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Calculus Volume 3

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Calculus Volume 3Index
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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
Symbols
δδ ball4.2 Limits and Continuity
δδ disk4.2 Limits and Continuity
A
acceleration vector3.4 Motion in Space
angular coordinate1.3 Polar Coordinates
angular frequency7.3 Applications
arc-length function3.3 Arc Length and Curvature
arc-length parameterization3.3 Arc Length and Curvature
Archimedean spiral1.3 Polar Coordinates
B
Bessel functions7.4 Series Solutions of Differential Equations
binormal vector3.3 Arc Length and Curvature
boundary conditions7.1 Second-Order Linear Equations
boundary point4.2 Limits and Continuity
boundary-value problem7.1 Second-Order Linear Equations
Brahe3.4 Motion in Space
C
cardioid1.3 Polar Coordinates
chain rule3.2 Calculus of Vector-Valued Functions
chambered nautilusIntroduction, 1.3 Polar Coordinates
characteristic equation7.1 Second-Order Linear Equations
circulation6.2 Line Integrals
cissoid of Diocles1.4 Area and Arc Length in Polar Coordinates
Clairaut’s theorem6.1 Vector Fields
closed curve6.2 Line Integrals, 6.3 Conservative Vector Fields
closed set4.2 Limits and Continuity
Cobb-Douglas function4.8 Lagrange Multipliers
Cobb-Douglas production function4.3 Partial Derivatives
complementary equation7.2 Nonhomogeneous Linear Equations
complex conjugates7.1 Second-Order Linear Equations
complex number7.1 Second-Order Linear Equations
component functions3.1 Vector-Valued Functions and Space Curves
components2.1 Vectors in the Plane
conic section1.5 Conic Sections
connected region6.3 Conservative Vector Fields
connected set4.2 Limits and Continuity
conservative field6.1 Vector Fields
constant multiple rule3.2 Calculus of Vector-Valued Functions
constraints4.8 Lagrange Multipliers
contour map4.1 Functions of Several Variables, 4.3 Partial Derivatives
coordinate planes2.2 Vectors in Three Dimensions
critical point of a function of two variables4.7 Maxima/Minima Problems
cross product2.4 The Cross Product
cross-partial property6.1 Vector Fields
curl6.5 Divergence and Curl
curtate cycloid1.1 Parametric Equations
curvature3.3 Arc Length and Curvature
cusps1.1 Parametric Equations
cycloid1.1 Parametric Equations
cylinder2.6 Quadric Surfaces
cylindrical coordinate system2.7 Cylindrical and Spherical Coordinates
D
definite integral of a vector-valued function3.2 Calculus of Vector-Valued Functions
derivative3.2 Calculus of Vector-Valued Functions
derivative of a vector-valued function3.2 Calculus of Vector-Valued Functions
determinant2.4 The Cross Product
differentiable4.4 Tangent Planes and Linear Approximations
direction angles2.3 The Dot Product
direction cosines2.3 The Dot Product
direction vector2.5 Equations of Lines and Planes in Space
directional cosines4.6 Directional Derivatives and the Gradient
directional derivative4.6 Directional Derivatives and the Gradient
directrix1.5 Conic Sections
discriminant1.5 Conic Sections, 4.7 Maxima/Minima Problems
divergence6.5 Divergence and Curl
divergence theorem6.8 The Divergence Theorem
domain6.1 Vector Fields
dot product2.3 The Dot Product
double integral5.1 Double Integrals over Rectangular Regions
double Riemann sum5.1 Double Integrals over Rectangular Regions
E
Earth’s orbit1.1 Parametric Equations
eccentricity1.5 Conic Sections
electrical potential4.6 Directional Derivatives and the Gradient
Electrical power4.4 Tangent Planes and Linear Approximations
electrical resistance4.4 Tangent Planes and Linear Approximations
electrostatic fields6.8 The Divergence Theorem
ellipsoid2.6 Quadric Surfaces
Elliptic Cone2.6 Quadric Surfaces
elliptic paraboloid2.6 Quadric Surfaces, 2.6 Quadric Surfaces
