.

May A Concise Course in Algebraic Topology cover.jpg

A Concise Course in Algebraic Topology

A Concise Course in Algebraic Topology by Peter May.


Here are some more awesome books.

Quantum Fields Beyond Landau

Dewitt global qft 1 cover.jpg

The Global Approach to Quantum Field Theory

The Global Approach to Quantum Field Theory by Bryce DeWitt.

Nima grassmannian scattering cover.jpg

Grassmannian Geometry of Scattering Amplitudes

Grassmannian Geometry of Scattering Amplitudes by Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka .


Hermann Geometric Computing Science cover.jpg

Geometric Computing Science

Geometric Computing Science, Interdisciplinary Mathematics XXV by Robert Hermann.

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)