Vector Calculus (Textbooks in Mathematics)

✍ Scribed by Steven G. Krantz, Harold Parks

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 682
• Edition: 1
• Category: Library

✦ Synopsis

Using meaningful examples, credible applications, and incisive technology, Vector Calculus strives to empower students, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the major or discipline they ultimately choose to study. This text is intended to be a cornerstone of that process. An engaging style and clear writing make the language of mathematics accessible, understandable, and enjoyable, with a high standard for mathematical rigor.

A calculus book must tell the truth. This book is carefully written in the accepted language of mathematics in a readable exposition. It includes useful and fascinating applications, acquaints students with the history of the subject, and offers a sense of what mathematics is all about.

Technique is presented, yet so are ideas. The authors help students to master basic methods and discover and build their own concepts in a scientific subject. There is an emphasis on using modeling and numerical calculation.

Additional features include:

• A Quick Quiz and Problems for Practice, Further Theory and Practice, and Calculator/Computer Exercises appear at the end of each section
• All exercise sets are step laddered
• A Look Back and A Look Forward help students put the ideas in context
• Every chapter ends with a Genesis and Development section, giving history and perspective on key topics in the evolution of calculus
• Boxed Insights clear up points or answer commonly asked questions
• The text has an extra-large offering of examples
• Examples are illustrated with meaningful and useful graphics

The pedagogical features make the subject more interesting and accessible to students than other texts, while maintaining an appropriate rigor. ―Daniel Cunningham, CSU-Fresno

This text is truly well written and organized. I do like the fact the book is quite rigorous, yet full of illustrative examples. ―Bob Devaney, Boston University

