CONTENTS
Part I Single Variable Calculus
Chapter 7. Techniques of Integration
7.3 Trigonometric Substitution
7.4 Integration of Rational Functions by the Method of Partial Fractions
7.5 Guidelines for Integration
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 L¡¯Hospital¡¯s Rule
4.4 Optimization Problems
4.5 Newton¡¯s 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 Green¡¯s 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 :
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2011-How to use Sage 1: ¿¬°á
2011-How to use Sage 2: ¿¬°á
2011-How to use Sage 3: ¿¬°á
William Stein demos sage math: ¿¬°á
2011-Mobile Math with Sage: ¿¬°á
Sage Interact / ODE and Mandelbrot: ¿¬°á
Sage Multivariable Calculus (1 of 2) by Travis: ¿¬°á
Sage Multivariable Calculus (2 of 2) by Travis: ¿¬°á
http://matrix.skku.ac.kr/sglee
http://matrix.skku.ac.kr/2013-Calculus-Sage/Cal-lab-0-3/cal-lab-ch0to3.htm
http://matrix.skku.ac.kr/2013-Calculus-Sage/Web-Cover/CH-0-Cover.pdf
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)
2013.
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 http://math1.skku.ac.kr/ 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.