epitrochoid1.1 Parametric Equations
equivalent vectors2.1 Vectors in the Plane
Ernest Rutherford4.3 Partial Derivatives
error term4.4 Tangent Planes and Linear Approximations
Euler’s formula7.1 Second-Order Linear Equations
expected values5.2 Double Integrals over General Regions
F
Faraday’s law6.7 Stokes’ Theorem
flow line6.1 Vector Fields
flux6.2 Line Integrals
flux integral6.6 Surface Integrals
focal parameter1.5 Conic Sections
focus1.5 Conic Sections
force2.1 Vectors in the Plane
Fourier’s law of heat transfer6.8 The Divergence Theorem
Frenet frame of reference3.3 Arc Length and Curvature
Fubini’s theorem5.1 Double Integrals over Rectangular Regions
Fubini’s thereom5.4 Triple Integrals
function of two variables4.1 Functions of Several Variables
Fundamental Theorem for Line Integrals6.8 The Divergence Theorem
Fundamental Theorem for Line Integrals.6.3 Conservative Vector Fields
Fundamental Theorem of Calculus6.8 The Divergence Theorem
G
Gauss’ law6.8 The Divergence Theorem
Gauss’s law for magnetism6.5 Divergence and Curl
general bounded region5.4 Triple Integrals
general form1.5 Conic Sections
general form of the equation of a plane2.5 Equations of Lines and Planes in Space
general solution to a differential equation7.1 Second-Order Linear Equations
generalized chain rule4.5 The Chain Rule
gradient4.6 Directional Derivatives and the Gradient
gradient field6.1 Vector Fields
graph of a function of two variables4.1 Functions of Several Variables
gravitational force6.3 Conservative Vector Fields
Green’s theorem6.4 Green’s Theorem, 6.8 The Divergence Theorem
grid curves6.6 Surface Integrals
H
harmonic function6.4 Green’s Theorem
heat equation4.3 Partial Derivatives
heat flow6.6 Surface Integrals
helix3.1 Vector-Valued Functions and Space Curves, 3.1 Vector-Valued Functions and Space Curves
higher-order partial derivatives4.3 Partial Derivatives
homogeneous functions4.5 The Chain Rule
homogeneous linear equation7.1 Second-Order Linear Equations
Hooke’s law7.3 Applications
hurricanes6.1 Vector Fields
Hyperboloid of One Sheet2.6 Quadric Surfaces
Hyperboloid of Two Sheets2.6 Quadric Surfaces
hypocycloid1.1 Parametric Equations
I
implicit differentiation4.5 The Chain Rule
improper double integral5.2 Double Integrals over General Regions
indefinite integral of a vector-valued function3.2 Calculus of Vector-Valued Functions
independent of path6.3 Conservative Vector Fields
independent random variables5.2 Double Integrals over General Regions
independent variables4.5 The Chain Rule
initial point2.1 Vectors in the Plane, 3.1 Vector-Valued Functions and Space Curves
initial-value problems7.1 Second-Order Linear Equations
interior point4.2 Limits and Continuity
intermediate variables4.5 The Chain Rule
inverse-square law6.8 The Divergence Theorem
iterated integral5.1 Double Integrals over Rectangular Regions
J
Jacobian5.7 Change of Variables in Multiple Integrals
joint density function5.2 Double Integrals over General Regions
K
Kepler3.4 Motion in Space
Kepler’s laws of planetary motion3.4 Motion in Space
L
Lagrange multiplier4.8 Lagrange Multipliers
Laplace operator6.5 Divergence and Curl
Laplace’s equation4.3 Partial Derivatives, 6.4 Green’s Theorem
level curve of a function of two variables4.1 Functions of Several Variables
level surface of a function of three variables4.1 Functions of Several Variables
limaçon1.3 Polar Coordinates
limit of a function of two variables4.2 Limits and Continuity
limit of a vector-valued function3.1 Vector-Valued Functions and Space Curves
line integral6.2 Line Integrals
linear approximation4.4 Tangent Planes and Linear Approximations
linearly dependent7.1 Second-Order Linear Equations
linearly independent7.1 Second-Order Linear Equations
local extremum4.7 Maxima/Minima Problems