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
1. Vectors
1.1. Vectors in the Plane
1.1.1. The Idea of Vector
1.1.2. Vector Algebra
1.1.3. The Length of a Vector
1.1.4. Unit Vectors and Directions
1.1.5. An Application to Physics
1.1.6. The Special Unit Vectors i and j
1.1.7. The Triangle Inequality
1.1.8. Problems for Practice
1.1.9. Further Theory and Practice
1.1.10. Calculator/Computer Exercises
1.2. Vectors in Three-Dimensional Space
1.2.1. Distance
1.2.2. Vectors in Space
1.2.3. Vector Operations
1.2.4. Vector Length
1.2.5. Unit Vectors and Directions
1.2.6. The Special Unit Vectors i, j, and k
1.2.7. Relations among Addition, Length, and Scalar Product
1.2.8. Problems for Practice
1.2.9. Further Theory and Practice
1.2.10. Calculator and Computer Exercises
1.3. The Dot Product and Applications
1.3.1. Algebraic Rules for the Dot Product
1.3.2. A Geometric Formula for the Dot Product
1.3.3. Projection
1.3.4. The Standard Basis Vectors
1.3.5. Direction Cosines and Direction Angles
1.3.6. Applications
1.3.7. A Final Remark
1.3.8. Problems for Practice
1.3.9. Further Theory and Practice
1.3.10. Calculator and Computer Exercises
1.4. The Cross Product and Triple Product
1.4.1. The Relationship between Cross Products and Determinants
1.4.2. A Geometric Understanding of the Cross Product
1.4.3. Cross Products and the Calculation of Area
1.4.4. A Physical Application of the Cross Product
1.4.5. The Triple Scalar Product
1.4.6. Problems for Practice
1.4.7. Further Theory and Practice
1.4.8. Calculator and Computer Exercises
1.5. Lines and Planes in Space
1.5.1. Cartesian Equations of Planes in Space
1.5.2. Parametric Equations of Planes in Space
1.5.3. Parametric Equations of Lines in Space
1.5.4. Cartesian Equations of Lines in Space
1.5.5. Triple Vector Product
1.5.6. Problems for Practice
1.5.7. Further Theory and Practice
1.5.8. Calculator and Computer Exercises
1.6. Summary of Key Topics
1.6.1. Points in Space (Section 1.2)
1.6.2. Vectors (Sections 1.1, 1.2)
1.6.3. Dot Product (Section 1.3)
1.6.4. Projection (Section 1.3)
1.6.5. Direction Vectors (Sections 1.1, 1.2, 1.3)
1.6.6. Cross Product (Section 1.4)
1.6.7. Lines and Planes (Section 1.5)
1.6.8. Triple Scalar Product (Section 1.4)
2. Vector-Valued Functions
2.1. Vector-Valued Functions
2.1.1. Limits of Vector-Valued Functions
2.1.2. Continuity
2.1.3. Derivatives of Vector-Valued Functions
2.1.4. Antidifferentiation
2.1.5. Problems for Practice
2.1.6. Further Theory and Practice
2.1.7. Calculator and Computer Exercises
2.2. Velocity and Acceleration
2.2.1. The Tangent Line to a Curve in Space
2.2.2. Acceleration
2.2.3. The Physics of Baseball
2.2.4. Problems for Practice
2.2.5. Further Theory and Practice
2.2.6. Calculator/Computer Exercises
2.3. Tangent Vectors and Arc Length
2.3.1. Unit Tangent Vectors
2.3.2. Arc Length
2.3.3. Reparameterization
2.3.4. Parameterizing a Curve by Arc Length
2.3.5. Unit Normal Vectors
2.3.6. Problems for Practice
2.3.7. Further Theory and Practice
2.3.8. Calculator and Computer Exercises
2.4. Curvature
2.4.1. Calculating Curvature without Reparameterizing
2.4.2. The Osculating Circle
2.4.3. Planar Curves
2.4.4. Problems for Practice
2.4.5. Further Theory and Practice
2.4.6. Calculator and Computer Exercises
2.5. Applications to Motion
2.5.1. Central Force Fields
2.5.2. Ellipses
2.5.3. Applications to Planetary Motion
2.5.4. Problems for Practice
2.5.5. Further Theory and Practice
2.5.6. Calculator and Computer Exercises
2.6. Summary of Key Topics
2.6.1. Vector-Valued Functions (Section 2.1)
2.6.2. Properties of Limits (Section 2.1)
2.6.3. Velocity and Acceleration (Section 2.2)
2.6.4. Tangent Vectors and Arc Length (Section 2.3)
2.6.5. Curvature (Section 2.4)
2.6.6. Tangential and Normal Components of Acceleration (Section 2.5)
2.6.7. Kepler’s Three Laws of Planetary Motion (Section 2.5)
3. Functions of Several Variables
3.1. Functions of Several Variables
3.1.1. Combining Functions
3.1.2. Graphing Functions of Several Variables
3.1.3. More on Level Sets
3.1.4. Problems for Practice
3.1.5. Further Theory and Practice
3.1.6. Calculator/Computer Exercises
