Free Online Calculus Text Books

I’ve spent years collecting free math resources, including a variety of calculus text books. Not the sketchy PDFs from torrent sites. Not the scanned copies with missing pages. Real Calculus textbooks that professors and universities have made freely available.

In this comprehensive guide, you will find a selection of calculus textbooks that cover everything from first-semester calculus through graduate-level stochastic analysis.

This list covers everything. First-semester calculus through graduate-level stochastic analysis. Every link works. Every book is legitimately free.

Calculus Books for a Solid Foundation

  1. Elementary Calculus: An Approach Using Infinitesimals by H.J. Keisler https://www.math.wisc.edu/~keisler/keislercalc-12-23-18.pdf
  2. Multivariable Calculus by Jim Herod and George Cain http://people.math.gatech.edu/~cain/notes/calculus.html
  3. Calculus by Gilbert Strang http://ocw.mit.edu/ans7870/textbooks/Strang/strangtext.htm
  4. Calculus Bible by Neveln http://www.cs.widener.edu/~neveln/Calcbible.pdf
  5. Lecture Notes for Applied Calculus by Karl Heinz Dovermann http://www.math.hawaii.edu/~heiner/calculus.pdf
  6. A Summary of Calculus by Karl Heinz Dovermann http://www.math.hawaii.edu/~heiner/short.pdf
  7. First Year Calculus Notes by Paul Garrett http://www.math.umn.edu/~garrett/calculus/
  8. The Calculus of Functions of Several Variables by Dan Sloughter https://tutorial.math.lamar.edu/classes/calciii/multivrblefcns.aspx
  9. Difference Equations to Differential Equations by Dan Sloughter http://math.furman.edu/~dcs/courses/math41/
  10. Visual Calculus by Lawrence S. Husch http://archives.math.utk.edu/visual.calculus/
  11. A Problem Text in Advanced Calculus by John Erdman http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf
  12. Understanding Calculus by Faraz Hussain https://understandingcalculus.com/
  13. Advanced Calculus by Lynn Loomis and Shlomo Sternberg http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
  14. The Calculus Wikibook http://upload.wikimedia.org/wikipedia/commons/a/a3/Calculus.pdf
  15. Vector Calculus http://www.mecmath.net/calc3book.pdf
  16. The Calculus for Engineers by John Perry http://books.google.com/books?id=FQE3AAAAMAAJ
  17. Calculus Unlimited by J.E. Marsden & A. Weinstein http://resolver.caltech.edu/CaltechBOOK:1981.001
  18. Advanced Calculus by E.B. Wilson https://archive.org/details/ost-math-advanced_calculus
  19. Differential and Integral Calculus by Daniel A. Murray http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=00870001;view=image;seq=6;page=root;size=100
  20. Elements of Differential and Integral Calculus by W.A. Granville & P.F. Smith http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf
  21. Calculus by Raja Almukkahal et al. https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp53791
  22. A Course of Pure Mathematics by G.H. Hardy http://archive.org/details/coursepuremath00hardrich
  23. Calculus Volumes I, II, III by Marsden and Weinstein http://authors.library.caltech.edu/25030/
  24. Calculus Early Transcendentals by David Guichard http://www.whitman.edu/mathematics/calculus/
  25. Calculus in Context by Callahan et al. http://www.e-booksdirectory.com/details.php?ebook=8262
  26. Introduction to Calculus Volumes 1 & 2 by J.H. Heinbockel http://www.math.odu.edu/~jhh/Volume-1.PDF
  27. Concepts in Calculus (1, 2, 3) by Miklos Bona & Sergei Shabanov https://people.clas.ufl.edu/kees/files/MAA4212Spring2016.pdf

Single-Variable Calculus

The bread and butter. Limits, derivatives, integrals, series. If you’re just starting out, grab one of these.

28. APEX Calculus (Volumes 1-3) by Gregory Hartman et al. https://www.apexcalculus.com/downloads

Three-semester coverage with interactive 3D graphics. AIM-approved. One of the best free options I’ve found.

