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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index
A
acceleration vector3.4 Motion in Space
angular coordinate1.3 Polar Coordinates
angular frequency7.3 Applications
arc-length function3.3 Arc Length and Curvature
arc-length parameterization3.3 Arc Length and Curvature
Archimedean spiral1.3 Polar Coordinates
C
characteristic equation7.1 Second-Order Linear Equations
Clairaut’s theorem6.1 Vector Fields
Cobb-Douglas function4.8 Lagrange Multipliers
Cobb-Douglas production function4.3 Partial Derivatives
conic section1.5 Conic Sections
conservative field6.1 Vector Fields
critical point of a function of two variables4.7 Maxima/Minima Problems
cross-partial property6.1 Vector Fields
cylindrical coordinate system2.7 Cylindrical and Spherical Coordinates
F
Faraday’s law6.7 Stokes’ Theorem
focal parameter1.5 Conic Sections
Fourier’s law of heat transfer6.8 The Divergence Theorem
Frenet frame of reference3.3 Arc Length and Curvature
Fubini’s thereom5.4 Triple Integrals
function of two variables4.1 Functions of Several Variables
Fundamental Theorem for Line Integrals6.8 The Divergence Theorem
Fundamental Theorem for Line Integrals.6.3 Conservative Vector Fields
Fundamental Theorem of Calculus6.8 The Divergence Theorem
G
Gauss’s law for magnetism6.5 Divergence and Curl
general bounded region5.4 Triple Integrals
general form1.5 Conic Sections
general form of the equation of a plane2.5 Equations of Lines and Planes in Space
general solution to a differential equation7.1 Second-Order Linear Equations
generalized chain rule4.5 The Chain Rule
gradient field6.1 Vector Fields
graph of a function of two variables4.1 Functions of Several Variables
I
implicit differentiation4.5 The Chain Rule
indefinite integral of a vector-valued function3.2 Calculus of Vector-Valued Functions
independent random variables5.2 Double Integrals over General Regions
independent variables4.5 The Chain Rule
initial-value problems7.1 Second-Order Linear Equations
intermediate variables4.5 The Chain Rule
inverse-square law6.8 The Divergence Theorem
K
Kepler’s laws of planetary motion3.4 Motion in Space
L
Lagrange multiplier4.8 Lagrange Multipliers
Laplace operator6.5 Divergence and Curl
level curve of a function of two variables4.1 Functions of Several Variables
level surface of a function of three variables4.1 Functions of Several Variables
limit of a function of two variables4.2 Limits and Continuity
limit of a vector-valued function3.1 Vector-Valued Functions and Space Curves
line integral6.2 Line Integrals
M
mass of a wire6.2 Line Integrals
method of Lagrange multipliers4.8 Lagrange Multipliers
method of undetermined coefficients7.2 Nonhomogeneous Linear Equations
method of variation of parameters7.2 Nonhomogeneous Linear Equations
mixed partial derivatives4.3 Partial Derivatives
N
nonhomogeneous linear equation7.1 Second-Order Linear Equations
normal component of acceleration3.4 Motion in Space
normal form of Green’s theorem6.4 Green’s Theorem
Q
Quadric surfaces2.6 Quadric Surfaces
S
scalar line integral6.2 Line Integrals
scalar multiplication2.1 Vectors in the Plane
scalar projection2.3 The Dot Product
simple harmonic motion7.3 Applications
simply connected region6.3 Conservative Vector Fields
space-filling curve1.3 Polar Coordinates
space-filling curves1.1 Parametric Equations
spring-mass system7.3 Applications
standard equation of a sphere2.2 Vectors in Three Dimensions
standard form1.5 Conic Sections
standard unit vectors2.1 Vectors in the Plane
standard-position vector2.1 Vectors in the Plane
steady-state solution7.3 Applications
stream function6.4 Green’s Theorem
superposition principle7.1 Second-Order Linear Equations
surface independent6.7 Stokes’ Theorem
surface integral6.6 Surface Integrals
surface integral of a scalar-valued function6.6 Surface Integrals
surface integral of a vector field6.6 Surface Integrals
symmetric equations of a line2.5 Equations of Lines and Planes in Space
U
unit vector field6.1 Vector Fields
W
William Thomson (Lord Kelvin)4.3 Partial Derivatives
work done by a vector field6.2 Line Integrals
work done by the force2.3 The Dot Product
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