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[[File:Linmechfieldsfolds.jpg|thumb|alt=Linear algebra, Mechanics, Relativity and Fields, Differential Geometry|The starter pack to physics and differential geometry.]]

The starter pack to physics and differential geometry.

== Philosophy ==
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 [[Watch|video lectures]] concisely summarize various algebraic and geometric constructions that commonly appear in theoretical physics.

A [https://sheafification.com/the-fast-track/ 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 decalogy 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. Books in the top rows are generally more basic and can be read at the same time, giving important insight into the structure of the subjects.

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 ==
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = === Basic Mathematics ===
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
}}
{{BookListing
| cover = Apostol Calculus V1 Cover.jpg
| link = Calculus (Book)
| title = === Calculus ===
| desc = Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.
}}
</div>
== Landau ==
<div class="flex-container">
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| link = Mechanics (Book)
| title = === Mechanics ===
| desc = Classical mechanics of particles by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Mechmath.jpg
| link = Mechanics (Book)#Applications
| title = 
| desc = Symplectic geometry and other mathematical Structures of Classical Mechanics
}}
</div>
}}
{{BookListing
| cover = Landau Course in Theoretical Physics V2 Cover.jpg
| link = The Classical Theory of Fields (Book)
| title = === The Classical Theory of Fields ===
| desc = Classical field theory of electromagnetism and general relativity by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Fieldsmath3.jpg
| link = The Classical Theory of Fields (Book)#Applications
| title = 
| desc = Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
}}
</div>
}}
{{BookListing
| cover = Landau Quantum Mechanics.jpg
| link = Quantum Mechanics (Book)
| title = === Quantum Mechanics ===
| desc = Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Quantmathv3.jpg
| link = Quantum Mechanics (Book)#Applications
| title = 
| desc = Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics
}}
</div>
}}
{{BookListing
| cover = Landau 4 Quantum Electrodynamics cover.jpg
| link = Quantum Electrodynamics (Book)
| title = === Quantum Electrodynamics ===
| desc = Quantum Electrodynamics by Landau, written by Berestetskii, Lifshitz, and Pitaevskii.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = QFTmathv2.jpg
| link = Quantum Electrodynamics (Book)#Applications
| title = 
| desc = Axiomatic and Geometric Structures in Quantum Field Theory
}}
</div>
}}
{{BookListing
| cover = Landau statistical physics.jpg
| link = Statistical Physics (Book)
| title = === Statistical Physics ===
| desc = Statistical Physics by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Statmechmath.jpg
| link = Statistical Physics (Book)#Applications
| title = 
| desc = Symplectic statistical mechanics, lattice, and stochastic quantization methods
}}
</div>
}}
{{BookListing
| cover = Landau 6 fluid mechanics cover.jpg
| link = Fluid Mechanics (Book)
| title = === Fluid Mechanics ===
| desc = Fluid Mechanics by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Fluidsmath.jpg
| link = Fluid Mechanics (Book)#Applications
| title = 
| desc = Topological hydrodynamics, statistical methods of turbulence, cloud physics, geometric and tensor numerical analysis
}}
</div>
}}
{{BookListing
| cover = Landau 7 elasticity cover.jpg
| link = Theory of Elasticity (Book)
| title = === Theory of Elasticity ===
| desc = Theory of Elasticity by Landau and Lifshitz.
}}
{{BookListing
| cover = Landau 8 electrodynamics of continuous media cover.jpg
| link = Electrodynamics of Continuous Media (Book)
| title = === Electrodynamics of Continuous Media ===
| desc = Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.
}}
{{BookListing
| cover = Landau 9 statistical physics part 2 cover.jpg
| link = Statistical Physics part 2 - quantum theory (Book)
| title = === Statistical Physics part 2 ===
| desc = Statistical Physics part 2 by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Condensedmath.jpg
| link = Statistical Physics part 2 - quantum theory (Book)#Applications
| title = 
| desc = High Tc superconductivity, holography and string theory in condensed matter, emergent gauge fields/Chern-Simons, random matrix theory, the Quantum Hall Effect
}}
</div>
}}
{{BookListing
| cover = Landau 10 physical kinetics cover.jpg
| link = Physical Kinetics (Book)
| title = === Physical Kinetics ===
| desc = Physical Kinetics by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Kineticsmath.jpg
| link = Physical Kinetics (Book)#Applications
| title = 
| desc = Open quantum systems, stochastic differential equations, plasma physics, combustion, cloud physics, and stochastic processes on manifolds
}}
</div>
}}
</div>

