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Calculus Volume 3

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Calculus Volume 3Index
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Table of contents
  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
Symbols
δ δ ball 4.2 Limits and Continuity
δ δ disk 4.2 Limits and Continuity
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
B
Bessel functions 7.4 Series Solutions of Differential Equations
binormal vector 3.3 Arc Length and Curvature
boundary conditions 7.1 Second-Order Linear Equations
boundary point 4.2 Limits and Continuity
boundary-value problem 7.1 Second-Order Linear Equations
Brahe 3.4 Motion in Space
C
cardioid 1.3 Polar Coordinates
chain rule 3.2 Calculus of Vector-Valued Functions
chambered nautilus Introduction, 1.3 Polar Coordinates
characteristic equation 7.1 Second-Order Linear Equations
circulation 6.2 Line Integrals
cissoid of Diocles 1.4 Area and Arc Length in Polar Coordinates
Clairaut’s theorem 6.1 Vector Fields
closed curve 6.2 Line Integrals, 6.3 Conservative Vector Fields
closed set 4.2 Limits and Continuity
Cobb-Douglas function 4.8 Lagrange Multipliers
Cobb-Douglas production function 4.3 Partial Derivatives
complementary equation 7.2 Nonhomogeneous Linear Equations
complex conjugates 7.1 Second-Order Linear Equations
complex number 7.1 Second-Order Linear Equations
component functions 3.1 Vector-Valued Functions and Space Curves
components 2.1 Vectors in the Plane
conic section 1.5 Conic Sections
connected region 6.3 Conservative Vector Fields
connected set 4.2 Limits and Continuity
conservative field 6.1 Vector Fields
constant multiple rule 3.2 Calculus of Vector-Valued Functions
constraints 4.8 Lagrange Multipliers
contour map 4.1 Functions of Several Variables, 4.3 Partial Derivatives
coordinate planes 2.2 Vectors in Three Dimensions
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
curl 6.5 Divergence and Curl
curtate cycloid 1.1 Parametric Equations
curvature 3.3 Arc Length and Curvature
cusps 1.1 Parametric Equations
cycloid 1.1 Parametric Equations
cylinder 2.6 Quadric Surfaces
cylindrical coordinate system 2.7 Cylindrical and Spherical Coordinates
D
definite integral of a vector-valued function 3.2 Calculus of Vector-Valued Functions
derivative 3.2 Calculus of Vector-Valued Functions
derivative of a vector-valued function 3.2 Calculus of Vector-Valued Functions
determinant 2.4 The Cross Product
differentiable 4.4 Tangent Planes and Linear Approximations
direction angles 2.3 The Dot Product
direction cosines 2.3 The Dot Product
direction vector 2.5 Equations of Lines and Planes in Space
directional cosines 4.6 Directional Derivatives and the Gradient
directional derivative 4.6 Directional Derivatives and the Gradient
directrix 1.5 Conic Sections
discriminant 1.5 Conic Sections, 4.7 Maxima/Minima Problems
divergence 6.5 Divergence and Curl
divergence theorem 6.8 The Divergence Theorem
domain 6.1 Vector Fields
dot product 2.3 The Dot Product
double integral 5.1 Double Integrals over Rectangular Regions
double Riemann sum 5.1 Double Integrals over Rectangular Regions
E
Earth’s orbit 1.1 Parametric Equations
eccentricity 1.5 Conic Sections
electrical potential 4.6 Directional Derivatives and the Gradient
Electrical power 4.4 Tangent Planes and Linear Approximations
electrical resistance 4.4 Tangent Planes and Linear Approximations
electrostatic fields 6.8 The Divergence Theorem
ellipsoid 2.6 Quadric Surfaces
Elliptic Cone 2.6 Quadric Surfaces
elliptic paraboloid 2.6 Quadric Surfaces
Elliptic Paraboloid 2.6 Quadric Surfaces
epitrochoid 1.1 Parametric Equations
equivalent vectors 2.1 Vectors in the Plane
Ernest Rutherford 4.3 Partial Derivatives
error term 4.4 Tangent Planes and Linear Approximations
Euler’s formula 7.1 Second-Order Linear Equations
expected values 5.2 Double Integrals over General Regions
F
Faraday’s law 6.7 Stokes’ Theorem
flow line 6.1 Vector Fields
flux 6.2 Line Integrals
flux integral 6.6 Surface Integrals
focal parameter 1.5 Conic Sections
focus 1.5 Conic Sections
force 2.1 Vectors in the Plane
Fourier’s law of heat transfer 6.8 The Divergence Theorem
Frenet frame of reference 3.3 Arc Length and Curvature
Fubini’s theorem 5.1 Double Integrals over Rectangular Regions
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’ law 6.8 The Divergence Theorem
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 4.6 Directional Derivatives and the Gradient
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
Green’s theorem 6.4 Green’s Theorem, 6.8 The Divergence Theorem
grid curves 6.6 Surface Integrals
H
harmonic function 6.4 Green’s Theorem
heat equation 4.3 Partial Derivatives
heat flow 6.6 Surface Integrals
helix 3.1 Vector-Valued Functions and Space Curves, 3.1 Vector-Valued Functions and Space Curves
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
hypocycloid 1.1 Parametric Equations
I
implicit differentiation 4.5 The Chain Rule
improper double integral 5.2 Double Integrals over General Regions
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 point 2.1 Vectors in the Plane, 3.1 Vector-Valued Functions and Space Curves
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
iterated integral 5.1 Double Integrals over Rectangular Regions
J
Jacobian 5.7 Change of Variables in Multiple Integrals
joint density function 5.2 Double Integrals over General Regions
K
Kepler 3.4 Motion in Space
