Free Online Calculus Text Books
Calculus textbooks are expensive. A new copy of Stewart’s Calculus runs over $250. Spivak’s is $80 used. And most students only use about 40% of the book before the semester ends. That’s a terrible deal.
The good news is you don’t have to pay for any of it. Not from torrent sites. Not scanned copies with missing pages. Real calculus textbooks that professors and universities have made legitimately free, covering everything from first-semester derivatives to graduate-level stochastic analysis.
I’ve spent years collecting these resources. This list has over 100 free calculus textbooks, every link works, and they’re organized by level so you can find exactly what you need without scrolling through titles you’ll never open.
Which Section to Start With
With 100+ books across 15 sections, you don’t need to read everything. You need to find the right shelf. Here’s a 4-path decision guide based on where you actually are.
First-semester student? Go straight to Single-Variable Calculus. Start with OpenStax Calculus Volume 1 (it’s free, peer-reviewed, and used at hundreds of colleges) or Active Calculus if you like activity-based learning. Don’t touch the Foundation section yet. It’s denser and will confuse more than it helps at that stage.
Engineer or scientist? You need calculus that connects to real problems. Head to Applied Calculus for Specific Fields for a domain-relevant entry point, then come back to Differential Equations and Numerical Methods. Chasnov’s HKUST series shows up in multiple sections here for good reason. It’s practical and clear.
Math major building rigor? Start with Real Analysis and Foundations. Lebl’s Basic Analysis or Trench’s Introduction to Real Analysis are the standard entry points. Once you’re comfortable with epsilon-delta proofs and metric spaces, the Calculus of Variations and Differential Forms sections become accessible.
Graduate student? You probably already know which topic you need. Jump directly to your section: Stochastic Calculus for quant finance or ML theory, Functional Analysis for PDEs and operator theory, Tensor Calculus for GR. The Foundation section also has some graduate-level material (Loomis-Sternberg, Marsden-Weinstein) worth knowing about.
Calculus Books for a Solid Foundation
This section is the grab-bag of the list. It pulls from multiple levels: some books here are true introductory texts (Keisler, Strang, Dovermann), others are upper-division undergraduate (Loomis-Sternberg, Erdman), and a few are classic historical texts. If you’re not sure where to start, scroll down to Single-Variable Calculus first. Come back here once you have a sense of what you’re looking for. That said, if you’re an advanced student wanting breadth, this section is worth a slow read.
- Elementary Calculus: An Approach Using Infinitesimals by H.J. Keisler Introductory https://www.math.wisc.edu/~keisler/keislercalc-12-23-18.pdf
- Multivariable Calculus by Jim Herod and George Cain Undergraduate http://people.math.gatech.edu/~cain/notes/calculus.html
- Calculus by Gilbert Strang Introductory http://ocw.mit.edu/ans7870/textbooks/Strang/strangtext.htm
- Calculus Bible by Neveln Introductory http://www.cs.widener.edu/~neveln/Calcbible.pdf
- Lecture Notes for Applied Calculus by Karl Heinz Dovermann Introductory http://www.math.hawaii.edu/~heiner/calculus.pdf
- A Summary of Calculus by Karl Heinz Dovermann Introductory http://www.math.hawaii.edu/~heiner/short.pdf
- First Year Calculus Notes by Paul Garrett Introductory http://www.math.umn.edu/~garrett/calculus/
- The Calculus of Functions of Several Variables by Dan Sloughter Undergraduate https://tutorial.math.lamar.edu/classes/calciii/multivrblefcns.aspx
- Difference Equations to Differential Equations by Dan Sloughter Undergraduate http://math.furman.edu/~dcs/courses/math41/
- Visual Calculus by Lawrence S. Husch Introductory http://archives.math.utk.edu/visual.calculus/
- A Problem Text in Advanced Calculus by John Erdman Undergraduate http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf
- Understanding Calculus by Faraz Hussain Introductory https://understandingcalculus.com/
- Advanced Calculus by Lynn Loomis and Shlomo Sternberg Graduate http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
- The Calculus Wikibook Introductory http://upload.wikimedia.org/wikipedia/commons/a/a3/Calculus.pdf
- Vector Calculus Undergraduate http://www.mecmath.net/calc3book.pdf
- The Calculus for Engineers by John Perry Introductory http://books.google.com/books?id=FQE3AAAAMAAJ
- Calculus Unlimited by J.E. Marsden & A. Weinstein Undergraduate http://resolver.caltech.edu/CaltechBOOK:1981.001
- Advanced Calculus by E.B. Wilson Undergraduate https://archive.org/details/ost-math-advanced_calculus
- Differential and Integral Calculus by Daniel A. Murray Introductory http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=00870001;view=image;seq=6;page=root;size=100
- Elements of Differential and Integral Calculus by W.A. Granville & P.F. Smith Introductory http://djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf
- Calculus by Raja Almukkahal et al. Introductory https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp53791
- A Course of Pure Mathematics by G.H. Hardy Undergraduate http://archive.org/details/coursepuremath00hardrich