29. Active Calculus (Single Variable) by Matthew Boelkins, David Austin, Steven Schlicker https://activecalculus.org/

Activity-driven approach with about 200 exercises. Grand Valley State uses this. CC BY-SA licensed, so you can adapt it.

30. CLP-1 Differential Calculus by Joel Feldman, Andrew Rechnitzer, Elyse Yeager https://www.math.ubc.ca/~CLP/

University of British Columbia’s rigorous treatment. Clean source files available if you want to customize.

31. CLP-2 Integral Calculus by Feldman, Rechnitzer, Yeager https://www.math.ubc.ca/~CLP/

Picks up where CLP-1 ends. Integration techniques, applications, sequences, series.

32. OpenStax Calculus Volume 1 by Gilbert Strang and Edwin Herman https://openstax.org/details/books/calculus-volume-1

The standard open textbook for Calc I. Functions, limits, derivatives, integration basics.

33. OpenStax Calculus Volume 2 by Strang and Herman https://openstax.org/details/books/calculus-volume-2

Integration techniques, differential equations, sequences, series, parametric equations.

34. Yet Another Calculus Text by Dan Sloughter https://synechism.org/wp/yet-another-calculus-text/

Uses hyperreal numbers and infinitesimals. Different approach than most. Worth exploring if the standard epsilon-delta approach never clicked for you.

35. Calculus for Team-Based Inquiry Learning by TBIL Institute Fellows https://teambasedinquirylearning.github.io/calculus/

Designed for group work. Good for study groups or tutoring situations.

Multivariable and Vector Calculus

Calc III territory. Multiple integrals, vector fields, Stokes’ theorem. Things get interesting here.

36. Active Calculus Multivariable by Steven Schlicker, David Austin, Matthew Boelkins https://activecalculus.org/acm/

Same activity-driven style as the single variable version. 3D graphics and embedded Sage cells.

37. OpenStax Calculus Volume 3 by Strang and Herman https://openstax.org/details/books/calculus-volume-3

Vectors, functions of several variables, multiple integration, vector calculus.

38. CLP-3 Multivariable Calculus by Feldman, Rechnitzer, Yeager https://www.math.ubc.ca/~CLP/

Vectors and geometry, partial derivatives, multivariable integrals.

39. CLP-4 Vector Calculus by Feldman, Rechnitzer, Yeager https://www.math.ubc.ca/~CLP/

Curves, vector fields, surface integrals, divergence theorem. Completes the UBC series.

40. Vector Calculus (Cambridge) by David Tong http://www.damtp.cam.ac.uk/user/tong/vc/vc.pdf

Cambridge undergraduate notes. Curves, surfaces, grad/div/curl, integral theorems. Concise and clear.

41. Vector Calculus for Engineers by Jeffrey R. Chasnov https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf

Engineering focus. If you need practical applications, this one delivers.

Differential Equations

ODEs and PDEs. The math that describes how things change over time. Essential for physics, engineering, and quantitative finance.

42. Elementary Differential Equations with Boundary Value Problems by William F. Trench http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_FREE_DIFFEQ_I.PDF

The standard undergraduate ODE text. Science and engineering applications throughout.

43. Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl https://www.jirka.org/diffyqs/

ODEs, Laplace transforms, Fourier series, intro to PDEs. Excellent modern text. I recommend this one often.

44. Ordinary Differential Equations and Dynamical Systems by Gerald Teschl https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf

Graduate level. Dynamical systems, stability theory, Sturm-Liouville problems. Vienna produces excellent math texts.

45. The Ordinary Differential Equations Project by Thomas W. Judson https://judsonbooks.org/ode-project/

Open source with embedded Sage cells. Interactive approach that works well for self-study.

46. Differential Equations (HKUST) by Jeffrey R. Chasnov https://www.math.hkust.edu.hk/~machas/differential-equations.pdf

First course lecture notes with linked YouTube videos. Good if you learn better from video.

