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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
double integral
of the function f(x,y)f(x,y) over the region RR in the xyxy-plane is defined as the limit of a double Riemann sum, Rf(x,y)dA=limm,ni=1mj=1nf(xij*,yij*)ΔA.Rf(x,y)dA=limm,ni=1mj=1nf(xij*,yij*)ΔA.
double Riemann sum
of the function f(x,y)f(x,y) over a rectangular region RR is i=1mj=1nf(xij*,yij*)ΔAi=1mj=1nf(xij*,yij*)ΔA where RR is divided into smaller subrectangles RijRij and (xij*,yij*)(xij*,yij*) is an arbitrary point in RijRij
Fubini’s theorem
if f(x,y)f(x,y) is a function of two variables that is continuous over a rectangular region R={(x,y)2|axb,cyd},R={(x,y)2|axb,cyd}, then the double integral of ff over the region equals an iterated integral, Rf(x,y)dydx=abcdf(x,y)dxdy=cdabf(x,y)dxdyRf(x,y)dydx=abcdf(x,y)dxdy=cdabf(x,y)dxdy
improper double integral
a double integral over an unbounded region or of an unbounded function
iterated integral
for a function f(x,y)f(x,y) over the region RR is
  1. abcdf(x,y)dxdy=ab[cdf(x,y)dy]dx,abcdf(x,y)dxdy=ab[cdf(x,y)dy]dx,
  2. cdbaf(x,y)dxdy=cd[abf(x,y)dx]dy,cdbaf(x,y)dxdy=cd[abf(x,y)dx]dy,
where a,b,c,a,b,c, and dd are any real numbers and R=[a,b]×[c,d]R=[a,b]×[c,d]
Jacobian
the Jacobian J(u,v)J(u,v) in two variables is a 2×22×2 determinant:
J(u,v)=|xuyuxvyv|;J(u,v)=|xuyuxvyv|;

the Jacobian J(u,v,w)J(u,v,w) in three variables is a 3×33×3 determinant:
J(u,v,w)=|xuyuzuxvyvzvxwywzw|J(u,v,w)=|xuyuzuxvyvzvxwywzw|
one-to-one transformation
a transformation T:GRT:GR defined as T(u,v)=(x,y)T(u,v)=(x,y) is said to be one-to-one if no two points map to the same image point
planar transformation
a function TT that transforms a region GG in one plane into a region RR in another plane by a change of variables
polar rectangle
the region enclosed between the circles r=ar=a and r=br=b and the angles θ=αθ=α and θ=β;θ=β; it is described as R={(r,θ)|arb,αθβ}R={(r,θ)|arb,αθβ}
radius of gyration
the distance from an object’s center of mass to its axis of rotation
transformation
a function that transforms a region GG in one plane into a region RR in another plane by a change of variables
triple integral
the triple integral of a continuous function f(x,y,z)f(x,y,z) over a rectangular solid box BB is the limit of a Riemann sum for a function of three variables, if this limit exists
triple integral in cylindrical coordinates
the limit of a triple Riemann sum, provided the following limit exists:
liml,m,ni=1lj=1mk=1nf(rijk*,θijk*,zijk*)rijk*ΔrΔθΔzliml,m,ni=1lj=1mk=1nf(rijk*,θijk*,zijk*)rijk*ΔrΔθΔz
triple integral in spherical coordinates
the limit of a triple Riemann sum, provided the following limit exists:
liml,m,ni=1lj=1mk=1nf(ρijk*,θijk*,φijk*)(ρijk*)2sinφΔρΔθΔφliml,m,ni=1lj=1mk=1nf(ρijk*,θijk*,φijk*)(ρijk*)2sinφΔρΔθΔφ
Type I
a region DD in the xyxy-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions g1(x)g1(x) and g2(x)g2(x)
Type II
a region DD in the xyxy-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions h1(y)andh2(y)h1(y)andh2(y)
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