Pages
The calculus courses at MIT are 18.01 Single Variable Calculus and 18.02 Multivariable Calculus. Videos of those courses are on OpenCourseWare (OCW) along with lots of other useful materials. This site is about a completely separate calculus textbook by Gilbert Strang, and it will be helpful to viewers of OCW who would like to have online access to a textbook.
Professor Strang’s many contributions to OCW are mostly about linear algebra:
- 18.06 Linear Algebra,
- 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, and
- 18.085 Computational Science and Engineering I
Each of those courses has a full set of video lectures recorded on the MIT campus. Those videos have been watched by millions of viewers around the world (especially 18.06 Linear Algebra). Calculus and linear algebra are the two principal lead-ins to pure and applied mathematics.
Professor Strang’s Highlights of Calculus course on OCW is a series of short videos. They focus on the main idea of the subject, involving two functions: function 2 is the “derivative” of function 1, and function 1 is the “integral” of function 2. If you are given one of those functions, then this calculus textbook and the Highlights of Calculus videos show how to derive the other function. The heart of calculus is to use those functions to solve real problems, as described in Lecture 1: Big Picture of Calculus.
Below, Professor Strang shares some thoughts on the history of the calculus textbook:
For the textbook itself, its first printing was in 1991. That was a time of active “rethinking” of the course. I remember being in the audience at a discussion organized by the US National Science Foundation, about needed changes in typical calculus courses (to make them more relevant and interesting to students). Sitting at the back, I thought one necessary step would be a new textbook. That is the book you see here on OpenCourseWare, published by Wellesley-Cambridge Press.
Chapter 0 of the book came in a later edition. Its purposes were to add more figures to illustrate the key ideas of calculus, and also to develop in a new way the most important function in this subject. That function is the exponential \(e^x\). It has a very special feature, because it is both function 1 and function 2! In other words, the derivative of \(e^x\)is \(e^x\), and the integral of \(e^x\)is \(e^x\) (plus any constant C). This becomes the most important function in many applications, including the enormous field of “differential equations” (because it solves the most fundamental differential equation dy/dx = y).
I hope the book and the exercises and the videos and even Chapter 0 will be useful to you! Best wishes in all your work.
Gilbert Strang
Instructor’s Manual Components
ChapterS | FILES |
---|---|
1: Introduction to Calculus 1.1 Velocity and Distance 1.2 Calculus Without Limits 1.3 The Velocity at an Instant 1.4 Circular Motion 1.5 A Review of Trigonometry 1.6 A Thousand Points of Light |
Chapter 1 Manual (PDF) |
2: Derivatives 2.1 The Derivative of a Function 2.2 Powers and Polynomials 2.3 The Slope and the Tangent Line 2.4 Derivative of the Sine and Cosine 2.5 The Product and Quotient and Power Rules 2.6 Limits 2.7 Continuous Functions |
Chapter 2 Manual (PDF) |
3: Applications of the Derivative 3.1 Linear Approximation 3.2 Maximum and Minimum Problems 3.3 Second Derivatives: Bending and Acceleration 3.4 Graphs 3.5 Parabolas, Ellipses, and Hyperbolas 3.6 Iterations \(x_{n+1}=F(x_n)\) 3.7 Newton’s Method (and Chaos) 3.8 The Mean Value Theorem and 1’Hôpital’s Rule |
Chapter 3 Manual (PDF) |
4: Derivatives by the Chain Rule 4.1 The Chain Rule 4.2 Implicit Differentiation and Related Rates 4.3 Inverse Functions and Their Derivatives 4.4 Inverses of Trigonometric Functions |
Chapter 4 Manual (PDF) |
5: Integrals 5.1 The Idea of an Integral 5.2 Antiderivatives 5.3 Summation versus Integration 5.4 Indefinite Integrals and Substitutions 5.5 The Definite Integral 5.6 Properties of the Integral and Average Value 5.7 The Fundamental Theorem and Its Applications 5.8 Numerical Integration |
Chapter 5 Manual (PDF) |
6: Exponentials and Logarithms 6.1 An Overview 6.2 The Exponential \(e^x\) 6.3 Growth and Decay in Science and Economics 6.4 Logarithms 6.5 Separable Equations Including the Logistic Equation 6.6 Powers Instead of Exponentials 6.7 Hyperbolic Functions |
Chapter 6 Manual (PDF) |
7: Techniques of Integration 7.1 Integration by Parts 7.2 Trigonometric Integrals 7.3 Trigonometric Substitutions 7.4 Partial Fractions 7.5 Improper Integrals |
Chapter 7 Manual (PDF) |
