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{{Grid2Col
 
| content =
This list of books provides the most direct and rigorous route to understanding differential geometry. 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 read several at a time to acquire the necessary language and understand the motivations for each idea.
{{CardBook
 
| img = Lang Basic Mathematics Cover.jpg
See the image on the right for a visual treatment of its dependencies.
 
The '''Royal Road to Differential Geometry and Physics''' is the focus of the list. While on that track, you should refer to the "Fill in Gaps" and "Backbone" sections as needed.
 
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.
 
The greatest hurdles are motivation and coming to understand the language of mathematics.
 
See also this list of videos.
 
== Fill in Gaps ==
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| link = Basic Mathematics (Book)
| title = === Basic Mathematics ===
| title = Basic Mathematics
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| followon =
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
}}
{{BookListing
{{CardBook
| cover = Apostol Calculus V1 Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Calculus (Book)
| link = Basic Mathematics (Book)
| title = === Calculus ===
| title = Basic Mathematics
| desc = Overview of Calculus by Tom Apostol.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| followon =
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{CardBook
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| followon =
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
</div>
{{CardBookFollowOnListItem
== Royal Road to Differential Geometry and Physics ==
| img = Lang Basic Mathematics Cover.jpg
<div class="flex-container">
| link = Basic Mathematics (Book)
{{BookListing
| title = Basic Mathematics
| cover = Lawvere Sets for Mathematics Cover.jpg
| authors = Serge Lang
| link = Sets for Mathematics (Book)
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| 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
{{CardBookFollowOnListItem
| cover = Shilov Linear Algebra Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Linear Algebra (Book)
| link = Basic Mathematics (Book)
| title = === Linear Algebra ===
| title = Basic Mathematics
| desc = Overview of linear algebra by Georgi Shilov.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| link = Mechanics (Book)
| title = === Mechanics ===
| desc = Classical mechanics of physics by Lev Landau.<br>
Prerequisite:
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
Backbone reference:
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
}}
}}
{{BookListing
{{CardBook
| cover = Landau Course in Theoretical Physics V2 Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = The Classical Theory of Fields (Book)
| link = Basic Mathematics (Book)
| title = === The Classical Theory of Fields ===
| title = Basic Mathematics
| desc = Physics by Lev Landau.<br>
| authors = Serge Lang
Prerequisite:
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
| followon =  
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
{{CardBookFollowOnListItem
| cover = Bishop Tensor Analysis Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Tensor Analysis on Manifolds (Book)
| link = Basic Mathematics (Book)
| title = === Tensor Analysis on Manifolds ===
| title = Basic Mathematics
| desc = Tensor analysis by Richard Bishop and Samuel Goldberg.<br>
| authors = Serge Lang
Prerequisite:
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
* [[{{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
{{CardBookFollowOnListItem
| cover = Sternberg Differential Geometry Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Lectures on Differential Geometry (Book)
| link = Basic Mathematics (Book)
| title = === Lectures on Differential Geometry ===
| title = Basic Mathematics
| desc = Differential geometry by Shlomo Sternberg.<br>
| authors = Serge Lang
Prerequisite:
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
* [[{{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.<br>
Backbone reference:
* [[{{FULLPAGENAME}}#Algebra: Chapter 0|Algebra: Chapter 0]]
* [[{{FULLPAGENAME}}#Algebra|Algebra]]
}}
}}
</div>
{{CardBook
== Backbone ==
| img = Lang Basic Mathematics Cover.jpg
<div class="flex-container">
| link = Basic Mathematics (Book)
{{BookListing
| title = Basic Mathematics
| cover = Kaplansky Set Theory and Metric Spaces Cover.jpg
| authors = Serge Lang
| link = Set Theory and Metric Spaces (Book)
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| title = === Set Theory and Metric Spaces ===
| followon =  
| desc = Set theory and metric spaces by Irving Kaplansky.
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
{{CardBookFollowOnListItem
| cover = E Landau Foundations of Analysis Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Foundations of Analysis (Book)
| link = Basic Mathematics (Book)
| title = === Foundations of Analysis ===
| title = Basic Mathematics
| desc = Analysis, intro to numbers, by Edmund Landau.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
{{CardBookFollowOnListItem
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Principles of Mathematical Analysis (Book)
| link = Basic Mathematics (Book)
| title = === Principles of Mathematical Analysis ===
| title = Basic Mathematics
| desc = Mathematical analysis by Walter Rudin.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{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
{{CardBook
| cover = Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Topology: A Categorical Approach (Book)
| link = Basic Mathematics (Book)
| title = === Topology: A Categorical Approach ===
| title = Basic Mathematics
| desc = Mathematical analysis by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
| followon =  
{{CardBookFollowOnListItem
| img = Lang Basic Mathematics Cover.jpg
| link = Basic Mathematics (Book)
| title = Basic Mathematics
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
{{CardBookFollowOnListItem
| cover = Ahlfors Complex Analysis Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Complex Analysis (Book)
| link = Basic Mathematics (Book)
| title = === Complex Analysis ===
| title = Basic Mathematics
| desc = Complex analysis by Lars Ahlfors.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{BookListing
{{CardBookFollowOnListItem
| cover = Olver Applications of Lie Groups to Differential Equations Cover.jpg
| img = Lang Basic Mathematics Cover.jpg
| link = Applications of Lie Groups to Differential Equations (Book)
| link = Basic Mathematics (Book)
| title = === Applications of Lie Groups to Differential Equations ===
| title = Basic Mathematics
| desc = Applications of Lie Groups to Differential Equations by Peter Olver.
| authors = Serge Lang
| description = Review of arithmetic, algebra, trigonometry, logic, and geometry
}}
}}
{{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>
<details>
__NOTOC__
<summary>Test</summary>
test
</details>

Latest revision as of 23:59, 28 November 2023

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Serge Lang

Review of arithmetic, algebra, trigonometry, logic, and geometry

Follow-on Books:
  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

  • Lang Basic Mathematics Cover.jpg

    Basic Mathematics

    Serge Lang

    Review of arithmetic, algebra, trigonometry, logic, and geometry

<details> <summary>Test</summary> test </details>

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