== General Mathematics ==
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Sets for Mathematics (Book)
| title = === Sets for Mathematics ===
| desc = Categorical approach to set theory by F. William Lawvere.
}}
{{BookListing
| cover = Arnold Ordinary Differential Equations Cover.jpg
| link = Ordinary Differential Equations (Book)
| title = === Ordinary Differential Equations ===
| desc = Ordinary differential equations by Vladimir Arnold.
}}
{{BookListing
| cover = Ahlfors Complex Analysis Cover.jpg
| link = Complex Analysis (Book)
| title = === Complex Analysis ===
| desc = Complex analysis by Lars Ahlfors.
}}
{{BookListing
| cover = Olver Applications of Lie Groups to Differential Equations Cover.jpg
| link = Applications of Lie Groups to Differential Equations (Book)
| title = === Applications of Lie Groups to Differential Equations ===
| desc = Applications of Lie Groups to Differential Equations by Peter Olver.
}}
{{BookListing
| cover = Brown groupoids cover.jpg
| link = Topology and Groupoids (Book)
| title = === Topology and Groupoids ===
| desc = Topology and Groupoids by Ronald Brown. [https://groupoids.org.uk/pdffiles/topgrpds-e.pdf Click here for the Open Access version.]
}}
{{BookListing
| cover = Sternberg Differential Geometry Cover.jpg
| link = Lectures on Differential Geometry (Book)
| title = === Lectures on Differential Geometry ===
| desc = Differential geometry by Shlomo Sternberg.
}}
{{BookListing
| cover = Gelfand Generalized Functions vol 1 cover.png
| link = Generalized Functions (Book Series)
| title = === Generalized Functions: Properties and Operations ===
| desc = Generalized Functions: Properties and Operations by Israel Gel'fand and Georgi Shilov.
}}
{{BookListing
| cover = Gelfand Generalized Functions vol 2 cover.png
| link = Generalized Functions (Book Series)
| title = === Generalized Functions: Spaces of Fundamental and Generalized Functions ===
| desc = Generalized Functions: Spaces of Fundamental and Generalized Functions by Israel Gel'fand and Georgi Shilov.
}}
{{BookListing
| cover = Lang Algebra Cover.jpg
| link = Algebra (Book)
| title = === Algebra ===
| desc = Algebra by Serge Lang. The most direct approach to the subject.
}}
{{BookListing
| cover = Bott and Tu Differential Forms in Algebraic Topology.jpg
| link = Differential Forms in Algebraic Topology (Book)
| title = === Differential Forms in Algebraic Topology ===
| desc = Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.
}}
{{BookListing
| cover = Fulton-Harris Representation Theory cover.jpg
| link = Representation Theory (Book)
| title = === Representation Theory ===
| desc = Representation Theory by William Fulton and Joe Harris.
}}
{{BookListing
| cover = Manin algebra v cover.jpg
| link = Algebra V: Homological Algebra (Book)
| title = === Algebra V: Homological Algebra ===
| desc = Algebra V: Homological Algebra by Sergei Gelfand and Yuri Manin.
}}
{{BookListing
| cover = Hartshorne Algebraic Geometry cover.jpg
| link = Algebraic Geometry (Book)
| title = === Algebraic Geometry ===
| desc = Algebraic Geometry by Robin Hartshorne.
}}
{{BookListing
| cover = Manin number theory cover.jpg
| link = Introduction to Modern Number Theory (Book)
| title = === Introduction to Modern Number Theory ===
| desc = Introduction to Modern Number Theory by Yuri Manin and Alexei Panchishkin.
}}
{{BookListing
| cover = Vaisman Cohomology and Differential Forms Cover.jpg
| link = Cohomology & Differential Forms (Book)
| title = === Cohomology & Differential Forms ===
| desc = Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.
}}
{{BookListing
| cover = May A Concise Course in Algebraic Topology cover.jpg
| link = A Concise Course in Algebraic Topology (Book)
| title = === A Concise Course in Algebraic Topology ===
| desc = A Concise Course in Algebraic Topology by Peter May.
}}