47. Partial Differential Equations (Toronto) by Victor Ivrii https://www.math.utoronto.ca/ivrii/PDE-textbook/PDE-textbook.pdf

Heat, wave, and Laplace equations. Fourier methods. Comprehensive treatment.

48. Introduction to Partial Differential Equations (UNCW) by Russell Herman https://people.uncw.edu/hermanr/pde1/pdebook/PDE_Main.pdf

First-order PDEs, wave equations, heat equations, Green’s functions.

49. MIT OCW 18.03 Differential Equations by Arthur Mattuck, Haynes Miller, Jeremy Orloff https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/

Complete MIT course. Lectures, videos, problem sets, supplementary notes. Free MIT education.

50. Oxford Physics Lectures: Ordinary Differential Equations by Alexander Schekochihin http://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/ODE/2018/ODELectureNotes.pdf

Physics-oriented. Phase portraits, stability analysis. Different perspective than pure math texts.

51. Numerical Methods for Ordinary Differential Equations by Kees Vuik et al. https://open.umn.edu/opentextbooks/textbooks/numerical-methods-for-ordinary-differential-equations

Euler, Runge-Kutta, multistep methods. MATLAB examples included.

52. An Introduction to Partial Differential Equations (arXiv) by Per Kristen Jakobsen https://arxiv.org/abs/1901.03022

Master-class lecture notes. Covers both analytical and numerical approaches.

Real Analysis and Foundations

The rigorous underpinnings of calculus. Epsilon-delta proofs, measure theory, all the stuff that makes calculus actually work. Not for the faint of heart.

53. Basic Analysis: Introduction to Real Analysis (Vol I & II) by Jiří Lebl https://www.jirka.org/ra/

AIM-approved. Takes you from real numbers through metric spaces and Fourier series. One of my favorites in this category.

54. Introduction to Real Analysis by William F. Trench http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

Two-term course. AIM-approved. Real numbers through metric spaces.

55. Mathematical Analysis I by Elias Zakon (Trillia Group) http://www.trillia.com/zakon-analysisI.html

Award-winning text. Metric spaces, convergent sequences, compact sets. 500+ exercises.

56. Mathematical Analysis II by Elias Zakon (Trillia Group) http://www.trillia.com/zakon-analysisII.html

Graduate-level sequel. Measure theory, calculus on Banach spaces.

57. Measure, Integration & Real Analysis by Sheldon Axler https://measure.axler.net/MIRA.pdf

Springer Open Access. Lebesgue measure, Banach and Hilbert spaces, Fourier analysis. Axler writes clearly.

58. Basic Real Analysis by Anthony W. Knapp https://www.math.stonybrook.edu/~aknapp/download/b2-realanal-inside.pdf

840 pages. Graduate-level treatment. Real variable theory, Lebesgue measure, Fourier analysis. Comprehensive doesn’t begin to describe it.

59. An Introduction to Measure Theory by Terence Tao https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf

Tao’s clarity applied to Lebesgue measure and integration. If you’ve read any of his blog posts, you know he explains things well.

60. Elementary Real Analysis by Brian Thomson, Judith Bruckner, Andrew Bruckner https://www.classicalrealanalysis.com

Historical perspective. 13 chapters covering real numbers through integration.

61. How We Got from There to Here: A Story of Real Analysis by Eugene Boman & Robert Rogers https://milneopentextbooks.org/wp-content/uploads/2014/12/A_Story_of_Real_Analysis_ebook-pdf.pdf

Real analysis through historical development. Different approach. Worth reading even if you’ve studied analysis before.

62. An Introduction to Real Analysis (UC Davis) by John K. Hunter https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf

Clear introduction. Sequences, limits, continuity. Upper-division undergraduate level.

