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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
chambered nautilus Introduction, 1.3 Polar Coordinates
characteristic equation 7.1 Second-Order Linear Equations
circulation 6.2 Line Integrals
Clairaut’s theorem 6.1 Vector Fields
Cobb-Douglas function 4.8 Lagrange Multipliers
Cobb-Douglas production function 4.3 Partial Derivatives
complementary equation 7.2 Nonhomogeneous Linear Equations
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 product 2.4 The Cross Product
cross-partial property 6.1 Vector Fields
curtate cycloid 1.1 Parametric Equations
cylindrical coordinate system 2.7 Cylindrical and Spherical Coordinates
F
Faraday’s law 6.7 Stokes’ Theorem
flow line 6.1 Vector Fields
flux integral 6.6 Surface Integrals
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 1.5 Conic Sections
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
gravitational force 6.3 Conservative Vector Fields
grid curves 6.6 Surface Integrals
H
harmonic function 6.4 Green’s Theorem
heat equation 4.3 Partial Derivatives
higher-order partial derivatives 4.3 Partial Derivatives
homogeneous functions 4.5 The Chain Rule
homogeneous linear equation 7.1 Second-Order Linear Equations
Hooke’s law 7.3 Applications
hurricanes 6.1 Vector Fields
Hyperboloid of One Sheet 2.6 Quadric Surfaces
Hyperboloid of Two Sheets 2.6 Quadric Surfaces
I
implicit differentiation 4.5 The Chain Rule
indefinite integral of a vector-valued function 3.2 Calculus of Vector-Valued Functions
independent of path 6.3 Conservative Vector Fields
independent random variables 5.2 Double Integrals over General Regions
independent variables 4.5 The Chain Rule
initial-value problems 7.1 Second-Order Linear Equations
interior point 4.2 Limits and Continuity
intermediate variables 4.5 The Chain Rule
inverse-square law 6.8 The Divergence Theorem
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
linearly independent 7.1 Second-Order Linear Equations
M
major axis 1.5 Conic Sections
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
minor axis 1.5 Conic Sections
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
normalization 2.1 Vectors in the Plane
O
objective function 4.8 Lagrange Multipliers
optimization problem 4.8 Lagrange Multipliers
orientation of a curve 6.2 Line Integrals
orientation of a surface 6.6 Surface Integrals
orthogonal 2.3 The Dot Product
orthogonal vectors 2.3 The Dot Product
osculating circle 3.3 Arc Length and Curvature
overdamped 7.3 Applications
P
parallelepiped 2.4 The Cross Product
parallelogram method 2.1 Vectors in the Plane
parameter domain 6.6 Surface Integrals
parameter space 6.6 Surface Integrals
parameterization of a curve 1.1 Parametric Equations
parameterized surface 6.6 Surface Integrals
parametric curve 1.1 Parametric Equations
parametric equations 1.1 Parametric Equations
parametric equations of a line 2.5 Equations of Lines and Planes in Space
parametric surface 6.6 Surface Integrals
partial derivative 4.3 Partial Derivatives
partial differential equation 4.3 Partial Derivatives
perpendicular 2.3 The Dot Product
piecewise smooth curve 6.2 Line Integrals
polar coordinate system 1.3 Polar Coordinates
polar equations 1.3 Polar Coordinates
potential function 6.1 Vector Fields
principal unit normal vector 3.3 Arc Length and Curvature
principal unit tangent vector 3.2 Calculus of Vector-Valued Functions
projectile motion 3.4 Motion in Space
prolate cycloid 1.1 Parametric Equations
Q
Quadric surfaces 2.6 Quadric Surfaces
R
radial coordinate 1.3 Polar Coordinates
radial field 6.1 Vector Fields
radius of curvature 3.3 Arc Length and Curvature
regular parameterization 6.6 Surface Integrals
resolution of a vector into components 2.3 The Dot Product
resonance 7.3 Applications
RLC series circuit 7.3 Applications
rotational field 6.1 Vector Fields
S
scalar equation of a plane 2.5 Equations of Lines and Planes in Space
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
spherical coordinate system 2.7 Cylindrical and Spherical Coordinates
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
sum and difference rules 3.2 Calculus of Vector-Valued Functions
superposition principle 7.1 Second-Order Linear Equations
surface area 6.6 Surface Integrals
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
T
Tacoma Narrows Bridge 7.3 Applications
tangential component of acceleration 3.4 Motion in Space
tangential form of Green’s theorem 6.4 Green’s Theorem
three-dimensional rectangular coordinate system 2.2 Vectors in Three Dimensions
transient solution 7.3 Applications
tree diagram 4.5 The Chain Rule
triangle inequality 2.1 Vectors in the Plane
triangle method 2.1 Vectors in the Plane
triple integral 5.4 Triple Integrals
triple integral in cylindrical coordinates 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
triple integral in spherical coordinates 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
triple scalar product 2.4 The Cross Product
U
unit vector field 6.1 Vector Fields
V
vector addition 2.1 Vectors in the Plane
vector difference 2.1 Vectors in the Plane
vector equation of a line 2.5 Equations of Lines and Planes in Space
vector equation of a plane 2.5 Equations of Lines and Planes in Space
vector field 6.1 Vector Fields
vector line integral 6.2 Line Integrals
vector product 2.4 The Cross Product
vector projection 2.3 The Dot Product
vector-valued functions 3.4 Motion in Space
velocity vector 3.4 Motion in Space
W
wave equation 4.3 Partial Derivatives
William Thomson (Lord Kelvin) 4.3 Partial Derivatives
witch of Agnesi 1.1 Parametric Equations
work done by a vector field 6.2 Line Integrals
work done by the force 2.3 The Dot Product
Z
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