8: Applications of the Integral 8.1 Areas and Volumes by Slices 8.2 Length of a Plane Curve 8.3 Area of a Surface of Revolution 8.4 Probability and Calculus 8.5 Masses and Moments 8.6 Force, Work, and Energy |
Chapter 8 Manual (PDF) |
9: Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 9.2 Polar Equations and Graphs 9.3 Slope, Length, and Area for Polar Curves 9.4 Complex Numbers |
Chapter 9 Manual (PDF) |
10: Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for \(e^x\), \(\sin{x}\), and \(\cos{x}\) 10.5 Power Series |
Chapter 10 Manual (PDF) |
11: Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra |
Chapter 11 Manual (PDF) |
12: Motion along a Curve 12.1 The Position Vector 12.2 Plane Motion: Projectiles and Cycloids 12.3 Curvature and Normal Vector 12.4 Polar Coordinates and Planetary Motion |
Chapter 12 Manual (PDF) |
13: Partial Derivatives 13.1 Surface and Level Curves 13.2 Partial Derivatives 13.3 Tangent Planes and Linear Approximations 13.4 Directional Derivatives and Gradients 13.5 The Chain Rule 13.6 Maxima, Minima, and Saddle Points 13.7 Constraints and Lagrange Multipliers |
Chapter 13 Manual (PDF) |
14: Multiple Integrals 14.1 Double Integrals 14.2 Changing to Better Coordinates 14.3 Triple Integrals 14.4 Cylindrical and Spherical Coordinates |
Chapter 14 Manual (PDF) |
15: Vector Calculus 15.1 Vector Fields 15.2 Line Integrals 15.3 Green’s Theorem 15.4 Surface Integrals 15.5 The Divergence Theorem 15.6 Stokes’ Theorem and the Curl of F |
Chapter 15 Manual (PDF) |
16: Mathematics after Calculus | Chapter 16 Manual (PDF) |
Professor Strang’s Related Courses on OCW
Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.
Professor Strang’s foundational course 18.06 Linear Algebra has long been one of the most popular courses on OCW. It has received more than 10 million visits since its first publication in 2002. Professor Strang also has a website dedicated to his linear algebra teaching. In Spring 2023, Professor Strang taught 18.06 Linear Algebra for the final time before retiring. The final class was live streamed and recorded.
A new version of Professor Strang’s classic Linear Algebra was released in 2011 in the innovative OCW Scholar format designed for independent learners. 18.06SC Linear Algebra includes 35 lecture videos and 36 short (and highly-praised) problem-solving help videos by teaching assistants.
Professor Strang has continued to offer new insights into key mathematics subjects. In 2014, he published the new textbook Differential Equations and Linear Algebra. In 2016, that textbook was developed into a series of 55 short videos supported by MathWorks, with parallel videos about numerical solutions by Dr. Cleve Moler, the creator of MATLAB®. The textbook and video lectures help students in a basic ordinary differential equations course. This new series, Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, is also available on the MathWorks website.
In 2017, Professor Strang launched a new undergraduate course at MIT: 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Published on the OCW site in 2019, the course uses linear algebra concepts for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization and, above all, a full explanation of deep learning.
Professor Raj Rao was the inspiration for the course 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, and co-taught it with Professor Strang in the first year it was offered at MIT. Since then, he has developed a very successful course on computational linear algebra.
Professor Strang recorded a new series of videos called A 2020 Vision of Linear Algebra. These six videos contain ideas and suggestions from Professor Strang about the recommended order of topics in teaching and learning linear algebra.
Professor Strang’s Related Textbooks, Notes, and Videos
Strang, Gilbert. Introduction to Linear Algebra. 5th ed. 2016. Wellesley-Cambridge Press. ISBN: 9780980232776.
Strang, Gilbert. Linear Algebra and Learning from Data. 2019. Wellesley-Cambridge Press. ISBN: 9780692196380.
Strang, Gilbert. Linear Algebra for Everyone. 2020. Wellesley-Cambridge Press. ISBN: 9781733146630.
Strang, Gilbert. Differential Equations and Linear Algebra. 2014. Wellesley-Cambridge Press. ISBN: 9780980232790.
ZoomNotes for Linear Algebra (PDF). Professor Strang created these notes in 2020 and 2021 when many MIT classes were moved online (using Zoom) due to the COVID-19 Pandemic. He hopes that faculty who are planning a linear algebra course and students who are reading for themselves will see these notes.