- Calculus Volumes I, II, III by Marsden and Weinstein Undergraduate http://authors.library.caltech.edu/25030/
- Calculus Early Transcendentals by David Guichard Introductory http://www.whitman.edu/mathematics/calculus/
- Calculus in Context by Callahan et al. Undergraduate http://www.e-booksdirectory.com/details.php?ebook=8262
- Introduction to Calculus Volumes 1 & 2 by J.H. Heinbockel Introductory http://www.math.odu.edu/~jhh/Volume-1.PDF
- Concepts in Calculus (1, 2, 3) by Miklos Bona & Sergei Shabanov Undergraduate https://people.clas.ufl.edu/kees/files/MAA4212Spring2016.pdf
Single-Variable Calculus: Start Here if You’re a First-Semester Student
Limits, derivatives, integrals, series. This is the bread and butter. If you’re just starting out, one of these books is where you should be spending your time. All of them cover a standard three-semester sequence; the differences are in approach and presentation style.
28. APEX Calculus (Volumes 1-3) by Gregory Hartman et al. Introductory 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 Introductory 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 Introductory 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 Introductory 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 Introductory 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 Introductory https://openstax.org/details/books/calculus-volume-2
Integration techniques, differential equations, sequences, series, parametric equations.
34. Yet Another Calculus Text by Dan Sloughter Introductory 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 Introductory 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 Undergraduate 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 Undergraduate 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 Undergraduate https://www.math.ubc.ca/~CLP/
Vectors and geometry, partial derivatives, multivariable integrals.
39. CLP-4 Vector Calculus by Feldman, Rechnitzer, Yeager Undergraduate https://www.math.ubc.ca/~CLP/
Curves, vector fields, surface integrals, divergence theorem. Completes the UBC series.
40. Vector Calculus (Cambridge) by David Tong Undergraduate 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 Undergraduate 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 Undergraduate 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 Undergraduate 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 Graduate 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 Undergraduate 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 Graduate 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 Undergraduate 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 Undergraduate 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 Undergraduate 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. Undergraduate 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 Graduate https://arxiv.org/abs/1901.03022
Master-class lecture notes. Covers both analytical and numerical approaches.
Real Analysis and Foundations: Where Calculus Gets Honest
The rigorous underpinnings of calculus. Epsilon-delta proofs, measure theory, all the stuff that makes calculus actually work. Not for the faint of heart. If you want to do math seriously, at any level beyond straightforward computation, you need to spend time here. Start with Lebl’s Basic Analysis or Trench’s Introduction to Real Analysis. Both are AIM-approved, freely available, and widely used in undergraduate analysis courses. If you’re on a graduate track, Axler’s Measure, Integration & Real Analysis (Springer Open Access) is where things get serious.
53. Basic Analysis: Introduction to Real Analysis (Vol I & II) by Jiří Lebl Undergraduate 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 Undergraduate 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) Undergraduate 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) Graduate http://www.trillia.com/zakon-analysisII.html
Graduate-level sequel. Measure theory, calculus on Banach spaces.
57. Measure, Integration & Real Analysis by Sheldon Axler Graduate 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 Graduate 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 Graduate 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 Undergraduate 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 Undergraduate 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 Undergraduate 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 Undergraduate 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 Graduate 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 Graduate 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 Graduate 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 Research 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Research 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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 Research 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 Undergraduate 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. Undergraduate 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 Undergraduate 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 Graduate 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 Undergraduate 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 Undergraduate https://open.umn.edu/opentextbooks/textbooks/925
Python-based. NumPy, SciPy, Matplotlib examples.
