Difference between revisions of "User:Aardvark/Read"

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Pretty pictures version of [[Read]].
[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]


__NOTOC__
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.


== Books ==
== Fill in Gaps ==
=== Fill in Gaps ===
<div class="flex-container" style="clear: both;">
<br>
{{BookListing
<div class="iframe-container books" style="display: flex; flex-wrap: wrap; grid-gap: 1%; align-items: flex-start; justify-content: space-evenly;">
| cover = Lang Basic Mathematics Cover.jpg
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| link = Basic Mathematics (Book)
[[File:Lang Basic Mathematics Cover.jpg|center|300px]]
| title = === Basic Mathematics ===
==== Basic Mathematics ====
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
Review of arithmetic, basic algegra, trigonemetry, and geometry by Serge Lang.
}}
    </div>
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Apostol Calculus V1 Cover.jpg
[[File:Apostol Calculus V1 Cover.jpg|center|300px]]
| link = Calculus (Book)
==== Calculus ====
| title = === Calculus ===
Introduction to Calculus by Tom Apostol.
| desc = Overview of Calculus by Tom Apostol.
    </div>
}}
</div>
</div>
=== Road to Differential Geometry and Physics ===
== Royal Road to Differential Geometry and Physics ==
<br>
<div class="flex-container">
<div class="iframe-container books" style="display: flex; flex-wrap: wrap; grid-gap: 1%; align-items: flex-start; justify-content: space-evenly;">
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Lawvere Sets for Mathematics Cover.jpg
[[File:Lawvere Sets for Mathematics Cover.jpg|center|300px]]
| link = Sets for Mathematics (Book)
==== Sets for Mathematics ====
| title = === Sets for Mathematics ===
Categorical approach to set theory by F. William Lawvere. Backbone reference: Set Theory and Metric Spaces by Kaplansky, Foundations of Analysis by Edmund Landau.
| desc = Categorical approach to set theory by F. William Lawvere.<br>
    </div>
Backbone reference:
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
[[File:Shilov Linear Algebra Cover.jpg|center|300px]]
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
==== Linear Algebra ====
}}
Linear Algabra by Georgi Shilov.
{{BookListing
    </div>
| cover = Shilov Linear Algebra Cover.jpg
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| link = Linear Algebra (Book)
[[File:Landau Course in Theoretical Physics V1 Cover.jpg|center|300px]]
| title = === Linear Algebra ===
==== Mechanics ====
| desc = Overview of linear algebra by Georgi Shilov.
Physics by Lev Landau.
}}
    </div>
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
[[File:Landau Course in Theoretical Physics V2 Cover.jpg|center|300px]]
| link = Mechanics (Book)
==== The Classical Theory of Fields ====
| title = === Mechanics ===
Physics by Lev Landau.
| desc = Classical mechanics of physics by Lev Landau.<br>
    </div>
Prerequisite:
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
[[File:Bishop Tensor Analysis Cover.jpg|center|300px]]
Backbone reference:
==== Tensor Analysis on Manifolds ====
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
Tensor analysis by Richard Bishop and Samuel Goldberg.
}}
    </div>
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Landau Course in Theoretical Physics V2 Cover.jpg
[[File:Sternberg Differential Geometry Cover.jpg|center|300px]]
| link = The Classical Theory of Fields (Book)
==== Lectures on Differential Geometry ====
| title = === The Classical Theory of Fields ===
Differential geometry by Shlomo Sternberg.
| desc = Physics by Lev Landau.<br>
    </div>
Prerequisite:
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
[[File:Vaisman Cohomology and Differential Forms Cover.jpg|center|300px]]
}}
==== Cohomology & Differential Forms ====
{{BookListing
Cohomology and differential forms by Isu Vaisman.
| cover = Bishop Tensor Analysis Cover.jpg
    </div>
| 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>
</div>
 
== Backbone ==
=== Backbone ===
<div class="flex-container">
<div class="iframe-container books" style="display: flex; flex-wrap: wrap; grid-gap: 1%; align-items: flex-start; justify-content: space-evenly;">
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Kaplansky Set Theory and Metric Spaces Cover.jpg
[[File:Arnold Ordinary Differential Equations Cover.jpg|center|300px]]
| link = Set Theory and Metric Spaces (Book)
==== Ordinary Differential Equations ====
| title = === Set Theory and Metric Spaces ===
Ordinary differential equations by Vladimir Arnold.
| desc = Set theory and metric spaces by Irving Kaplansky.
    </div>
}}
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
{{BookListing
[[File:Kaplansky Set Theory and Metric Spaces Cover.jpg|center|300px]]
| cover = E Landau Foundations of Analysis Cover.jpg
==== Set Theory and Metric Spaces ====
| link = Foundations of Analysis (Book)
Set theory and metric spaces by Irving Kaplansky.
| title = === Foundations of Analysis ===
    </div>
| desc = Analysis, intro to numbers, by Edmund Landau.
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
}}
[[File:E Landau Foundations of Analysis Cover.jpg|center|300px]]
{{BookListing
==== Foundations of Analysis ====
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
Analysis, intro to numbers, by Edmund Landau.
| link = Principles of Mathematical Analysis (Book)
    </div>
| title = === Principles of Mathematical Analysis ===
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| desc = Mathematical analysis by Walter Rudin.
[[File:Rudin Principles of Mathematical Analysis Cover.jpg|center|300px]]
}}
==== Principles of Mathematical Analysis ====
{{BookListing
Mathematical analysis by Walter Rudin.
| cover = Arnold Ordinary Differential Equations Cover.jpg
    </div>
| link = Ordinary Differential Equations (Book)
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| title = === Ordinary Differential Equations ===
[[File:Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg|center|300px]]
| desc = Ordinary differential equations by Vladimir Arnold.
==== Topology: A Categorical Approach ====
}}
Mathematical analysis by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
{{BookListing
    </div>
| cover = Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| link = Topology: A Categorical Approach (Book)
[[File:Ahlfors Complex Analysis Cover.jpg|center|300px]]
| title = === Topology: A Categorical Approach ===
==== Complex Analysis ====
| desc = Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
Complex analysis by Lars Ahlfors.
}}
    </div>
{{BookListing
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
| cover = Ahlfors Complex Analysis Cover.jpg
[[File:Olver Applications of Lie Groups to Differential Equations Cover.jpg|center|300px]]
| link = Complex Analysis (Book)
==== Applications of Lie Groups to Differential Equations ====
| title = === Complex Analysis ===
Complex analysis by Peter Olver.
| desc = Complex analysis by Lars Ahlfors.
    </div>
}}
    <div class="iframe-item" style="flex: 1 1 25%; margin-bottom: 1em;">
{{BookListing
[[File:Aluffi Algebra Chapter 0 Cover.jpg|center|300px]]
| cover = Olver Applications of Lie Groups to Differential Equations Cover.jpg
==== Algebra Chapter 0 ====
| link = Applications of Lie Groups to Differential Equations (Book)
Complex analysis by Paolo Aluffi.
| title = === Applications of Lie Groups to Differential Equations ===
    </div>
| 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>
__NOTOC__

Latest 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.