Guest speaker at the weekly OLSUME (Online Seminar on Undergraduate Mathematics Education) on the topic Linear Algebra and Deep Learning (MP4)
“Linear Algebra, Teaching, and MIT OpenCourseWare” (YouTube) on Lex Fridman Podcast
ChapterS | FILES |
---|---|
1: Introduction to Calculus 1.1 Velocity and Distance 1.2 Calculus Without Limits 1.3 The Velocity at an Instant 1.4 Circular Motion 1.5 A Review of Trigonometry 1.6 A Thousand Points of Light |
Chapter 1 Study Guide (PDF) |
2: Derivatives 2.1 The Derivative of a Function 2.2 Powers and Polynomials 2.3 The Slope and the Tangent Line 2.4 Derivative of the Sine and Cosine 2.5 The Product and Quotient and Power Rules 2.6 Limits 2.7 Continuous Functions |
Chapter 2 Study Guide (PDF) |
3: Applications of the Derivative 3.1 Linear Approximation 3.2 Maximum and Minimum Problems 3.3 Second Derivatives: Bending and Acceleration 3.4 Graphs 3.5 Parabolas, Ellipses, and Hyperbolas 3.6 Iterations \(x_{n+1}=F(x_n)\) 3.7 Newton’s Method (and Chaos) 3.8 The Mean Value Theorem and 1’Hôpital’s Rule |
Chapter 3 Study Guide (PDF) |
4: Derivatives by the Chain Rule 4.1 The Chain Rule 4.2 Implicit Differentiation and Related Rates 4.3 Inverse Functions and Their Derivatives 4.4 Inverses of Trigonometric Functions |
Chapter 4 Study Guide (PDF) |
5: Integrals 5.1 The Idea of an Integral 5.2 Antiderivatives 5.3 Summation versus Integration 5.4 Indefinite Integrals and Substitutions 5.5 The Definite Integral 5.6 Properties of the Integral and Average Value 5.7 The Fundamental Theorem and Its Applications 5.8 Numerical Integration |
Chapter 5 Study Guide (PDF) |
6: Exponentials and Logarithms 6.1 An Overview 6.2 The Exponential \(e^x\) 6.3 Growth and Decay in Science and Economics 6.4 Logarithms 6.5 Separable Equations Including the Logistic Equation 6.6 Powers Instead of Exponentials 6.7 Hyperbolic Functions |
Chapter 6 Study Guide (PDF) |
7: Techniques of Integration 7.1 Integration by Parts 7.2 Trigonometric Integrals 7.3 Trigonometric Substitutions 7.4 Partial Fractions 7.5 Improper Integrals |
Chapter 7 Study Guide (PDF) |
8: Applications of the Integral 8.1 Areas and Volumes by Slices 8.2 Length of a Plane Curve 8.3 Area of a Surface of Revolution 8.4 Probability and Calculus 8.5 Masses and Moments 8.6 Force, Work, and Energy |
Chapter 8 Study Guide (PDF) |
9: Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 9.2 Polar Equations and Graphs 9.3 Slope, Length, and Area for Polar Curves 9.4 Complex Numbers |
Chapter 9 Study Guide (PDF) |
10: Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for \(e^x\), \(\sin{x}\), and \(\cos{x}\) 10.5 Power Series |
Chapter 10 Study Guide (PDF) |
11: Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra |
Chapter 11 Study Guide (PDF) |
12: Motion along a Curve 12.1 The Position Vector 12.2 Plane Motion: Projectiles and Cycloids 12.3 Curvature and Normal Vector 12.4 Polar Coordinates and Planetary Motion |
Chapter 12 Study Guide (PDF) |
13: Partial Derivatives 13.1 Surface and Level Curves 13.2 Partial Derivatives 13.3 Tangent Planes and Linear Approximations 13.4 Directional Derivatives and Gradients 13.5 The Chain Rule 13.6 Maxima, Minima, and Saddle Points 13.7 Constraints and Lagrange Multipliers |
Chapter 13 Study Guide (PDF) |
14: Multiple Integrals 14.1 Double Integrals 14.2 Changing to Better Coordinates 14.3 Triple Integrals 14.4 Cylindrical and Spherical Coordinates |
Chapter 14 Study Guide (PDF) |
15: Vector Calculus 15.1 Vector Fields 15.2 Line Integrals 15.3 Green’s Theorem 15.4 Surface Integrals 15.5 The Divergence Theorem 15.6 Stokes’ Theorem and the Curl of F |
Chapter 15 Study Guide (PDF) |
First published in 1991 by Wellesley-Cambridge Press, this updated 3rd edition of the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor’s Manual and a student Study Guide.
The complete textbook (PDF) is also available as a single file.
