Collection of Free Online Math Books and Lecture Notes (part 1)

[malawi_glenn]

School starts soon, and I know students are looking to get their textbooks at bargain prices

Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of these are pre-prints of published books) (some you can download as a pdf, some you have to read online)

There are many many more out there, and I have for sure not read all of the ones in this post. If you have read any of these, or have other suggestions, feel free to share it in this thread (The ones that I have read, I will write about at some point).

Note: I have not included video lecture series, or interactive sites, only stuff you can read like a textbook.

I might do another thread in the future for physics books...

Anyway, here you go, enjoy

Introductory Mathematics (logic, set theory, reading and writing proofs, algebra, number theory) “Mathematical reasoning” https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1024&context=books more info https://www.tedsundstrom.com/mathematical-reasoning-3 “Book of proof” https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf more info https://www.people.vcu.edu/~rhammack/BookOfProof/ “A gentle introduction to the art of mathematics” https://github.com/osj1961/giam/blob/master/GIAM.pdf?raw=true more info https://osj1961.github.io/giam/ “An introduction to mathematical reasoning” https://sites.math.washington.edu/~conroy/m300-general/ConroyTaggartIMR.pdf more info https://sites.math.washington.edu/~conroy/2019/m300-win2019/index.htm “Proofs and concepts - the fundamentals of abstract mathematics” https://batch.libretexts.org/print/Finished/math-23870/Full.pdf more info math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts “Elementary Foundations: An Introduction to Topics in Discrete Mathematics” https://batch.libretexts.org/print/Finished/math-83395/Full.pdf more info https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations “Introduction to modern set theory” https://www.people.vcu.edu/~clarson/roitman-set-theory.pdf

Calculus and Real analysis “Calculus Volume 1, 2, 3 (Openstax)” https://assets.openstax.org/oscms-prodcms/media/documents/CalculusVolume1-OP.pdf more info https://openstax.org/details/books/calculus-volume-1 https://assets.openstax.org/oscms-prodcms/media/documents/CalculusVolume2-OP.pdf more info https://openstax.org/details/books/calculus-volume-2 https://assets.openstax.org/oscms-prodcms/media/documents/CalculusVolume3-OP.pdf more info https://openstax.org/details/books/calculus-volume-3 “Calculus Volume 1, 2, 3 (Paul’s online notes)” https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx more info https://tutorial.math.lamar.edu/ “Introduction to real analysis” http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF more info http://ramanujan.math.trinity.edu/wtrench/misc/index.shtml “Basic analysis: Introduction to real analysis” Volume 1: https://www.jirka.org/ra/realanal.pdf Volume 2: https://www.jirka.org/ra/realanal2.pdf more info https://www.jirka.org/ra/ “Real analysis” http://mathonline.wikidot.com/real-analysis “Vector calculus for Engineers” https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf Even though the title says “for engineers” the stuff should be pretty good for for pure math and physics too! “Vector calculus (Corral)” https://math.libretexts.org/Bookshelves/Calculus/Book:_Vector_Calculus_(Corral) “CLP Calculus textbooks” https://personal.math.ubc.ca/~CLP/

Linear Algebra “A first course in linear algebra (Kuttler)” https://batch.libretexts.org/print/Finished/math-14495/Full.pdf more info math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler) “A first course in linear algebra (Beezer)” http://linear.pugetsound.edu/download/fcla-3.50-tablet.pdf more info http://linear.pugetsound.edu/ “Linear algebra (Hefferon)” https://joshua.smcvt.edu/linearalgebra/book.pdf more info https://joshua.smcvt.edu/linearalgebra/ “Linear algebra with applications (Nicholson)” https://lila1.lyryx.com/textbooks/OPEN_LAWA_1/marketing/Nicholson-OpenLAWA-2021A.pdf more info https://lyryx.com/linear-algebra-applications/ “Linear algebra done wrong” https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-04.pdf more info https://www.math.brown.edu/streil/papers/LADW/LADW.html Do not let the title discourage you. It is a hint to a very good and popular linear algebra book called “Linear Algebra done right” by Sheldon Axler. In that book, the author avoids the usage of determinants as much as possible. “Linear algebra (Math online)” http://mathonline.wikidot.com/linear-algebra “Introduction to Matrix algebra” http://autarkaw.com/books/matrixalgebra/index.html

Differential equations “Notes of Diffy Qs: Differential equations for engineers” https://www.jirka.org/diffyqs/diffyqs.pdf more info https://www.jirka.org/diffyqs/ Even though the title says “for engineers” the stuff should be pretty good for for pure math and physics too! “Differential equations (Paul’s online notes)” https://tutorial.math.lamar.edu/Classes/DE/DE.aspx “Elementary differential equations” https://digitalcommons.trinity.edu/cgi/viewcontent.cgi?article=1007&context=mono “Differential equations (Math online)” http://mathonline.wikidot.com/differential-equations "Ordinary Differential Equations and Dynamical Systems (Teschl)" https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf more info https://www.mat.univie.ac.at/~gerald/ftp/book-ode/