</div>

== Aspirational ==
Here are some more awesome books.
=== Quantum Fields Beyond Landau ===
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Weinberg 1 quantum fields cover.jpg
| link = The Quantum Theory of Fields 1, Foundations (Book)
| title = === The Quantum Theory of Fields 1, Foundations ===
| desc = The Quantum Theory of Fields 1, Foundations by Steven Weinberg.
}}
{{BookListing
| cover = Weinberg 2 QFT gauge theory cover.jpg
| link = The Quantum Theory of Fields 2, Gauge Theory (Book)
| title = === The Quantum Theory of Fields 2, Gauge Theory ===
| desc = The Quantum Theory of Fields 2, Gauge Theory by Steven Weinberg.
}}
{{BookListing
| cover = Weinberg 3 QFT supersymmetry cover.jpg
| link = The Quantum Theory of Fields 3, Supersymmetry (Book)
| title = === The Quantum Theory of Fields 3, Supersymmetry ===
| desc = The Quantum Theory of Fields 3, Supersymmetry by Steven Weinberg.
}}
{{BookListing
| cover = Dewitt global qft 1 cover.jpg
| link = The Global Approach to Quantum Field Theory (Book Series)
| title = === The Global Approach to Quantum Field Theory ===
| desc = The Global Approach to Quantum Field Theory by Bryce DeWitt.
}}
{{BookListing
| cover = Connes Noncommutative Geometry, Quantum Fields and Motives cover.jpg
| link = Noncommutative Geometry, Quantum Fields and Motives (Book)
| title = === Noncommutative Geometry, Quantum Fields and Motives ===
| desc = Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli.
}}
{{BookListing
| cover = Nima grassmannian scattering cover.jpg
| link = Grassmannian Geometry of Scattering Amplitudes (Book)
| title = === Grassmannian Geometry of Scattering Amplitudes ===
| desc = Grassmannian Geometry of Scattering Amplitudes by Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka .
}}
</div>

=== Mathematics ===
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Brown NAT cover.jpg
| link = Nonabelian Algebraic Topology (Book)
| title = === Nonabelian Algebraic Topology ===
| desc = Nonabelian Algebraic Topology by Ronald Brown, Philip Higgins, and Rafael Sivera.
}}
{{BookListing
| cover = Kacvertex cover.jpg
| link = Vertex Algebras for Beginners (Book)
| title = === Vertex Algebras for Beginners ===
| desc = Vertex Algebras for Beginners by Victor Kac.
}}
{{BookListing
| cover = Frenkelvertex cover.jpg
| link = Vertex Algebras and Algebraic Curves (Book)
| title = === Vertex Algebras and Algebraic Curves ===
| desc = Vertex Algebras and Algebraic Curves by Edward Frenkel and David Ben-Zvi.
}}
{{BookListing
| cover = HHR Kervaire cover .jpg
| link = Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem (Book)
| title = === Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem ===
| desc = Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel.
}}
{{BookListing
| cover = Mcduff jholomorphic cover.jpg
| link = J-holomorphic Curves and Symplectic Topology (Book)
| title = === J-holomorphic Curves and Symplectic Topology ===
| desc = J-holomorphic Curves and Symplectic Topology By Dusa McDuff and Dietmar Salamon.
}}
{{BookListing
| cover = Fukaya intersection cover.jpg
| link = Lagrangian Intersection Floer Theory: Anomaly and Obstruction (Book Series)
| title = === Lagrangian Intersection Floer Theory: Anomaly and Obstruction ===
| desc = Lagrangian Intersection Floer Theory: Anomaly and Obstruction by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
}}
{{BookListing
| cover = Hermann Geometric Computing Science cover.jpg
| link = Geometric Computing Science (Book)
| title = === Geometric Computing Science ===
| desc = Geometric Computing Science, Interdisciplinary Mathematics XXV by [[Robert Hermann]].
}}
</div>

== 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)

{{SHORTDESC:The starter pack to physics and differential geometry.}}

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