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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
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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.
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 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.
E. Answers to Odd Exercises.
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