Complex analysis “A first course in complex analysis” http://math.sfsu.edu/beck/papers/complexorth.pdf more info https://matthbeck.github.io/complex.html “Guide to Cultivating Complex Analysis - Working the Complex Field” https://www.jirka.org/ca/ca.pdf more info https://www.jirka.org/ca/ “Complex analysis” http://mathonline.wikidot.com/complex-analysis

Abstract algebra “Abstract algebra: Theory and applications” http://abstract.ups.edu/download/aata-20220728.pdf more info http://abstract.ups.edu/ “Algebra: abstract and concrete” http://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/book.2.6.pdf more info http://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html “Elements of abstract and linear algebra” https://www.math.miami.edu/~ec/book/book.pdf more info https://www.math.miami.edu/~ec/book/ “Abstract algebra” http://mathonline.wikidot.com/abstract-algebra “Introduction to algebraic structures” https://batch.libretexts.org/print/Finished/math-666/Full.pdf more info https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures “Abstract algebra: the basic graduate year” https://faculty.math.illinois.edu/~r-ash/Algebra.html

Topology “Topology without tears” http://www.topologywithouttears.net/topbook.pdf more info https://www.topologywithouttears.net/ “Topology” http://mathonline.wikidot.com/topology

Measure theory and integration “Measure, Integration & Real Analysis” https://measure.axler.net/MIRA.pdf more info https://measure.axler.net/ “An introduction to measure theory” https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf more info https://terrytao.wordpress.com/books/an-introduction-to-measure-theory/ “Measure theory” http://mathonline.wikidot.com/measure-theory

Functional analysis “Lectures on functional analysis” http://facultymembers.sbu.ac.ir/shahrokhi/HasseOpeThe.pdf “Introduction to functional analysis (Daners)” https://www.maths.usyd.edu.au/u/athomas/FunctionalAnalysis/daners-functional-analysis-2017.pdf “Introduction to functional analysis (Kisil)” http://www1.maths.leeds.ac.uk/~kisilv/courses/math3263m.pdf “Functional analysis” http://mathonline.wikidot.com/functional-analysis

Differential geometry/ Mathematical Physics “Differential geometry in physics” http://people.uncw.edu/lugo/COURSES/DiffGeom/DG1.pdf more info http://people.uncw.edu/lugo/COURSES/DiffGeom/index.htm “Quick introduction to tensor analysis” https://arxiv.org/pdf/math/0403252.pdf “A geometrical approach to differential forms” https://maths.dur.ac.uk/users/mark.a.powell/Bachman_Geometric_Approach_to_Differential_Forms.pdf “A simple introduction to particle physics part II” https://arxiv.org/pdf/0908.1395.pdf part I is also worth to check out “Mathematics for Physics: An illustrated handbook” https://www.mathphysicsbook.com/intro/

 

[dextercioby]

Let there be known that links are valid as of 14_08_2022 and „link rots” are to be reported here, hopefully with the new valid links of the same documents/resources.

 

[MidgetDwarf]

Robert Ash has made books available for free download. I am only familiar with his basic probability book.

 

[malawi_glenn]

Robert Ash has made books available for free download. I am only familiar with his basic probability book.

Yeah I had one of his included in the OP, I have his "Real Variables with Basic Metric Space Topology" on my Read-list (gonna check it out after I am done with Axlers book on real analysis).

Anyway, here they are:

Mathbooks by Robert Ash: “Abstract algebra: the graduate year” Included in the OP "A course in algebraic number theory" https://faculty.math.illinois.edu/~r-ash/ANT.html "A course in commutative algebra" https://faculty.math.illinois.edu/~r-ash/ComAlg.html "Complex variables" (graduate course) https://faculty.math.illinois.edu/~r-ash/CV.html "Lectures on statistics" https://faculty.math.illinois.edu/~r-ash/Stat.html "Basic Probability theory" https://faculty.math.illinois.edu/~r-ash/BPT.html "Real Variables with Basic Metric Space Topology" https://faculty.math.illinois.edu/~r-ash/RV.html More info: https://faculty.math.illinois.edu/~r-ash/ Lecture notes by Robert Ash: "Differential equations" https://faculty.math.illinois.edu/~ash/DE.html "Linear Algebra" https://faculty.math.illinois.edu/~ash/LinearAlg.html "Discrete Math" https://faculty.math.illinois.edu/~ash/Discrete.html "Advanced Calculus" https://faculty.math.illinois.edu/~ash/AdvCalc.html More info: https://faculty.math.illinois.edu/~ash/

