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[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]
[[File:Linmechfieldsfolds.jpg|thumb|alt=Linear algebra, Mechanics, Relativity and Fields, Differential Geometry|The starter pack to physics and differential geometry.]]


This list of books provides the most direct and rigorous route to understanding differential geometry, the mathematical language of physics. Each selection thoroughly addresses its subject matter. The list does not need to be read linearly or only one book at a time. It is encouraged to go between books and/or read several together to acquire the necessary language and understand the motivations for each idea. The greatest hurdles are the motivation to learn and developing an understanding of the language of mathematics.
The starter pack to physics and differential geometry.


See the image on the right for a visual representation of its dependencies.
== 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:
* 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.  


Also see this [[Watch|list of video lectures]].
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.


== Related Lists ==


A further set of texts extending this one, but 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 [http://sheafification.com/the-fast-track/ here]
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 ==
== List Structure ==


The '''Royal Road to Differential Geometry and Physics''' is the list's core. While on that track, you should refer to the '''Fill in Gaps''' and '''Backbone''' sections as needed or as you desire.
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 '''Fill in Gaps''' section covers the knowledge acquired in a strong high school mathematics education. Refer to it as needed, or begin there to develop your core skills.
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 '''Backbone''' section supports the ideas in the '''Royal Road'''. Refer to it to strengthen your understanding of the ideas in the main track and to take those ideas further.
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.


