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[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]
[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]
<nowiki>**</nowiki>Fill in Gaps**:


Lang - Basic Mathematics
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.


Apostol - Calculus
See the image on the right for a visual treatment of its dependencies.


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.


<nowiki>**</nowiki>Royal Road To Differential Geometry and Physics**:
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.


Lawvere - Sets for Mathematics
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.
Shilov - Linear Algebra (and)


Lev Landau - courses in physics Volume 1+2 (then)
The greatest hurdles are motivation and coming to understand the language of mathematics.
Bishop - Tensor Analysis (or)\*           


Sternberg - Differential Geometry (then)\*\*
See also this list of videos.


Isu Vaisman - Cohomology and Differential Forms
== Fill in Gaps ==
 
<div class="flex-container" style="clear: both;">
 
{{BookListing
<nowiki>**</nowiki>Backbone**:
| cover = Lang Basic Mathematics Cover.jpg
 
| link = Basic Mathematics (Book)
Arnold - Ordinary Differential Equations
| title = === Basic Mathematics ===
 
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
Kaplansky - Set Theory and Metric Spaces (and)
}}
 
{{BookListing
Edmund Landau - Foundations of Analysis (then)
| cover = Apostol Calculus V1 Cover.jpg
Rudin - Principles of Mathematical Analysis
| link = Calculus (Book)
 
| title = === Calculus ===
Bradley, Bryson, Terrilla - Topology: A Categorical Approach
| desc = Overview of Calculus by Tom Apostol.
 
}}
Ahlfors - Complex Analysis
</div>
Olver - Applications of Lie Groups to Differential Equations
== Royal Road to Differential Geometry and Physics ==
 
<div class="flex-container">
Aluffi - Algebra Chapter 0 (or) \*
{{BookListing
 
| cover = Lawvere Sets for Mathematics Cover.jpg
Lang - Algebra\*\*
| link = Sets for Mathematics (Book)
 
| title = === Sets for Mathematics ===
 
| desc = Categorical approach to set theory by F. William Lawvere.<br>
\* = easier
Backbone reference:
 
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
\*\* = direct
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
 
}}
 
{{BookListing
<nowiki>**</nowiki>Good General Videos**:
| cover = Shilov Linear Algebra Cover.jpg
 
| link = Linear Algebra (Book)
https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic
| title = === Linear Algebra ===
 
| desc = Overview of linear algebra by Georgi Shilov.
notes:
}}
 
{{BookListing
https://drive.google.com/file/d/1nchF1fRGSY3R3rP1QmjUg7fe28tAS428/view
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
 
| link = Mechanics (Book)
 
| title = === Mechanics ===
https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ
| desc = Classical mechanics of physics 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 = Physics 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.<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.
}}
{{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>
__NOTOC__

Revision as of 16:12, 6 July 2021

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. 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.

See the image on the right for a visual treatment of its dependencies.

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.

The greatest hurdles are motivation and coming to understand the language of mathematics.

See also this list of videos.

Fill in Gaps

Lang Basic Mathematics Cover.jpg

Basic Mathematics

Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.

Apostol Calculus V1 Cover.jpg

Calculus

Overview of Calculus by Tom Apostol.

Royal Road to Differential Geometry and Physics

Lawvere Sets for Mathematics Cover.jpg

Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.
Backbone reference:

Shilov Linear Algebra Cover.jpg

Linear Algebra

Overview of linear algebra by Georgi Shilov.

Landau Course in Theoretical Physics V1 Cover.jpg

Mechanics

Classical mechanics of physics by Lev Landau.
Prerequisite:

Backbone reference:

Landau Course in Theoretical Physics V2 Cover.jpg

The Classical Theory of Fields

Physics by Lev Landau.
Prerequisite:

Bishop Tensor Analysis Cover.jpg

Tensor Analysis on Manifolds

Tensor analysis by Richard Bishop and Samuel Goldberg.
Prerequisite:

Backbone reference:

Sternberg Differential Geometry Cover.jpg

Lectures on Differential Geometry

Differential geometry by Shlomo Sternberg.
Prerequisite:

Backbone reference:

Vaisman Cohomology and Differential Forms Cover.jpg

Cohomology & Differential Forms

Cohomology and differential forms by Isu Vaisman.
Backbone reference:

Backbone

Kaplansky Set Theory and Metric Spaces Cover.jpg

Set Theory and Metric Spaces

Set theory and metric spaces by Irving Kaplansky.

E Landau Foundations of Analysis Cover.jpg

Foundations of Analysis

Analysis, intro to numbers, by Edmund Landau.

Rudin Principles of Mathematical Analysis Cover.jpg

Principles of Mathematical Analysis

Mathematical analysis by Walter Rudin.

Arnold Ordinary Differential Equations Cover.jpg

Ordinary Differential Equations

Ordinary differential equations by Vladimir Arnold.

Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg

Topology: A Categorical Approach

Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.

Ahlfors Complex Analysis Cover.jpg

Complex Analysis

Complex analysis by Lars Ahlfors.

Olver Applications of Lie Groups to Differential Equations Cover.jpg

Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations by Peter Olver.

Aluffi Algebra Chapter 0 Cover.jpg

Algebra Chapter 0

Algebra by Paolo Aluffi. Easier than Lang's, but less direct.

Lang Algebra Cover.jpg

Algebra

Algebra by Serge Lang. The most direct approach to the subject.

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