94. Numerical Methods for Engineers by Jeffrey R. Chasnov Undergraduate 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 Graduate 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 Graduate 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 Graduate 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 Graduate 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: For Engineers, Biologists, and Business Students
Calculus tailored for business, biology, and life sciences. Less proof-heavy, more application-focused. If the standard textbook approach feels disconnected from what you actually do, start here.
99. Applied Calculus by Shana Calaway, Dale Hoffman, David Lippman Undergraduate 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 Undergraduate 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.
| Title | Author | Year | URL |
|---|---|---|---|
| Calculus Made Easy | S.P. Thompson | 1910 | gutenberg.org/my-books/33283 |
| Introduction to Infinitesimal Analysis | Veblen & Lennes | 1907 | gutenberg.org/my-books/18741 |
| Elementary Illustrations of Calculus | A. De Morgan | 1842 | gutenberg.org/my-books/39041 |
| Treatise on Differential Calculus | I. Todhunter | 1860s | archive.org/details/atreatiseondiff04todhgoog |
| Treatise on Integral Calculus | I. Todhunter | 1886 | archive.org/details/treatiseontheint017146mbp |
| Elementary Treatise on Integral Calculus | B. Williamson | 1888 | archive.org/details/cu31924031264769 |
| Differential Calculus for Beginners | J. Edwards | 1890s | archive.org/details/differentialcal03edwagoog |
| First Course in Differential & Integral Calculus | W.F. Osgood | 1907 | archive.org/details/cu31924031252053 |
| Elementary Course of Infinitesimal Calculus | H. Lamb | 1897 | archive.org/details/elementarycourse00lambuoft |
| Elements of Integral Calculus | W.E. Byerly | 1880s | archive.org/details/elementsofintegr00byer_0 |
| Differential and Integral Calculus | A. De Morgan | 1836-42 | archive.org/details/differentialinte00demo_0 |
What to Do Next: Reading Paths by Goal
Don’t just download everything and stare at a folder. Pick a path. Here are four reading progressions based on where you’re headed, with 3 books each to keep it manageable.
Beginner path (first calculus course):
- OpenStax Calculus Volume 1 by Strang & Herman. The standard open textbook. Clear, well-illustrated, widely adopted. Free at openstax.org. Start here.
- Active Calculus (Single Variable) by Boelkins, Austin, Schlicker. Once you’re comfortable reading, switch to this for the exercise-driven approach. It builds understanding, not just technique.
- APEX Calculus (Volumes 1-3) by Hartman et al. When you want to cover all three semesters in one place with proper depth. The 3D graphics for Calc III are genuinely helpful.
Engineer/scientist path:
- Differential Equations for Engineers (HKUST) by Chasnov. ODEs plus Laplace transforms, with YouTube video links for every topic. Start here rather than a pure math course.
- Vector Calculus for Engineers by Chasnov. The same clear style applied to Calc III. Real physical applications, not abstract geometry exercises.
- Numerical Methods for Engineers by Chasnov. Closes the loop: now you can solve the equations numerically when analytical solutions aren’t available. The whole Chasnov series is underrated.
Math major / rigorous path:
- Basic Analysis: Introduction to Real Analysis by Jiří Lebl. Two volumes, AIM-approved, freely maintained. This is where you learn what calculus actually is. Takes you from real numbers through metric spaces and Fourier series.
- Measure, Integration & Real Analysis by Sheldon Axler. Springer Open Access. Once Lebl feels comfortable, Axler takes you into Lebesgue measure, Banach spaces, and Fourier analysis. He writes unusually clearly for this level.
- Ordinary Differential Equations and Dynamical Systems by Gerald Teschl. Vienna university lecture notes. Bridges real analysis into dynamical systems, stability theory, and Sturm-Liouville problems. Graduate-level but rigorous and clean.
Graduate path (topic-specific):
- Stochastic Calculus: An Introduction with Applications by Gregory F. Lawler (Chicago). If you need Itô calculus for quant finance or ML theory, this is the cleaner entry point before heavier texts.
- Calculus of Variations by Gelfand & Fomin. The classic that’s been in print since 1963. Euler-Lagrange equations, Hamilton-Jacobi theory, field theory. If you only read one book in this section, it’s this one.
- Functional Analysis (ETH Zürich) by Bühler & Salamon. 400+ pages of Banach/Hilbert spaces, Hahn-Banach, spectral theory. ETH puts out excellent free material, and this course proves it.
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.