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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.
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.  


See the image on the right for a visual representation of its dependencies.
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 [[Watch|list of video lectures]].


Also see this [[Watch|list of video lectures]], the lectures by Schuller concisely summarize various algebraic and geometric constructions commonly appearing in theoretical physics.


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].
 
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 [http://sheafification.com/the-fast-track/ here].


== 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 '''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 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 '''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 '''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.


== Fill in Gaps ==
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;">
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
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| 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
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}}
}}
</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 = Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
}}
{{BookListing
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
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| title = === Mechanics ===
| title = === Mechanics ===
| desc = Classical mechanics of particles 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)
| title = === Applications ===
| desc = Symplectic geometry and other mathematical Structures of Classical Mechanics
}}
</div>
}}
}}
{{BookListing
{{BookListing
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| title = === The Classical Theory of Fields ===
| title = === The Classical Theory of Fields ===
| desc = Classical field theory of electromagnetism and general relativity 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
{{BookListing
| cover = Bishop Tensor Analysis Cover.jpg
| cover = Fieldsmath.jpg
| link = Tensor Analysis on Manifolds (Book)
| link = The Classical Theory of Fields (Book)
| title = === Tensor Analysis on Manifolds ===
| title = === Applications ===
| desc = Tensor analysis by Richard Bishop and Samuel Goldberg.<br>
| desc = Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
'''Prerequisite:'''
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
}}
{{BookListing
| cover = Sternberg Differential Geometry Cover.jpg
| link = Lectures on Differential Geometry (Book)
| title = === Lectures on Differential Geometry ===
| desc = Differential geometry by Shlomo Sternberg.<br>
'''Prerequisite:'''
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
}}
{{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.<br>
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Algebra: Chapter 0|Algebra: Chapter 0]]
* [[{{FULLPAGENAME}}#Algebra|Algebra]]
}}
}}
</div>
</div>
== Backbone ==
<div class="flex-container">
{{BookListing
| cover = Kaplansky Set Theory and Metric Spaces Cover.jpg
| link = Set Theory and Metric Spaces (Book)
| title = === Set Theory and Metric Spaces ===
| desc = Set theory and metric spaces by Irving Kaplansky.
}}
}}
{{BookListing
{{BookListing
| cover = E Landau Foundations of Analysis Cover.jpg
| cover = Landau Quantum Mechanics.jpg
| link = Foundations of Analysis (Book)
| link = Quantum Mechanics (Book)
| title = === Foundations of Analysis ===
| title = === Quantum Mechanics ===
| desc = Analysis, intro to numbers, by Edmund Landau.
| desc = Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz<br>
}}
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
| cover = Quantmath.jpg
| link = Principles of Mathematical Analysis (Book)
| link = Quantum Mechanics (Book)
| title = === Principles of Mathematical Analysis ===
| title = === Applications ===
| desc = Mathematical analysis by Walter Rudin.
| desc = Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics
}}
}}
{{BookListing
</div>
| 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 = 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
| 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 = Aluffi Algebra Chapter 0 Cover.jpg
| link = Algebra Chapter 0 (Book)
| title = === Algebra Chapter 0 ===
| desc = Algebra by Paolo Aluffi. Easier than Lang's, but less direct.
}}
{{BookListing
| cover = Lang Algebra Cover.jpg
| link = Algebra (Book)
| title = === Algebra ===
| desc = Algebra by Serge Lang. The most direct approach to the subject.
}}
}}
</div>
</div>


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