lunes of Alhazen5.2 Double Integrals over General Regions
M
magnitude2.1 Vectors in the Plane, 2.1 Vectors in the Plane
major axis1.5 Conic Sections
mass flux6.6 Surface Integrals
mass of a wire6.2 Line Integrals
method of Lagrange multipliers4.8 Lagrange Multipliers
method of undetermined coefficients7.2 Nonhomogeneous Linear Equations
method of variation of parameters7.2 Nonhomogeneous Linear Equations
minor axis1.5 Conic Sections
mixed partial derivatives4.3 Partial Derivatives
N
nappes1.5 Conic Sections
nonhomogeneous linear equation7.1 Second-Order Linear Equations
normal2.3 The Dot Product
normal component of acceleration3.4 Motion in Space
normal form of Green’s theorem6.4 Green’s Theorem
normal plane3.3 Arc Length and Curvature
normal vector2.5 Equations of Lines and Planes in Space
normalization2.1 Vectors in the Plane
O
objective function4.8 Lagrange Multipliers
octants2.2 Vectors in Three Dimensions
one-to-one transformation5.7 Change of Variables in Multiple Integrals
open set4.2 Limits and Continuity
optimization problem4.8 Lagrange Multipliers
orientation1.1 Parametric Equations
orientation of a curve6.2 Line Integrals
orientation of a surface6.6 Surface Integrals
orthogonal2.3 The Dot Product
orthogonal vectors2.3 The Dot Product
osculating circle3.3 Arc Length and Curvature
overdamped7.3 Applications
P
parallelepiped2.4 The Cross Product
parallelogram method2.1 Vectors in the Plane
parameter1.1 Parametric Equations
parameter domain6.6 Surface Integrals
parameter space6.6 Surface Integrals
parameterization of a curve1.1 Parametric Equations
parameterized surface6.6 Surface Integrals
parametric curve1.1 Parametric Equations
parametric equations1.1 Parametric Equations
parametric equations of a line2.5 Equations of Lines and Planes in Space
parametric surface6.6 Surface Integrals
partial derivative4.3 Partial Derivatives
partial differential equation4.3 Partial Derivatives
particular solution7.2 Nonhomogeneous Linear Equations
path independent6.3 Conservative Vector Fields
perpendicular2.3 The Dot Product
piecewise smooth curve6.2 Line Integrals
planar transformation5.7 Change of Variables in Multiple Integrals
plane curve3.1 Vector-Valued Functions and Space Curves
polar axis1.3 Polar Coordinates
polar coordinate system1.3 Polar Coordinates
polar equations1.3 Polar Coordinates
polar rectangle5.3 Double Integrals in Polar Coordinates
pole1.3 Polar Coordinates
potential function6.1 Vector Fields
power series7.4 Series Solutions of Differential Equations
principal unit normal vector3.3 Arc Length and Curvature
principal unit tangent vector3.2 Calculus of Vector-Valued Functions
product rule3.2 Calculus of Vector-Valued Functions
projectile motion3.4 Motion in Space
prolate cycloid1.1 Parametric Equations
Q
Quadric surfaces2.6 Quadric Surfaces
R
radial coordinate1.3 Polar Coordinates
radial field6.1 Vector Fields
radius of curvature3.3 Arc Length and Curvature
radius of gyration5.6 Calculating Centers of Mass and Moments of Inertia
region4.2 Limits and Continuity
regular parameterization6.6 Surface Integrals
reparameterization3.1 Vector-Valued Functions and Space Curves
resolution of a vector into components2.3 The Dot Product
resonance7.3 Applications
Reuleaux triangle5.2 Double Integrals over General Regions
right-hand rule2.2 Vectors in Three Dimensions
RLC series circuit7.3 Applications
rose1.3 Polar Coordinates
rotational field6.1 Vector Fields
rulings2.6 Quadric Surfaces
S
saddle point4.7 Maxima/Minima Problems
scalar2.1 Vectors in the Plane
scalar equation of a plane2.5 Equations of Lines and Planes in Space
scalar line integral6.2 Line Integrals
scalar multiplication2.1 Vectors in the Plane
scalar projection2.3 The Dot Product
simple curve6.3 Conservative Vector Fields