63. Introduction to Mathematical Analysis I by Lafferriere, Lafferriere, Nguyen https://open.umn.edu/opentextbooks/textbooks/introduction-to-mathematical-analysis-i-second-edition

Portland State University. Rigorous 10-week course foundation.

64. Measure and Integration (ETH Zürich) by Dietmar A. Salamon https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf

Graduate treatment. Abstract measure theory, Lebesgue integration, Lp spaces. ETH consistently produces excellent free materials.

Differential Forms and Manifolds

This is where calculus gets geometric. Stokes’ theorem generalized. If you’re heading toward differential geometry or mathematical physics, you’ll need this.

65. Differential Forms (MIT 18.952) by Victor Guillemin & Peter J. Haine https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf

Multilinear algebra, differential forms, manifolds, Stokes’ theorem, de Rham cohomology.

66. A Geometric Approach to Differential Forms by David Bachman https://faculty.washington.edu/seattle/physics544/2011-lectures/bachman.pdf

Emphasizes geometric intuition over algebraic formalism. Good entry point.

67. Discrete Differential Geometry: An Applied Introduction by Keenan Crane https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf

Exterior calculus, discrete exterior calculus, curvature, Hodge theory. Modern text with computer graphics applications.

68. Introduction to Differential Geometry by Joel W. Robbin & Dietmar A. Salamon https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf

Graduate notes. Manifolds, Levi-Civita connections, geodesics, curvature.

69. Introduction to Differential Forms by Donu Arapura https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf

Connects 1-forms, exact/closed forms, and Stokes’ theorem back to vector calculus you already know.

70. Analysis on Manifolds (Vienna) by Andreas Kriegl https://www.mat.univie.ac.at/~kriegl/Skripten/2018SSe.pdf

Graduate notes. Tangent bundles, vector fields, differential forms, integration on manifolds.

Stochastic Calculus

Calculus meets probability. If you’re going into quantitative finance, mathematical physics, or theoretical machine learning, this is essential.

71. Stochastic Calculus: An Introduction with Applications by Gregory F. Lawler https://www.math.uchicago.edu/~lawler/finbook.pdf

Graduate introduction. Martingales, Brownian motion, Itô calculus, Black-Scholes. Chicago quality.

72. Introduction to Stochastic Calculus (Melbourne) by Xi Geng https://researchers.ms.unimelb.edu.au/~xgge@unimelb/Files/Notes/An%20Introductory%20Course%20on%20Stochastic%20Calculus.pdf

Graduate course. Filtrations, Brownian motion, Itô’s formula, SDEs.

73. Stochastic Calculus, Filtering, and Stochastic Control by Ramon van Handel https://web.math.princeton.edu/~rvan/acm217/ACM217.pdf

Princeton advanced notes. Integrates stochastic calculus with filtering and control theory.

74. Introduction to Stochastic Calculus (Duke) by Andrea Agazzi https://sites.math.duke.edu/~agazzi/notesSDEv1.03.pdf

Graduate notes. Random walks, Brownian motion, Markov processes, SDEs.

Fractional Calculus

Non-integer order derivatives and integrals. Niche but increasingly important for modeling memory effects in physical systems.

75. Fractional Calculus and Special Functions by Francesco Mainardi & Rudolf Gorenflo https://appliedmath.brown.edu/sites/default/files/fractional/21%20Fractional%20Calculus%20and%20Special%20Functions.pdf

Liouville-Weyl, Riemann-Liouville, Grünwald-Letnikov approaches. Mittag-Leffler functions.

76. Construction & Physical Application of the Fractional Calculus by Nicholas Wheeler https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Fractional%20Calculus/A.%20Fractional%20Calculus.pdf

Physics-oriented. Abel’s tautochrone problem, fractal curves.

77. A Compact Introduction to Fractional Calculus (arXiv) by Alexander I. Zhmakin https://arxiv.org/pdf/2301.00037

Modern concise introduction. Fractional differential equations, recent developments.

Tensor Calculus

Indices, covariant derivatives, and the mathematical machinery behind general relativity.

78. Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters https://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

Self-contained. Index notation, covariant derivatives, metric tensors. Physics focus.

79. Introduction to Tensor Calculus and Continuum Mechanics by John H. Heinbockel http://www.math.odu.edu/~jhh/counter2.html

Applications in dynamics, elasticity, fluids, and electromagnetism.

80. Introduction to Tensor Calculus for General Relativity by Edmund Bertschinger https://web.mit.edu/edbert/GR/gr1.pdf

MIT. Tensors in curved spacetime. If you’re trying to learn GR, start here.

81. A Primer on Tensor Calculus by David A. Clarke https://people.iith.ac.in/ashok/Maths_Lectures/TutorialB/tprimer.pdf

Christoffel symbols, covariant differentiation, curvature.

Calculus of Variations

Finding functions that minimize or maximize integrals. The original optimization, before anyone called it that.

82. Calculus of Variations by I.M. Gelfand & S.V. Fomin http://users.uoa.gr/~pjioannou/mech2/READING/Gelfand_Fomin_Calculus_of_Variations.pdf

The classic. Euler equations, Hamilton-Jacobi theory, field theory. If you only read one book on this topic, make it this one.

83. The Calculus of Variations (Minnesota) by Jeff Calder https://www-users.cse.umn.edu/~jwcalder/CalculusOfVariations.pdf

Graduate notes. Euler-Lagrange equations, direct methods, Sobolev spaces.

84. Calculus of Variations Lecture Notes by Peter J. Olver https://www-users.cse.umn.edu/~olver/ln_/cv.pdf

Geodesics, brachistochrone, Euler-Lagrange with physics applications.

Integral Equations

When the unknown function appears under an integral sign. More common than you’d think.

85. Linear and Nonlinear Integral Equations by Abdul-Majid Wazwaz http://ndl.ethernet.edu.et/bitstream/123456789/72533/1/551.pdf

Volterra, Fredholm, singular, and integro-differential equations.

86. Integral Equations and their Applications by M. Rahman https://simkosal04.files.wordpress.com/2013/11/integral-equation-and-their-application.pdf

Applied textbook. Abel’s problem, Hilbert transforms, Fourier methods.

87. Handbook of Integral Equations by Polyanin & Manzhirov https://dl.icdst.org/pdfs/files/82792ab1a4d80d9fdc4f47bc3a93e116.pdf

Reference with over 2100 integral equations and their solutions. Massive. Keep it handy.

Numerical Methods

When you need a computer to do your calculus. Which is most of the time in practice.

88. Tea Time Numerical Analysis by Leon Brin https://lqbrin.github.io/tea-time-numerical/

One-semester textbook. Root finding, interpolation, numerical calculus, ODEs.

89. Numerical Methods with Applications by Autar K. Kaw et al. https://nm.mathforcollege.com/textbook-numerical-methods-with-applications/

STEM undergraduate focus. Differentiation, integration, ODEs, PDEs. Engineering examples.

90. Numerical Computing with MATLAB by Cleve B. Moler https://www.mathworks.com/moler/chapters.html

Written by the creator of MATLAB. Interpolation, quadrature, ODEs, Fourier analysis.

91. Numerical Algorithms (MIT) by Justin Solomon https://people.csail.mit.edu/jsolomon/share/book/numerical_book.pdf

Modern text. Linear algebra, optimization, integration, ODEs, PDEs. Computer science focus.

92. First Semester in Numerical Analysis with Julia by Giray Ökten https://open.umn.edu/opentextbooks/textbooks/710

Julia-based. Quadrature, differentiation, approximation.

93. First Semester in Numerical Analysis with Python by Yaning Liu & Giray Ökten https://open.umn.edu/opentextbooks/textbooks/925

Python-based. NumPy, SciPy, Matplotlib examples.

