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I do math
I do math
[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]
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.
See the image on the right for a visual representation of its dependencies.
Also see this [[Watch|list of video lectures]].
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].
== 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.
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 '''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.
== Fill in Gaps ==
<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 = 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>
== Royal Road to Differential Geometry and Physics ==
<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
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| link = Mechanics (Book)
| title = === Mechanics ===
| desc = Classical mechanics of particles by Lev Landau.<br>
'''Prerequisite:'''
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
}}
{{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>
'''Prerequisite:'''
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
}}
{{BookListing
| cover = Bishop Tensor Analysis Cover.jpg
| link = Tensor Analysis on Manifolds (Book)
| title = === Tensor Analysis on Manifolds ===
| desc = Tensor analysis by Richard Bishop and Samuel Goldberg.<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 = 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>
== 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
| cover = E Landau Foundations of Analysis Cover.jpg
| link = Foundations of Analysis (Book)
| title = === Foundations of Analysis ===
| desc = Analysis, intro to numbers, by Edmund Landau.
}}
{{BookListing
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
| link = Principles of Mathematical Analysis (Book)
| title = === Principles of Mathematical Analysis ===
| desc = Mathematical analysis by Walter Rudin.
}}
{{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 = 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>
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