 

[Falgun]

Oxford has A TON OF lecture notes and problem sheets here:

https://courses.maths.ox.ac.uk/

 

[malawi_glenn]

Algebraic Topology (Hatcher)
https://pi.math.cornell.edu/~hatcher/AT/AT+.pdf
more info: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

 

[Muu9]

“Abstract algebra: Theory and applications”
http://abstract.ups.edu/download/aata-20210809.pdf
more info http://abstract.ups.edu/

Link rot. New link: http://abstract.ups.edu/download/aata-20220728.pdf

 

[malawi_glenn]

Link rot. New link: http://abstract.ups.edu/download/aata-20220728.pdf

Link works fine for me, but this is a newer edition so thanks for the heads up :)

 

[malawi_glenn]

Added https://www.mat.univie.ac.at/~gerald/ftp/book-ode/

 

[Office_Shredder]

I wanted to refresh my algebra skills. I'm about halfway through abstract algebra - the basic graduate year, and recommend it.

 

[WWGD]

This guy, Keith Conrad , from U of Connecticut, has some pretty great stuff.

https://kconrad.math.uconn.edu/blurbs/

 

[ShadowKraz]

Back in my youth I was a whiz with plane geometry and trig, not quite so much with algebra, less so with little white stones... um, calculus. I'm 62, trying to follow the maths in David Morin's "Special Relativity for the Enthusiastic Beginner", and have found I have forgotten a LOT of algebra and calculus. Can anyone recommend one or two good text books to help me get them back? If I can buy through PF, so much the better.
Please move this post if this is the wrong forum. Thank you.

 

[WWGD]

Back in my youth I was a whiz with plane geometry and trig, not quite so much with algebra, less so with little white stones... um, calculus. I'm 62, trying to follow the maths in David Morin's "Special Relativity for the Enthusiastic Beginner", and have found I have forgotten a LOT of algebra and calculus. Can anyone recommend one or two good text books to help me get them back? If I can buy through PF, so much the better.
Please move this post if this is the wrong forum. Thank you.

I'd suggest, if I may, separately, or jointly with other books, videos by Treffor Basset(Sp?)

 

[ShadowKraz]

I'd suggest, if I may, separately, or jointly with other books, videos by Treffor Basset(Sp?)

Thank you, I will check on those. I have some issues with video, however, I find them hard to 'rewind' properly when I miss something or don't understand. It's easier with a textbook to 'rewind', LOL. I will check them out as sometimes, the video author proves me wrong.

EDIT: Trefor Bazett. Obviously I found his videos.

 

[WWGD]

Give these a try

https://duckduckgo.com/?t=h_&q=trefor+bazett+youtube&iax=videos&ia=videos
Hope they help.

 

[WWGD]

If I may suggest, if you have a College, or otherwise ( Brick and Mortar) Math library within reach, spend a few hours there, browsing through the Algebra, Calculus, etc. books, see which feel right to you. As a rule of thumb, check out the Index, Notation sections, see if they are carefully-written, as a sign that the author wrote it more for the students than for themselves.

 

[mathwonk]

this is a hard question to answer with the little info you share.

I suggest lang's "short calculus"
https://www.amazon.com/Short-Calculus-Lang/dp/8181289749/?tag=pfamazon01-20

and suggest also that you give us some examples of things in your book that stump you and let us explain those directly.

about all there is to know about algebra is
1) how to manipulate (add, subtract, multiply and divide) polynomials,
2) the resulting Euclidean algorithm, and the theorems that
3) for any polynomial P, that c is a root if and only if (X-c) is a factor, and
4) if all coefficients of P are integers, then the only possible rational roots have form r/s where r divides the constant coefficient and s divides the leading coefficient.
5)If the coefficients are real numbers, then every polynomial factors into quadratic and linear factors, and
6) if P has odd degree then it has a real root.
7) oh yes, every real polynomial of degree d ≥1 has a complex root, hence factors completely into d linear factors with complex coefficients.
ok, also the quadratic formula,
8) that if x^2-bx + c = 0, has roots r,s, then x^2-bx + c = (x-r)(x-s), so b = r+s and c = rs, hence b^2-4c = (r-s)^2, so (r-s) = ± sqrt(b^2-4c), so (r+s) +(r-s) = 2r = b + sqrt(b^2-4c), and (r+s)-(r-s) = 2s = b - sqrt(b^2-4c).
I.e. r,s = (b ± sqrt(b^2-4c)/2. (note the coefficient of x is -b, not b.)

as to exponents, know that a^(r+s) = a^r.a^s, and (a^r)^s = a^(rs).

there are many algebra books, my favorite being Euler's but it is not short.