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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
acceleration vector
the second derivative of the position vector
arc-length function
a function s(t)s(t) that describes the arc length of curve C as a function of t
arc-length parameterization
a reparameterization of a vector-valued function in which the parameter is equal to the arc length
binormal vector
a unit vector orthogonal to the unit tangent vector and the unit normal vector
component functions
the component functions of the vector-valued function r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j are f(t)f(t) and g(t),g(t), and the component functions of the vector-valued function r(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)k are f(t),f(t), g(t)g(t) and h(t)h(t)
curvature
the derivative of the unit tangent vector with respect to the arc-length parameter
definite integral of a vector-valued function
the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function
derivative of a vector-valued function
the derivative of a vector-valued function r(t)r(t) is r(t)=limΔt0r(t+Δt)r(t)Δt,r(t)=limΔt0r(t+Δt)r(t)Δt, provided the limit exists
Frenet frame of reference
(TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector
helix
a three-dimensional curve in the shape of a spiral
indefinite integral of a vector-valued function
a vector-valued function with a derivative that is equal to a given vector-valued function
Kepler’s laws of planetary motion
three laws governing the motion of planets, asteroids, and comets in orbit around the Sun
limit of a vector-valued function
a vector-valued function r(t)r(t) has a limit L as t approaches a if limta|r(t)L|=0limta|r(t)L|=0
normal component of acceleration
the coefficient of the unit normal vector N when the acceleration vector is written as a linear combination of TT and NN
normal plane
a plane that is perpendicular to a curve at any point on the curve
osculating circle
a circle that is tangent to a curve C at a point P and that shares the same curvature
osculating plane
the plane determined by the unit tangent and the unit normal vector
plane curve
the set of ordered pairs (f(t),g(t))(f(t),g(t)) together with their defining parametric equations x=f(t)x=f(t) and y=g(t)y=g(t)
principal unit normal vector
a vector orthogonal to the unit tangent vector, given by the formula T(t)T(t)T(t)T(t)
principal unit tangent vector
a unit vector tangent to a curve C
projectile motion
motion of an object with an initial velocity but no force acting on it other than gravity
radius of curvature
the reciprocal of the curvature
reparameterization
an alternative parameterization of a given vector-valued function
smooth
curves where the vector-valued function r(t)r(t) is differentiable with a non-zero derivative
space curve
the set of ordered triples (f(t),g(t),h(t))(f(t),g(t),h(t)) together with their defining parametric equations x=f(t),x=f(t), y=g(t)y=g(t) and z=h(t)z=h(t)
tangent vector
to r(t)r(t) at t=t0t=t0 any vector v such that, when the tail of the vector is placed at point r(t0)r(t0) on the graph, vector v is tangent to curve C
tangential component of acceleration
the coefficient of the unit tangent vector T when the acceleration vector is written as a linear combination of TT and NN
vector parameterization
any representation of a plane or space curve using a vector-valued function
vector-valued function
a function of the form r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j or r(t)=f(t)i+g(t)j+h(t)k,r(t)=f(t)i+g(t)j+h(t)k, where the component functions f, g, and h are real-valued functions of the parameter t
velocity vector
the derivative of the position vector
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