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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
boundary conditions
the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times
boundary-value problem
a differential equation with associated boundary conditions
characteristic equation
the equation aλ2+bλ+c=0aλ2+bλ+c=0 for the differential equation ay+by+cy=0ay+by+cy=0
complementary equation
for the nonhomogeneous linear differential equation
a2(x)y+a1(x)y+a0(x)y=r(x),a2(x)y+a1(x)y+a0(x)y=r(x),

the associated homogeneous equation, called the complementary equation, is
a2(x)y+a1(x)y+a0(x)y=0a2(x)y+a1(x)y+a0(x)y=0
homogeneous linear equation
a second-order differential equation that can be written in the form a2(x)y+a1(x)y+a0(x)y=r(x),a2(x)y+a1(x)y+a0(x)y=r(x), but r(x)=0r(x)=0 for every value of xx
linearly dependent
a set of functions f1(x),f2(x),…,fn(x)f1(x),f2(x),…,fn(x) for which there are constants c1,c2,…cn,c1,c2,…cn, not all zero, such that c1f1(x)+c2f2(x)++cnfn(x)=0c1f1(x)+c2f2(x)++cnfn(x)=0 for all x in the interval of interest
linearly independent
a set of functions f1(x),f2(x),…,fn(x)f1(x),f2(x),…,fn(x) for which there are no constants c1,c2,…cn,c1,c2,…cn, such that c1f1(x)+c2f2(x)++cnfn(x)=0c1f1(x)+c2f2(x)++cnfn(x)=0 for all x in the interval of interest
method of undetermined coefficients
a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess
method of variation of parameters
a method that involves looking for particular solutions in the form yp(x)=u(x)y1(x)+v(x)y2(x),yp(x)=u(x)y1(x)+v(x)y2(x), where y1y1 and y2y2 are linearly independent solutions to the complementary equations, and then solving a system of equations to find u(x)u(x) and v(x)v(x)
nonhomogeneous linear equation
a second-order differential equation that can be written in the form a2(x)y+a1(x)y+a0(x)y=r(x),a2(x)y+a1(x)y+a0(x)y=r(x), but r(x)0r(x)0 for some value of xx
particular solution
a solution yp(x)yp(x) of a differential equation that contains no arbitrary constants
RLC series circuit
a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit
simple harmonic motion
motion described by the equation x(t)=c1cos(ωt)+c2sin(ωt),x(t)=c1cos(ωt)+c2sin(ωt), as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely
steady-state solution
a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution
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