== Fill in Gaps ==
== Basic Mathematics ==
<div class="flex-container" style="clear: both;">
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
Line 25: Line 35:
| title = === Basic Mathematics ===
| title = === Basic Mathematics ===
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
| 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
{{BookListing
Line 30: Line 46:
| link = Calculus (Book)
| link = Calculus (Book)
| title = === Calculus ===
| title = === Calculus ===
| desc = Overview of Calculus by Tom Apostol.
| desc = Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.
}}
}}
</div>
</div>
== Royal Road to Differential Geometry and Physics ==
== Landau ==
<div class="flex-container">
<div class="flex-container">
{{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.<br>
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Overview of linear algebra by Georgi Shilov.
}}
{{BookListing
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| link = Mechanics (Book)
| link = Mechanics (Book)
| title = === Mechanics ===
| title = === Mechanics ===
| desc = Classical mechanics of physics by Lev Landau.<br>
| desc = Classical mechanics of particles by Lev Landau.<br>
'''Prerequisite:'''
<div class="flex-container" style="clear: both;">
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
{{BookListing
'''Backbone reference:'''
| cover = Mechmath.jpg
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
| link = Mechanics (Book)#Applications
| title =
| desc = Symplectic geometry and other mathematical Structures of Classical Mechanics
}}
</div>
}}
}}
{{BookListing
{{BookListing
Line 64: Line 69:
| link = The Classical Theory of Fields (Book)
| link = The Classical Theory of Fields (Book)
| title = === The Classical Theory of Fields ===
| title = === The Classical Theory of Fields ===
| desc = Physics by Lev Landau.<br>
| desc = Classical field theory of electromagnetism and general relativity by Lev Landau.<br>
'''Prerequisite:'''
<div class="flex-container" style="clear: both;">
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
{{BookListing
| cover = Fieldsmath2.jpg
| link = The Classical Theory of Fields (Book)#Applications
| title =
| desc = Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
}}
</div>
}}
}}
{{BookListing
{{BookListing
| cover = Bishop Tensor Analysis Cover.jpg
| cover = Landau Quantum Mechanics.jpg
| link = Tensor Analysis on Manifolds (Book)
| link = Quantum Mechanics (Book)
| title = === Tensor Analysis on Manifolds ===
| title = === Quantum Mechanics ===
| desc = Tensor analysis by Richard Bishop and Samuel Goldberg.<br>
| desc = Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz<br>
'''Prerequisite:'''
<div class="flex-container" style="clear: both;">
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
{{BookListing
'''Backbone reference:'''
| cover = Quantmath.jpg
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
| link = Quantum Mechanics (Book)#Applications
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
| 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.
}}
}}
{{BookListing
{{BookListing
| cover = Sternberg Differential Geometry Cover.jpg
| cover = Landau statistical physics.jpg
| link = Lectures on Differential Geometry (Book)
| link = Statistical Physics (Book)
| title = === Lectures on Differential Geometry ===
| title = === Statistical Physics ===
| desc = Differential geometry by Shlomo Sternberg.<br>
| desc = Statistical Physics by Landau and Lifshitz.
'''Prerequisite:'''
}}
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
{{BookListing
'''Backbone reference:'''
| cover = Landau 6 fluid mechanics cover.jpg
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
| link = Fluid Mechanics (Book)
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
| title = === Fluid Mechanics ===
| desc = Fluid Mechanics by Landau and Lifshitz.
}}
{{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
{{BookListing
| cover = Vaisman Cohomology and Differential Forms Cover.jpg
| cover = Landau 8 electrodynamics of continuous media cover.jpg
| link = Cohomology & Differential Forms (Book)
| link = Electrodynamics of Continuous Media (Book)
| title = === Cohomology & Differential Forms ===
| title = === Electrodynamics of Continuous Media ===
| desc = Cohomology and differential forms by Isu Vaisman.<br>
| desc = Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Algebra: Chapter 0|Algebra: Chapter 0]]
* [[{{FULLPAGENAME}}#Algebra|Algebra]]
}}
}}
</div>
== Backbone ==
<div class="flex-container">
{{BookListing
{{BookListing
| cover = Kaplansky Set Theory and Metric Spaces Cover.jpg
| cover = Landau 9 statistical physics part 2 cover.jpg
| link = Set Theory and Metric Spaces (Book)
| link = Statistical Physics part 2 - quantum theory (Book)
| title = === Set Theory and Metric Spaces ===
| title = === Statistical Physics part 2 ===
| desc = Set theory and metric spaces by Irving Kaplansky.
| desc = Statistical Physics part 2 by Landau and Lifshitz.
}}
}}
{{BookListing
{{BookListing
| cover = E Landau Foundations of Analysis Cover.jpg
| cover = Landau 10 physical kinetics cover.jpg
| link = Foundations of Analysis (Book)
| link = Physical Kinetics (Book)
| title = === Foundations of Analysis ===
| title = === Physical Kinetics ===
| desc = Analysis, intro to numbers, by Edmund Landau.
| desc = Physical Kinetics by Landau and Lifshitz.
}}
}}
</div>
== General Mathematics ==
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Principles of Mathematical Analysis (Book)
| link = Sets for Mathematics (Book)
| title = === Principles of Mathematical Analysis ===
| title = === Sets for Mathematics ===
| desc = Mathematical analysis by Walter Rudin.
| desc = Categorical approach to set theory by F. William Lawvere.
}}
}}
{{BookListing
{{BookListing
Line 126: Line 150:
| title = === Ordinary Differential Equations ===
| title = === Ordinary Differential Equations ===
| desc = Ordinary differential equations by Vladimir Arnold.
| desc = Ordinary differential equations by Vladimir Arnold.
}}
{{BookListing
| cover = Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg
| link = Topology: A Categorical Approach (Book)
| title = === Topology: A Categorical Approach ===
| desc = Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. [https://topology.mitpress.mit.edu/ Click here for the Open Access version.]
}}
}}
{{BookListing
{{BookListing
Line 146: Line 164:
}}
}}
{{BookListing
{{BookListing
| cover = Aluffi Algebra Chapter 0 Cover.jpg
| cover = Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg
| link = Algebra Chapter 0 (Book)
| link = Topology: A Categorical Approach (Book)
| title = === Algebra Chapter 0 ===
| title = === Topology: A Categorical Approach ===
| desc = Algebra by Paolo Aluffi. Easier than Lang's, but less direct.
| desc = Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. [https://topology.mitpress.mit.edu/ 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
{{BookListing
Line 156: Line 192:
| title = === Algebra ===
| title = === Algebra ===
| desc = Algebra by Serge Lang. The most direct approach to the subject.
| 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 = Hartshorne Algebraic Geometry cover.jpg
| link = Algebraic Geometry (Book)
| title = === Algebraic Geometry ===
| desc = Algebraic Geometry by Robin Hartshorne.
}}
{{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>
</div>
=== Mathematics ===
<div class="flex-container" style="clear: both;">
{{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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