Part I  Single Variable Calculus


Chapter 7. Techniques of Integration


7.1 Integration by Parts


7.2 Trigonometric Integrals


7.3 Trigonometric Substitution


7.4 Integration of Rational Functions by the Method of Partial Fractions


7.5 Guidelines for Integration


7.6 Integration Using Tables


7.7 Approximate Integration


7.8 Improper Integrals




Chapter 1. Functions

1.1 History of Calculus

1.2 Symmetry

1.3 Common Functions

1.4 Translation, Stretching and Rotation of Functions

1.5 A Few Basic Concepts


Chapter 2. Limits and Continuity

2.1 Limits of functions

2.2 Continuity


Chapter 3. Theory of Differentiation

3.1 Definition of Derivatives, Differentiation

3.2 Derivatives of Polynomials, Exponential Functions, Trigonometric Functions, The product rule

3.3 The Chain Rule and Inverse Functions

3.4 Approximation and Related Rates


Chapter 4. Applications of Differentiation

4.1 Extreme values of a function

4.2 The Shape of a Graph

4.3 The Limit of Indeterminate Forms and LHospitals Rule

4.4 Optimization Problems

4.5 Newtons Method


Chapter 5. Integrals

5.1 Areas and Distances

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus

5.4 Indefinite Integrals and the Net Change Theorem

5.5 The Substitution Rule

5.6 The Logarithm Defined as an Integral


Chapter 6. Applications of Integration

6.1 Areas between Curves

6.2 Volumes

6.3 Volumes by Cylindrical Shells

6.4 Work

6.5 Average Value of a Function



Chapter 7. Techniques of Integration



Chapter 8. Further Applications of Integration

8.1 Arc Length

8.2 Area of a Surface of Revolution

8.3 Applications of Integral Calculus

8.4 Differential equations


Chapter 9. Parametric Equations and Polar Coordinates

9.1 Parametric Equations

9.2 Calculus with Parametric Curves

9.3 Polar Coordinates

9.4 Areas and Lengths in Polar Coordinates

9.5 Conic Section


Chapter 10. Infinite Sequences and Infinite Series

10.1 Sequences and Series

10.2 Tests for convergence of series with positive terms

10.3 Alternating Series and Absolute Convergence

10.4 Power Series

10.5 Taylor, Maclaurin, and Binomial Series



Part II  Multivariate Calculus


Chapter 11. Vectors and the Geometry of Space

11.1 Three-Dimensional Coordinate Systems

11.2 Vectors

11.3 The Scalar or Dot Product

11.4 The Vector or Cross Product

11.5 Equations of straight Lines and Planes

11.6 Cylinders and Quadric Surfaces


Chapter 12. Partial Derivatives and Local Maxima and Minima

12.1 Limits and Continuity of multi-variable functions

12.2 Partial Derivatives and Directional Derivatives

12.3 Tangent Plane and Differentiability

12.4 Local and global maximum and minimum


Chapter 13. Vector Functions

13.1 Vector-Valued Functions and Space Curves

13.2 Calculus of Vector Functions

13.3 Arc Length and Curvature

13.4 Motion Along A Space Curve: Velocity and Acceleration


Chapter 14. Multiple Integrals

14.1 Double Integrals

14.2 Cylindrical Coordinates and Spherical Coordinates

14.3 Volume Integrals


Chapter 15. Vector Calculus

15.1 Vector Differentiation

15.2 Line Integrals

15.3 Surface Integrals: Surface Area and Flux

15.4 Greens Theorem in Plane: Transformation between line integral and double integral

15.5 Stokes Theorem: Transformation between line integral and surface integral

15.6 Gauss Divergence Theorem: Transformation between surface integral and volume integral


Internet resources :


Sage Tutorial:  

2011-How to use Sage 1:

2011-How to use Sage 2:

2011-How to use Sage 3:

William Stein demos sage math:

2011-Mobile Math with Sage:

Sage Interact / ODE and Mandelbrot:

Sage Multivariable Calculus (1 of 2) by Travis:

Sage Multivariable Calculus (2 of 2) by Travis:


Main Author : Sang-Gu Lee


Co-Authors : Eung-Ki Kim, Yoonmee Ham, Ajit Kumar, Robert A. Beezer, ... (not  more than 10)


Reviewers : R. Sakthivel, K. Das, I. Hwang, J. Lee, ... (more than 10)






  Calculus is the mathematical foundation for much of university mathematics, science, and engineering curriculum. For the mathematics student, it is a first exposure to rigorous mathematics. For the engineer, it is an introduction to the modeling and approximation techniques used throughout an engineering curriculum. And for the future scientist, it is the mathematical language that will be used to express many of the most important scientific concepts.


  In the first semester, that is for the beginners of calculus, we start with differential and integral calculus on functions of single variable and then study L'Hospital's theorem, concavity, convexity, inflection points, optimization problems, ordinary differential equations as applications of differential and integral calculus, parameter equations, polar coordinates, infinite sequences and infinite series accordingly. In the second semester of calculus, we cover vector calculus that includes vectors, coordinate space, partial derivatives and multiple integrals. Concepts, definitions, terminology, and interpretation in calculus should be as current as possible. This book has many problems presenting calculus as the foundation of modern mathematics, science and engineering.


  Many recent calculus textbooks are using Computer Algebra System (CAS) including a variety of visual tools in it. But in most cases its use by students is limited. Therefore, for this book, we have adapted a wonderful free and open-source program, Sage. With the new learning environment of universities, students will take a full advantage of 21st century, state of the art technology to learn calculus easily and be better prepared for future careers. We can use Sage easily on popular web browsers such as Firefox or Chrome. The system language for Sage is Python, a powerful mainstream computer programming language.


  More content and related materials will be added to be viewed on the web. When you see a CAS or web mark in the book, this means you will be able to find relevant information by clicking on the address. This will save you a lot of work.


  Finally, the book also combines technology, reform, and tradition in a way that offers a wider view to students. Most importantly, we appreciate everyone who has contributed to the project of writing this book.