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**About this book :- **
**Elementary Calculus: An Infinitesimal Approach (2nd Edition) ** written by
** H. Jerome Keisler **.

This first-year calculus book is centered around the use of infinitesimals. It contains all the ordinary calculus topics, including the basic concepts of the derivative, continuity, and the integral, plus traditional limit concepts and approximation problems. Additional subjects include transcendental functions, series, vectors, partial derivatives, and multiple integrals. 2007 edition.

This book is concerned with the infinitesimal approach originally set forth by Newton and Leibnitz. The author has moved the theoretical material from Chapter One to an Appendix in this edition. A new chapter on differential equations has been added and the transcendental functions have been fully integrated into the first section. This book should be of interest to first and second year undergraduate mathematics students.

**Book Detail :- **
** Title: ** Elementary Calculus: An Infinitesimal Approach
** Edition: ** 2nd
** Author(s): ** H. Jerome Keisler
** Publisher: ** Brooks/Cole
** Series: ** Dover Books on Mathematics
** Year: ** 2012
** Pages: **
** Type: ** PDF
** Language: ** English
** ISBN: ** 0486484521,9780486484525
** Country: ** US
** Get this book from Amazon**

**About Author :- **

The author Howard Jerome Keisler (born 3 December 1936) as an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.

Keisler published Elementary Calculus: An Infinitesimal Approach, a first-year calculus textbook conceptually centered on the use of infinitesimals, rather than the epsilon, delta approach, for developing the calculus.

He is also known for extending the Henkin construction (of Leon Henkin) to what are now called Henkin–Keisler models. In 2012 he became a fellow of the American Mathematical Society.

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**Book Contents :- **
**Elementary Calculus: An Infinitesimal Approach (2nd Edition) ** written by
** H. Jerome Keisler **
cover the following topics.

INTRODUCTION
**1. REAL AND HVPERREAL NUMBERS**

1.1 The Real Line

1.2 Functions of Real Numbers

1.3 Straight Lines

1.4 Slope and Velocity; The Hyperreal Line

1.5 Infinitesimal, Finite, and Infinite Numbers

1.6 Standard Parts

Extra Problems for Chapter I
**2. DIFFERENTIATION**

2.1 Derivatives

2.2 Differentials and Tangent Lines

2.3 Derivatives of Rational Functions

2.4 Inverse Functions

2.5 Transcendental Functions

2.6 Chain Rule

2.7 Higher Derivatives

2.8 Implicit Functions

Extra Problems for Chapter 2
**3. CONTINUOUS FUNCTIONS**

3.1 How to Set Up a Problem

3.2 Related Rates

3.3 Limits

3.4 Continuity

3.5 Maxima and Minima

3.6 Maxima and Minima - Applications

3.7 Derivatives and Curve Sketching

3.8 Properties of Continuous Functions

Extra Problems for Chapter 3
**4. INTEGRATION**

4.1 The Definite Integral

4.2 Fundamental Theorem of Calculus

4.3 Indefinite Integrals

4.4 Integration by Change of Variables

4.5 Area between Two Curves

4.6 Numerical Integration

Extra Problems for Chapter 4
**5. LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS**

5.1 Infinite Limits

5.2 L

Hospital

s Rule

5.3 Limits and Curve Sketching

5.4 Parabolas

5.5 Ellipses and Hyperbolas

5.6 Second Degree Curves

5.7 Rotation of Axes

5.8 The e, 8 Condition for Limits

5.9 Newton

s Method

5.10 Derivatives and Increments

Extra Problems for Chapter 5
**6. APPLICATIONS OF THE INTEGRAL**

6.1 Infinite Sum Theorem

6.2 Volumes of Solids of Revolution

6.3 Length of a Curve

6.4 Area of a Surface of Revolution

6.5 Averages

6.6 Some Applications to Physics

6.7 Improper Integrals

Extra Problems for Chapter 6
**7. TRIGONOMETRIC FUNCTIONS**

7.1 Trigonometry

7.2 Derivatives of Trigonometric Functions

7.3 Inverse Trigonometric Functions

7.4 Integration by Parts

7.5 Integrals of Powers of Trigonometric Functions

7.6 Trigonometric Substitutions

7.7 Polar Coordinates

7.8 Slopes and Curve Sketching in Polar Coordinates

7.9 Area in Polar Coordinates

7.10 Length of a Curve in Polar Coordinates

Extra Problems for Chapter 7
**8. EXPONENTIAL AND LOGARITHMIC FUNCTIONS**

8.1 Exponential Functions

8.2 Logarithmic Functions

8.3 Derivatives of Exponential Functions and the Number e

8.4 Some Uses of Exponential Functions

8.5 Natural Logarithms

8.6 Some Differential Equations

8.7 Derivatives and Integrals Involving In x

8.8 Integration of Rational Functions

8.9 Methods of Integration

Extra Problems for Chapter 8
**9. INFINITE SERIES**

9.1 Sequences

9.2 Series

9.3 Properties of Infinite Series

9.4 Series with Positive Terms

9.5 Alternating Series

9.6 Absolute and Conditional Convergence

9.7 Power Series

9.8 Derivatives and Integrals of Power Series

9.9 Approximations by Power Series

9.10 Taylor

s Formula 547

9.11 Taylor Series 554

Extra Problems for Chapter 9
**10. VECTORS**

10.1 Vector Algebra

10.2 Vectors and Plane Geometry

10.3 Vectors and Lines in Space

10.4 Products of Vectors

10.5 Planes in Space

10.6 Vector Valued Functions

10.7 Vector Derivatives

10.8 Hyperreal Vectors

Extra Problems for Chapter I0
**11. PARTIAL DIFFERENTIATION**

II.1 Surfaces

11.2 Continuous Functions of Two or More Variables

11.3 Partial Derivatives

11.4 Total Differentials and Tangent Planes

11.5 Chain Rule

11.6 Implicit Functions

11.7 Maxima and Minima

11.8 Higher Partial Derivatives

Extra Problems for Chapter II
**12. MULTIPLE INTEGRALS**

12.1 Double Integrals

12.2 Iterated Integrals

12.3 Infinite Sum Theorem and Volume

12.4 Applications to Physics

12.5 Double Integrals in Polar Coordinates

12.6 Triple Integrals

12.7 Cylindrical and Spherical Coordinates

Extra Problems for Chapter 12
**13. VECTOR CALCULUS**

13.1 Directional Derivatives and Gradients

13.2 Line Integrals

13.3 Independence of Path

13.4 Green

s Theorem

13.5 Surface Area and Surface Integrals

13.6 Theorems of Stokes and Gauss

Extra Problems for Chapter 13
**14. DIFFERENTIAL EQUATIONS**

14.1 Equations with Separable Variables

14.2 First Order Homogeneous Linear Equations

14.3 First Order Linear Equations

14.4 Existence and Approximation of Solutions

14.5 Complex Numbers

14.6 Second Order Homogeneous Linear Equations

14.7 Second Order Linear Equations

Extra Problems for Chapter 14
**APPENDIX: Tables**

I Trigonometric Functions

II Greek Alphabet

III Exponential Functions

IV Natural Logarithms

V Powers And Roots

ANSWERS TO SELECTED PROBLEMS

Index

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