3.2. Cylinders and Quadric Surfaces
3.2.1. Cylinders
3.2.2. Quadric Surfaces
3.2.3. Recognizing the Graph of a Function
3.2.4. Problems for Practice
3.2.5. Further Theory and Practice
3.2.6. Calculator/Computar Exercises
3.3. Limits and Continuity
3.3.1. Limits
3.3.2. Continuity
3.3.3. Rules for Limits
3.3.4. Problems for Practice
3.3.5. Further Theory and Practice
3.3.6. Calculator/Computer Exercises
3.4. Partial Derivatives
3.4.1. Functions of Three Variables
3.4.2. Higher Partial Derivatives
3.4.3. Concluding Remarks about Partial Differentiation
3.4.4. Problems for Practice
3.4.5. Further Theory and Practice
3.4.6. Calculator/Computer Exercises
3.5. Differentiability and the Chain Rule
3.5.1. The Chain Rule
3.5.2. The Chain Rule for Two or More Independent Variables
3.5.3. Functions of Three or More Variables
3.5.4. Taylor’s Formula in Several Variables
3.5.5. Problems for Practice
3.5.6. Further Theory and Practice
3.5.7. Computer/Calculator Exercises
3.6. Gradients and Directional Derivatives
3.6.1. The Directional Derivative
3.6.2. The Gradient
3.6.3. The Directions of Greatest Increase and Decrease
3.6.4. The Gradient and Level Curves
3.6.5. Functions of Three or More Variables
3.6.6. Problems for Practice
3.6.7. Further Theory and Practice
3.6.8. Computer/Calculator Exercises
3.7. Tangent Planes
3.7.1. Level Surfaces
3.7.2. Normal Lines
3.7.3. Numerical Approximations Using the Tangent Plane
3.7.4. A Restatement of Theorem 3.7.7 Using Increments
3.7.5. Functions of Three or More Variables
3.7.6. Problems for Practice
3.7.7. Further Theory and Practice
3.7.8. Calculator/Computer Exercises
3.8. Maximum-Minimum Problems
3.8.1. The Analogue of Fermat’s Theorem
3.8.2. Saddle Points
3.8.3. The Second Derivative Test for Local Extrema
3.8.4. Applied Maximum-Minimum Problems
3.8.5. Least Squares Lines
3.8.6. Problems for Practice
3.8.7. Further Theory and Practice
3.8.8. Calculator/Computer Exercises
3.9. Lagrange Multipliers
3.9.1. Why the Method of Lagrange Multipliers Works
3.9.2. Lagrange Multipliers and Functions of Three Variables
3.9.3. Extremizing a Function Subject to Two Constraints
3.9.4. Problems for Practice
3.9.5. Further Theory and Practice
3.9.6. Computer/Calculator Exercises
3.10. Summary of Key Topics
3.10.1. Functions of Several Variables (Section 3.1)
3.10.2. Cylinders (Section 3.2)
3.10.3. Quadric Surfaces (Section 3.2)
3.10.4. Limits and Continuity (Section 3.3)
3.10.5. Partial Derivatives (Section 3.4)
3.10.6. The Chain Rule (Section 3.5)
3.10.7. Gradients and Directional Derivatives (Section 3.6)
3.10.8. Normal Vectors and Tangent Planes (Section 3.7)
3.10.9. Numerical Approximation (Section 3.7)
3.10.10. Critical Points (Section 3.8)
3.10.11. Lagrange Multipliers (Section 3.9)
4. Multiple Integrals
4.1. Double Integrals over Rectangular Regions
4.1.1. Iterated Integrals
4.1.2. Using Iterated Integrals to Calculate Double Integrals
4.1.3. Problems for Practice
4.1.4. Further Theory and Practice
4.1.5. Calculator/Computer Exercises
4.2. Integration over More General Regions
4.2.1. Planar Regions Bounded by Finitely Many Curves
4.2.2. Changing the Order of Integration
4.2.3. The Area of a Planar Region
4.2.4. Problems for Practice
4.2.5. Further Theory and Practice
4.2.6. Calculator/Computer Exercises
4.3. Calculation of Volumes of Solids
4.3.1. The Volume between Two Surfaces
4.3.2. Problems for Practice
4.3.3. Further Theory and Practice
4.3.4. Calculator/Computer Exercises
4.4. Polar Coordinates
4.4.1. The Polar Coordinate System
4.4.2. Negative Values of the Radial Variable
4.4.3. Relating Polar Coordinates to Rectangular Coordinates
4.4.4. Graphing in Polar Coordinates
4.4.5. Symmetry Principles in Graphing
4.4.6. Problems for Practice
4.4.7. Further Theory and Practice
4.4.8. Calculator/Computer Exercises
4.5. Integrating in Polar Coordinates
4.5.1. Areas of More General Regions
4.5.2. Using Iterated Integrals to Calculate Area in Polar Coordinates
4.5.3. Integrating Functions in Polar Coordinates
4.5.4. Change of Variable and the Jacobian
4.5.5. Problems for Practice
4.5.6. Further Theory and Practice
4.5.7. Calculator/Computer Exercises
4.6. Triple Integrals