simple harmonic motion7.3 Applications
simple pendulum4.4 Tangent Planes and Linear Approximations
simply connected region6.3 Conservative Vector Fields
skew lines2.5 Equations of Lines and Planes in Space
smooth3.3 Arc Length and Curvature
space curve3.1 Vector-Valued Functions and Space Curves
space-filling curve1.3 Polar Coordinates
space-filling curves1.1 Parametric Equations
speed2.1 Vectors in the Plane
sphere2.2 Vectors in Three Dimensions
spherical coordinate system2.7 Cylindrical and Spherical Coordinates
spring-mass system7.3 Applications
standard equation of a sphere2.2 Vectors in Three Dimensions
standard form1.5 Conic Sections
standard position3.1 Vector-Valued Functions and Space Curves
standard unit vectors2.1 Vectors in the Plane
standard-position vector2.1 Vectors in the Plane
steady-state solution7.3 Applications
Stokes’ theorem6.7 Stokes’ Theorem, 6.8 The Divergence Theorem
stream function6.4 Green’s Theorem
sum and difference rules3.2 Calculus of Vector-Valued Functions
superposition principle7.1 Second-Order Linear Equations
surface4.1 Functions of Several Variables
surface area6.6 Surface Integrals
surface independent6.7 Stokes’ Theorem
surface integral6.6 Surface Integrals
surface integral of a scalar-valued function6.6 Surface Integrals
surface integral of a vector field6.6 Surface Integrals
symmetric equations of a line2.5 Equations of Lines and Planes in Space
symmetry1.3 Polar Coordinates, 5.3 Double Integrals in Polar Coordinates
T
Tacoma Narrows Bridge7.3 Applications
tangent plane4.4 Tangent Planes and Linear Approximations
tangent vector3.2 Calculus of Vector-Valued Functions
tangential component of acceleration3.4 Motion in Space
tangential form of Green’s theorem6.4 Green’s Theorem
terminal point2.1 Vectors in the Plane, 3.1 Vector-Valued Functions and Space Curves
three-dimensional rectangular coordinate system2.2 Vectors in Three Dimensions
topographical map4.1 Functions of Several Variables
Torque2.4 The Cross Product
total differential4.4 Tangent Planes and Linear Approximations
traces2.6 Quadric Surfaces
transformation5.7 Change of Variables in Multiple Integrals
transient solution7.3 Applications
tree diagram4.5 The Chain Rule
triangle inequality2.1 Vectors in the Plane
triangle method2.1 Vectors in the Plane
triple integral5.4 Triple Integrals
triple integral in cylindrical coordinates5.5 Triple Integrals in Cylindrical and Spherical Coordinates
triple integral in spherical coordinates5.5 Triple Integrals in Cylindrical and Spherical Coordinates
triple scalar product2.4 The Cross Product
Type I5.2 Double Integrals over General Regions
Type II5.2 Double Integrals over General Regions
U
unit vector2.1 Vectors in the Plane
unit vector field6.1 Vector Fields
V
vector2.1 Vectors in the Plane
vector addition2.1 Vectors in the Plane
vector difference2.1 Vectors in the Plane
vector equation of a line2.5 Equations of Lines and Planes in Space
vector equation of a plane2.5 Equations of Lines and Planes in Space
vector field6.1 Vector Fields
vector line integral6.2 Line Integrals
vector parameterization3.1 Vector-Valued Functions and Space Curves
vector product2.4 The Cross Product
vector projection2.3 The Dot Product
vector sum2.1 Vectors in the Plane
vector-valued function3.1 Vector-Valued Functions and Space Curves
vector-valued functions3.4 Motion in Space
velocity vector3.4 Motion in Space
vertex1.5 Conic Sections
vertical trace4.1 Functions of Several Variables
W
wave equation4.3 Partial Derivatives
William Thomson (Lord Kelvin)4.3 Partial Derivatives
witch of Agnesi1.1 Parametric Equations
work done by a vector field6.2 Line Integrals
work done by the force2.3 The Dot Product
Z
zero vector2.1 Vectors in the Plane
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