S. Colley – Vector Calculus (2016)

Автор: S. Colley
Название книги: Vector Calculus
Формат: PDF
Жанр: Математика
Страницы: 625
Качество: Изначально компьютерное, E-book

Physical and natural phenomena depend on a complex array of factors. The sociologist or psychologist who studies group behavior, the economist who endeavors to understand the vagaries of a nation’s employment cycles, the physicist who observes the trajectory of a particle or planet, or indeed anyone who seeks to understand geometry in two, three, or more dimensions recognizes the need to analyze changing quantities that depend on more than a single variable. Vector calculus is the essential mathematical tool for such analysis. Moreover, it is an exciting and beautiful subject in its own right, a true adventure in many dimensions.
The only technical prerequisite for this text, which is intended for a sophomore-level course in multivariable calculus, is a standard course in the calculus of functions of one variable. In particular, the necessary matrix arithmetic and algebra (not linear algebra) are developed as needed. Although the mathematical background assumed is not exceptional, the reader will still be challenged in places.
Preface
To the Student: Some Preliminary Notation
Vectors
Vectors in Two and Three Dimensions
More About Vectors
The Dot Product
The Cross Product
Equations for Planes; Distance Problems
Some n-dimensional Geometry
New Coordinate Systems
True/False Exercises for Chapter 1
Miscellaneous Exercises for Chapter 1
Differentiation in Several Variables
Functions of Several Variables; Graphing Surfaces
Limits
The Derivative
Properties; Higher-order Partial Derivatives
The Chain Rule
Directional Derivatives and the Gradient
Newton’s Method (optional)
True/False Exercises for Chapter 2
Miscellaneous Exercises for Chapter 2
Vector-Valued Functions
Parametrized Curves and Kepler’s Laws
Arclength and Differential Geometry
Vector Fields: An Introduction
Gradient, Divergence, Curl, and the Del Operator
True/False Exercises for Chapter 3
Miscellaneous Exercises for Chapter 3
Maxima and Minima in Several Variables
Differentials and Taylor’s Theorem
Extrema of Functions
Lagrange Multipliers
Some Applications of Extrema
True/False Exercises for Chapter 4
Miscellaneous Exercises for Chapter 4
Multiple Integration
Introduction: Areas and Volumes
Double Integrals
Changing the Order of Integration
Triple Integrals
Change of Variables
Applications of Integration
Numerical Approximations of Multiple Integrals (optional)
True/False Exercises for Chapter 5
Miscellaneous Exercises for Chapter 5
Line Integrals
Scalar and Vector Line Integrals
Green’s Theorem
Conservative Vector Fields
True/False Exercises for Chapter 6
Miscellaneous Exercises for Chapter 6
Surface Integrals and Vector Analysis
Parametrized Surfaces
Surface Integrals
Stokes’s and Gauss’s Theorems
Further Vector Analysis; Maxwell’s Equations
True/False Exercises for Chapter 7
Miscellaneous Exercises for Chapter 7
Vector Analysis in Higher Dimensions
An Introduction to Differential Forms
Manifolds and Integrals of k-forms
The Generalized Stokes’s Theorem
True/False Exercises for Chapter 8
Miscellaneous Exercises for Chapter 8
Suggestions for Further Reading
Answers to Selected Exercises