4.6.1. The Concept of the Triple Integral
4.6.2. Problems for Practice
4.6.3. Further Theory and Practice
4.6.4. Computer/Calculator Exercises
4.7. Physical Applications
4.7.1. Mass
4.7.2. First Moments
4.7.3. Center of Mass
4.7.4. Moment of Inertia
4.7.5. Mass, First Moment, Moment of Inertia, and Center of Mass in Three Dimensions
4.7.6. Problems for Practice
4.7.7. Further Theory and Practice
4.7.8. Calculator/Computer Exercises
4.8. Other Coordinate Systems
4.8.1. Cylindrical Coordinates
4.8.2. Spherical Coordinates
4.8.3. Problems for Practice
4.8.4. Further Theory and Practice
4.8.5. Computer/Calculator Exercises
4.9. Summary of Key Topics
4.9.1. Double Integrals over Rectangular Regions (Section 4.1)
4.9.2. Integration over More General Planar Regions (Section 4.2)
4.9.3. Simple Regions (Section 4.2)
4.9.4. Area and Volume (Section 4.3)
4.9.5. Polar Coordinates (Section 4.5)
4.9.6. Graphing in Polar Coordinates (Section 4.4)
4.9.7. Area in Polar Coordinates (Section 4.5)
4.9.8. Double Integrals in Polar Coordinates (Section 4.5)
4.9.9. Change of Variable in Double Integrals (Section 4.5)
4.9.10. Triple Integrals (Section 4.6)
4.9.11. Mass, Moment of Inertia, and Center of Mass (Section 4.7)
4.9.12. Cylindrical Coordinates (Section 4.8)
4.9.13. Spherical Coordinates (Section 4.8)
4.9.14. The Lebesgue Integral
5. Vector Calculus
5.1. Vector Fields
5.1.1. Vector Fields in Physics
5.1.2. Integral Curves (Streamlines)
5.1.3. Gradient Vector Fields and Potential Functions
5.1.4. Continuously Differentiable Vector Fields
5.1.5. Problems for Practice
5.1.6. Further Theory and Practice
5.1.7. Calculator/Computer Exercises
5.2. Line Integrals
5.2.1. Work along a Curved Path
5.2.2. Line Integrals
5.2.3. Dependence on Path
5.2.4. Other Notation for the Line Integral
5.2.5. Closed Curves
5.2.6. Other Applications of Line Integrals
5.2.7. Problems for Practice
5.2.8. Further Theory and Practice
5.2.9. Calculator/Computer Exercises
5.3. Conservative Vector Fields
5.3.1. Path-Independent Vector Fields
5.3.2. Closed Vector Fields
5.3.3. Vector Fields in Space
5.3.4. Summary of Principal Ideas
5.3.5. Problems for Practice
5.3.6. Further Theory and Practice
5.3.7. Calculator/Computer Exercises
5.4. Divergence, Gradient, and Curl
5.4.1. Divergence of a Vector Field
5.4.2. The Curl of a Vector Field
5.4.3. The Del Notation
5.4.4. Identities Involving div, curl, grad, and △
5.4.5. Problems for Practice
5.4.6. Further Theory and Practice
5.4.7. Calculator/Computer Exercises
5.5. Green’s Theorem
5.5.1. A Vector Form of Green’s Theorem
5.5.2. Green’s Theorem for More General Regions
5.5.3. Problems for Practice
5.5.4. Further Theory and Practice
5.5.5. Calculator/Computer Exercises
5.6. Surface Integrals
5.6.1. The Integral for Surface Area
5.6.2. Integrating a Function over a Surface
5.6.3. An Application
5.6.4. The Element of Area for a Surface That Is Given Parametrically
5.6.5. Surface Integrals over Parameterized Surfaces
5.6.6. Problems for Practice
5.6.7. Further Theory and Practice
5.6.8. Calculator/Computer Exercises
5.7. Stokes’s Theorem
5.7.1. Orientable Surfaces and Their Boundaries
5.7.2. The Component of Curl in the Normal Direction
5.7.3. Stokes’s Theorem
5.7.4. Stokes’s Theorem on a Region with Piecewise-Smooth Boundary
5.7.5. An Application
5.7.6. Problems for Practice
5.7.7. Further Theory and Practice
5.7.8. Calculator/Computer Exercises
5.8. Flux and the Divergence Theorem
5.8.1. The Divergence Theorem
5.8.2. Some Applications
5.8.3. Proof of the Divergence Theorem
5.8.4. Problems for Practice
5.8.5. Further Theory and Practice
5.8.6. Calculator/Computer Exercises
5.9. Summary of Key Topics
5.9.1. Line Integrals (Section 5.2)
5.9.2. Conservative and Path-Independent Vector Fields (Section 5.3)
5.9.3. Closed Vector Fields (Section 5.3)
5.9.4. Divergence (Section 5.4)
5.9.5. Curl (Section 5.4)
5.9.6. The Del Notation (Section 5.4)
5.9.7. Operations with Div and Curl (Section 5.4)
5.9.8. Green’s Theorem (Section 5.5)
5.9.9. Surface Area and Surface Integrals (Section 5.6)
5.9.10. Stokes’s Theorem (Section 5.7)
5.9.11. The Divergence Theorem (Section 5.8)
Index