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
Laplace’s equation 4.3 Partial Derivatives, 6.4 Green’s Theorem
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
limaçon 1.3 Polar Coordinates
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
linear approximation 4.4 Tangent Planes and Linear Approximations
linearly dependent 7.1 Second-Order Linear Equations
linearly independent 7.1 Second-Order Linear Equations
local extremum 4.7 Maxima/Minima Problems
lunes of Alhazen 5.2 Double Integrals over General Regions
M
magnitude 2.1 Vectors in the Plane, 2.1 Vectors in the Plane
major axis 1.5 Conic Sections
mass flux 6.6 Surface Integrals
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
nappes 1.5 Conic Sections
nonhomogeneous linear equation 7.1 Second-Order Linear Equations
normal 2.3 The Dot Product
normal component of acceleration 3.4 Motion in Space
normal form of Green’s theorem 6.4 Green’s Theorem
normal plane 3.3 Arc Length and Curvature
normal vector 2.5 Equations of Lines and Planes in Space
normalization 2.1 Vectors in the Plane
O
objective function 4.8 Lagrange Multipliers
octants 2.2 Vectors in Three Dimensions
one-to-one transformation 5.7 Change of Variables in Multiple Integrals
open set 4.2 Limits and Continuity
optimization problem 4.8 Lagrange Multipliers
orientation 1.1 Parametric Equations
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 1.1 Parametric Equations
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
particular solution 7.2 Nonhomogeneous Linear Equations
path independent 6.3 Conservative Vector Fields
perpendicular 2.3 The Dot Product
piecewise smooth curve 6.2 Line Integrals
planar transformation 5.7 Change of Variables in Multiple Integrals
plane curve 3.1 Vector-Valued Functions and Space Curves
polar axis 1.3 Polar Coordinates
polar coordinate system 1.3 Polar Coordinates
polar equations 1.3 Polar Coordinates
polar rectangle 5.3 Double Integrals in Polar Coordinates
pole 1.3 Polar Coordinates
potential function 6.1 Vector Fields
power series 7.4 Series Solutions of Differential Equations
principal unit normal vector 3.3 Arc Length and Curvature
principal unit tangent vector 3.2 Calculus of Vector-Valued Functions
product rule 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
radius of gyration 5.6 Calculating Centers of Mass and Moments of Inertia
region 4.2 Limits and Continuity
regular parameterization 6.6 Surface Integrals
reparameterization 3.1 Vector-Valued Functions and Space Curves
resolution of a vector into components 2.3 The Dot Product
resonance 7.3 Applications
Reuleaux triangle 5.2 Double Integrals over General Regions
right-hand rule 2.2 Vectors in Three Dimensions
RLC series circuit 7.3 Applications
rose 1.3 Polar Coordinates
rotational field 6.1 Vector Fields
rulings 2.6 Quadric Surfaces
S
saddle point 4.7 Maxima/Minima Problems
scalar 2.1 Vectors in the Plane
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 curve 6.3 Conservative Vector Fields
simple harmonic motion 7.3 Applications
simple pendulum 4.4 Tangent Planes and Linear Approximations
simply connected region 6.3 Conservative Vector Fields
skew lines 2.5 Equations of Lines and Planes in Space
smooth 3.3 Arc Length and Curvature
space curve 3.1 Vector-Valued Functions and Space Curves
space-filling curves 1.1 Parametric Equations
speed 2.1 Vectors in the Plane
sphere 2.2 Vectors in Three Dimensions
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 position 3.1 Vector-Valued Functions and Space Curves
standard unit vectors 2.1 Vectors in the Plane
standard-position vector 2.1 Vectors in the Plane
steady-state solution 7.3 Applications
Stokes’ theorem 6.7 Stokes’ Theorem, 6.8 The Divergence Theorem
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 4.1 Functions of Several Variables
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
symmetry 1.3 Polar Coordinates, 5.3 Double Integrals in Polar Coordinates
T
Tacoma Narrows Bridge 7.3 Applications
tangent plane 4.4 Tangent Planes and Linear Approximations
tangent vector 3.2 Calculus of Vector-Valued Functions
tangential component of acceleration 3.4 Motion in Space
tangential form of Green’s theorem 6.4 Green’s Theorem
terminal point 2.1 Vectors in the Plane, 3.1 Vector-Valued Functions and Space Curves
three-dimensional rectangular coordinate system 2.2 Vectors in Three Dimensions
topographical map 4.1 Functions of Several Variables
Torque 2.4 The Cross Product
total differential 4.4 Tangent Planes and Linear Approximations
traces 2.6 Quadric Surfaces
transformation 5.7 Change of Variables in Multiple Integrals
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
Type I 5.2 Double Integrals over General Regions
Type II 5.2 Double Integrals over General Regions
U
unit vector 2.1 Vectors in the Plane
unit vector field 6.1 Vector Fields
V
vector 2.1 Vectors in the Plane
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 parameterization 3.1 Vector-Valued Functions and Space Curves
vector product 2.4 The Cross Product
vector projection 2.3 The Dot Product
vector sum 2.1 Vectors in the Plane
vector-valued function 3.1 Vector-Valued Functions and Space Curves
vector-valued functions 3.4 Motion in Space
velocity vector 3.4 Motion in Space
vertex 1.5 Conic Sections
vertical trace 4.1 Functions of Several Variables
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
zero vector 2.1 Vectors in the Plane
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