![]() |
Highlights of Calculus MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. The videos, which include real-life examples to illustrate the concepts, are ideal for high school students, college students, and anyone interested in learning the basics of calculus. › Watch the videos |
Textbook Components
Chapter 0: Highlights of Calculus (PDF)
0.1 Distance and Speed // Height and Slope
0.2 The Changing Slope of \(y=x^2\) and \(y=x^n\)
0.3 The Exponential \(y=e^x\)
0.4 Video Summaries and Practice Problems
0.5 Graphs and Graphing Calculators
Chapter 1: Introduction to Calculus (PDF)
1.1 Velocity and Distance
1.2 Calculus Without Limits
1.3 The Velocity at an Instant
1.4 Circular Motion
1.5 A Review of Trigonometry
1.6 A Thousand Points of Light
2.1 The Derivative of a Function
2.2 Powers and Polynomials
2.3 The Slope and the Tangent Line
2.4 Derivative of the Sine and Cosine
2.5 The Product and Quotient and Power Rules
2.6 Limits
2.7 Continuous Functions
Chapter 3: Applications of the Derivative (PDF)
3.1 Linear Approximation
3.2 Maximum and Minimum Problems
3.3 Second Derivatives: Bending and Acceleration
3.4 Graphs
3.5 Parabolas, Ellipses, and Hyperbolas
3.6 Iterations \(x_{n+1}=F(x_n)\)
3.7 Newton’s Method (and Chaos)
3.8 The Mean Value Theorem and 1’Hôpital’s Rule
Chapter 4: Derivatives by the Chain Rule (PDF)
4.1 The Chain Rule
4.2 Implicit Differentiation and Related Rates
4.3 Inverse Functions and Their Derivatives
4.4 Inverses of Trigonometric Functions
5.1 The Idea of an Integral
5.2 Antiderivatives
5.3 Summation versus Integration
5.4 Indefinite Integrals and Substitutions
5.5 The Definite Integral
5.6 Properties of the Integral and Average Value
5.7 The Fundamental Theorem and Its Applications
5.8 Numerical Integration
Chapter 6: Exponentials and Logarithms (PDF)
6.1 An Overview
6.2 The Exponential \(e^x\)
6.3 Growth and Decay in Science and Economics
6.4 Logarithms
6.5 Separable Equations Including the Logistic Equation
6.6 Powers Instead of Exponentials
6.7 Hyperbolic Functions
Chapter 7: Techniques of Integration (PDF)
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Partial Fractions
7.5 Improper Integrals
Chapter 8: Applications of the Integral (PDF)
8.1 Areas and Volumes by Slices
8.2 Length of a Plane Curve
8.3 Area of a Surface of Revolution
8.4 Probability and Calculus
8.5 Masses and Moments
8.6 Force, Work, and Energy
Chapter 9: Polar Coordinates and Complex Numbers (PDF)
9.1 Polar Coordinates
9.2 Polar Equations and Graphs
9.3 Slope, Length, and Area for Polar Curves
9.4 Complex Numbers
Chapter 10: Infinite Series (PDF)
10.1 The Geometric Series
10.2 Convergence Tests: Positive Series
10.3 Convergence Tests: All Series
10.4 The Taylor Series for \(e^x\), \(\sin{x}\), and \(\cos{x}\)
10.5 Power Series
Chapter 11: Vectors and Matrices (PDF)
11.1 Vectors and Dot Products
11.2 Planes and Projections
11.3 Cross Products and Determinants
11.4 Matrices and Linear Equations
11.5 Linear Algebra
Chapter 12: Motion Along a Curve (PDF)
12.1 The Position Vector
12.2 Plane Motion: Projectiles and Cycloids
12.3 Curvature and Normal Vector
12.4 Polar Coordinates and Planetary Motion
Chapter 13: Partial Derivatives (PDF)
13.1 Surface and Level Curves
13.2 Partial Derivatives
13.3 Tangent Planes and Linear Approximations
13.4 Directional Derivatives and Gradients
13.5 The Chain Rule
13.6 Maxima, Minima, and Saddle Points
13.7 Constraints and Lagrange Multipliers
Chapter 14: Multiple Integrals (PDF)
14.1 Double Integrals
14.2 Changing to Better Coordinates
14.3 Triple Integrals
14.4 Cylindrical and Spherical Coordinates
Chapter 15: Vector Calculus (PDF)
15.1 Vector Fields
15.2 Line Integrals
15.3 Green’s Theorem
15.4 Surface Integrals
15.5 The Divergence Theorem
15.6 Stokes’ Theorem and the Curl of F