## Calculus with Early Vectors, 1/e

Phillip Zenor, Auburn University
Edward E. Slaminka, Auburn University
Donald Thaxton, Auburn University

Published September, 1998 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-791203-X

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Calculus-Mathematics

Designed for students who are taking a concurrent calculus-based physics course— primarily engineering, science, and mathematics majors. By focusing on the requirements of a specific group of students, the text has been structured such that calculus is presented as a single subject rather than a collection of topics. With a user-friendly approach that keeps the student in mind, the material is organized so that vector calculus is thoroughly covered over a four-quarter or three-semester sequence without neglecting topics usually covered in a calculus course.

Introduces vector valued functions in Chapter 1 (on Precalculus Review) and treats vector calculus and real valued calculus (i.e., calculus of real variables) as a single subject throughout the text.
Establishes a solid mathematical foundation for understanding Newtonian physics, electricity, and magnetism.
Offers an appropriate amount of physics so that, throughout the text, ideas such as velocity, acceleration, force, energy, and work may be used for illustrations and motivation. This enforces the underlying theme of building mathematical tools to model and solve physical and geometric problems.
Presents line, surface and volume integrals, and the gradient, curl and divergence operators in the flow of the text rather than as optional, isolated topics. This organization allows for these topics to be covered with a thoroughness not matched in other calculus texts.
Introduces the Jacobian as the rate of change of length, area, or volume permitting students to do line integrals, surface integrals, and volume integrals by first parametrizing the region and then using change of variables theorems to translate the integral to one over an interval, rectangle, or a rectangular box.
Emphasizes the power of calculus as a tool for modeling complex physical problems in order to present the methods of differentiation and integration as necessary skills needed to solve problems that arise from mathematical models.
Integrates sequences and series into related topics and treats them together with improper integrals.
Approaches the theoretical aspects of calculus with the belief that, at the introductory level, it is important to understand the geometric basis for theorems and develop an intuitive understanding for the statements of the theorems and their implications.
Exposition is concise, correct, and to-the-point.
Ample examples and illustrations.
Sets off definitions, theorems, and corollaries in boxes to aid in finding important information.
Utilizes theme exercises throughout the text to show the student how to begin to model a seemingly difficult problem in a series of steps and to anticipate more advanced concepts and techniques that appear later in the text.

• Helps present the entire body of calculus as a tool for modeling complex processes. *
Includes optional calculator/computer exercises where they arise naturally in context and when they are pedagogically useful.

1. N-Dimensional Space.

An Introduction to R^n. Graphs in R^2 and R^3 and Their Equations. Algebra in R^n. The Dot Product. Determinants, Areas and Volumes. Equations of Lines and Planes.

2. Functions.

Functions. Functions and Graphing Technology. Functions from R into R^n. The Wrapping Function and Other Functions. Sketching Parametrized Curves. Compositions of Functions. Building New Functions.

3. Limits, Continuity, and Derivatives.

Average Velocity and Average Rate of Change. Limits: An Intuitive Approach. Instantaneous Rate of Change: The Derivative. Linear Approximations of Functions and Newton's Method. More on Limits. Limits: A Formal Approach.

4. Differentiation Rules.

The Sum and Product Rules and Higher Derivatives. The Quotient Rule. The Chain Rule. Implicit Differentiation. Higher Taylor Polynomials.

5. The Geometry of Functions and Curves.

Horizontal and Vertical Asymptotes. Increasing and Decreasing Functions. Increasing and Decreasing Curves in the Plane. Concavity. Tangential and Normal Components of Acceleration. Circular Motion and Curvature. Applications of Maxima and Minima. The Remainder Theorem for Taylor Polynomials.

6. Antiderivatives.

Antiderivatives and the Integral. The Chain Rule in Reverse. Acceleration, Velocity, and Position. Antiderivatives and Area. Area and Riemann Sums. The Definite Integral. Volumes. The Fundamental Theorem of Calculus.

7. Some Transcendental Functions.

Antiderivatives Revisited. Numerical Methods. The ln Function. The Function e^x. Exponents and Logarithms. Euler's Formula (Optional). Inverse Trigonometric Functions. Derivatives of Inverse Trigonometric Functions. Tables of Integrals.

8. Applications of Separation of Variables.

Separation of Variables and Exponential Growth. Equations of the Form y^1 = ky + b. The Logistic Equation.

9. L'Hospital's Rule, Improper Integrals, and Series.

L'Hopital's Rule (0/0 and …à/…à) Improper Integrals. Series. Alternating Series and Absolute Convergence. The Ration Test and Power Series. Power Series of Functions. Radius of Convergence for Rational Functions.

10. Techniques of Integration.

Integration by Parts. Trigonometric Substitutions. Rational Functions. Integration Factors.

11. Work, Energy, and The Line Integral.

Work. The Work-Energy Theorem. Fundamental Curves. Line Integrals of Type I and Arc Length. Center of Mass and Moment of Inertia. Vector Fields. Line Integrals of Type II and Work. Partial Derivatives. Potential Functions and the Gradient. The Gradient and Directional Derivatives. The Curl and Iterated Partial Derivatives.

12. Optimization of Functions From Rn Into R.

Tests for Local Extrema. Extrema on Closed and Bounded Domains. Lagrange Multipliers.

13. Change of Coordinate Systems.

Translations and Linear Transformations. Other Transformations. The Derivative. Arc Length for Curves in Other Coordinate Systems. Change of Area with Linear Transformations. The Jacobian.

14. Multiple Integrals.

The Integral Over a Rectangle. Simple Surfaces. An Introduction to Surface Integrals. Some Applications of Surface Integrals. Change of Variables. Simple Solids. Triple Integrals. More on Triple Integrals.

15. Divergence and Stokes' Theorem.

Oriented Surfaces and Surface Integrals. Gaussian Surfaces. Divergence. Surfaces and Their Boundaries. Stokes' Theorem. Stokes' Theorem and Conservative Fields.

A. Mathematical Induction.
B. Continuity for Functions From R^n into R^m.
C. Conic Sections in R^n.

Parabolas, Ellipses, and Hyperbolas. Some 3-Dimensional Graphs. Translation in R^n.

D. Geometric Formulas.