94. Numerical Methods for Engineers by Jeffrey R. Chasnov https://www.math.hkust.edu.hk/~machas/numerical-methods.pdf

Root-finding, linear systems, integration, Euler and Runge-Kutta.

95. Numerical Analysis (U. Chicago) by Ridgway Scott https://people.cs.uchicago.edu/~ridg/newna/nalrs.pdf

Advanced. Iterative methods, Chebyshev approximation, ODE discretization.

Functional Analysis

Infinite-dimensional vector spaces. Where analysis and algebra merge. Graduate level.

96. Functional Analysis (ETH Zürich) by Theo Bühler & Dietmar A. Salamon https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf

Graduate course. Banach/Hilbert spaces, Hahn-Banach, spectral theory, semigroups. 400+ pages of quality material.

97. Topics in Linear and Nonlinear Functional Analysis by Gerald Teschl https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf

Unbounded operators, spectral theory, fixed point theorems, Navier-Stokes applications.

98. Introduction to Functional Analysis (Sydney) by Daniel Daners https://www.maths.usyd.edu.au/u/athomas/FunctionalAnalysis/daners-functional-analysis-2017.pdf

Normed spaces, Lp spaces, bounded linear operators, inner product spaces.

Applied Calculus for Specific Fields

Calculus tailored for business, biology, and life sciences.

99. Applied Calculus by Shana Calaway, Dale Hoffman, David Lippman https://www.opentextbookstore.com/details.php?id=14

Business, social sciences, life sciences focus. CC BY licensed.

100. Calculus for the Life Sciences: A Modeling Approach by James L. Cornette & Ralph A. Ackerman https://open.umn.edu/opentextbooks/textbooks/92

Two volumes. Integrates mathematical modeling with genetics, ecology, epidemiology.

Bonus: Classic Public Domain Texts

Pre-1928 texts that are surprisingly good. Different era, same math.

TitleAuthorYearURL
Calculus Made EasyS.P. Thompson1910gutenberg.org/ebooks/33283
Introduction to Infinitesimal AnalysisVeblen & Lennes1907gutenberg.org/ebooks/18741
Elementary Illustrations of CalculusA. De Morgan1842gutenberg.org/ebooks/39041
Treatise on Differential CalculusI. Todhunter1860sarchive.org/details/atreatiseondiff04todhgoog
Treatise on Integral CalculusI. Todhunter1886archive.org/details/treatiseontheint017146mbp
Elementary Treatise on Integral CalculusB. Williamson1888archive.org/details/cu31924031264769
Differential Calculus for BeginnersJ. Edwards1890sarchive.org/details/differentialcal03edwagoog
First Course in Differential & Integral CalculusW.F. Osgood1907archive.org/details/cu31924031252053
Elementary Course of Infinitesimal CalculusH. Lamb1897archive.org/details/elementarycourse00lambuoft
Elements of Integral CalculusW.E. Byerly1880sarchive.org/details/elementsofintegr00byer_0
Differential and Integral CalculusA. De Morgan1836-42archive.org/details/differentialinte00demo_0

What to Do Next

If you’re just starting calculus, grab Active Calculus or OpenStax Volume 1.

If you want rigorous foundations, read Lebl’s Basic Analysis or Trench’s Real Analysis.

If you’re preparing for graduate school, work through Teschl’s ODE book and Axler’s Measure Theory.

If you need applied methods, Chasnov’s engineering-focused notes are practical and clear.

Download them all if you want. They’re free. Worst case, you have 100+ calculus textbooks sitting on a hard drive. Best case, you find the one that finally makes some concept click.

Disclaimer: My content is reader-supported, meaning that if you click on some of the links in my posts and make a purchase, I may earn a small commission at no extra cost to you. These affiliate links help me keep the content on gauravtiwari.org free and full of valuable insights. I only recommend products and services that I trust and believe will genuinely benefit you. Your support through these links is greatly appreciated—it helps me continue to create helpful content and resources for you. Thank you! ~ Gaurav Tiwari

Previous Post:

Next Post: