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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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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 vector 3.4 Motion in Space
angular coordinate 1.3 Polar Coordinates
angular frequency 7.3 Applications
arc-length function 3.3 Arc Length and Curvature
arc-length parameterization 3.3 Arc Length and Curvature
Archimedean spiral 1.3 Polar Coordinates
C
characteristic equation 7.1 Second-Order Linear Equations
Clairaut’s theorem 6.1 Vector Fields
Cobb-Douglas function 4.8 Lagrange Multipliers
Cobb-Douglas production function 4.3 Partial Derivatives
conic section 1.5 Conic Sections
conservative field 6.1 Vector Fields
critical point of a function of two variables 4.7 Maxima/Minima Problems
cross-partial property 6.1 Vector Fields
cylindrical coordinate system 2.7 Cylindrical and Spherical Coordinates
F
Faraday’s law 6.7 Stokes’ Theorem
focal parameter 1.5 Conic Sections
Fourier’s law of heat transfer 6.8 The Divergence Theorem
Frenet frame of reference 3.3 Arc Length and Curvature
Fubini’s thereom 5.4 Triple Integrals
function of two variables 4.1 Functions of Several Variables
Fundamental Theorem for Line Integrals 6.8 The Divergence Theorem
Fundamental Theorem for Line Integrals. 6.3 Conservative Vector Fields
Fundamental Theorem of Calculus 6.8 The Divergence Theorem
G
Gauss’s law for magnetism 6.5 Divergence and Curl
general bounded region 5.4 Triple Integrals
general form of the equation of a plane 2.5 Equations of Lines and Planes in Space
general solution to a differential equation 7.1 Second-Order Linear Equations
generalized chain rule 4.5 The Chain Rule
gradient field 6.1 Vector Fields
graph of a function of two variables 4.1 Functions of Several Variables
K
Kepler’s laws of planetary motion 3.4 Motion in Space
L
Lagrange multiplier 4.8 Lagrange Multipliers
Laplace operator 6.5 Divergence and Curl
level curve of a function of two variables 4.1 Functions of Several Variables
level surface of a function of three variables 4.1 Functions of Several Variables
limit of a function of two variables 4.2 Limits and Continuity
limit of a vector-valued function 3.1 Vector-Valued Functions and Space Curves
line integral 6.2 Line Integrals
M
mass of a wire 6.2 Line Integrals
method of Lagrange multipliers 4.8 Lagrange Multipliers
method of undetermined coefficients 7.2 Nonhomogeneous Linear Equations
method of variation of parameters 7.2 Nonhomogeneous Linear Equations
mixed partial derivatives 4.3 Partial Derivatives
N
nonhomogeneous linear equation 7.1 Second-Order Linear Equations
normal component of acceleration 3.4 Motion in Space
normal form of Green’s theorem 6.4 Green’s Theorem
Q
Quadric surfaces 2.6 Quadric Surfaces
S
scalar line integral 6.2 Line Integrals
scalar multiplication 2.1 Vectors in the Plane
scalar projection 2.3 The Dot Product
simple harmonic motion 7.3 Applications
simply connected region 6.3 Conservative Vector Fields
space-filling curve 1.3 Polar Coordinates
space-filling curves 1.1 Parametric Equations
spring-mass system 7.3 Applications
standard equation of a sphere 2.2 Vectors in Three Dimensions
standard form 1.5 Conic Sections
standard unit vectors 2.1 Vectors in the Plane
standard-position vector 2.1 Vectors in the Plane
steady-state solution 7.3 Applications
stream function 6.4 Green’s Theorem
superposition principle 7.1 Second-Order Linear Equations
surface independent 6.7 Stokes’ Theorem
surface integral 6.6 Surface Integrals
surface integral of a scalar-valued function 6.6 Surface Integrals
surface integral of a vector field 6.6 Surface Integrals
symmetric equations of a line 2.5 Equations of Lines and Planes in Space
U
unit vector field 6.1 Vector Fields
W
William Thomson (Lord Kelvin) 4.3 Partial Derivatives
work done by a vector field 6.2 Line Integrals
work done by the force 2.3 The Dot Product
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