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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 point
a point P0P0 of RR is a boundary point if every δδ disk centered around P0P0 contains points both inside and outside RR
closed set
a set SS that contains all its boundary points
connected set
an open set SS that cannot be represented as the union of two or more disjoint, nonempty open subsets
constraint
an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem
contour map
a plot of the various level curves of a given function f(x,y)f(x,y)
critical point of a function of two variables
the point (x0,y0)(x0,y0) is called a critical point of f(x,y)f(x,y) if one of the two following conditions holds:
  1. fx(x0,y0)=fy(x0,y0)=0fx(x0,y0)=fy(x0,y0)=0
  2. At least one of fx(x0,y0)fx(x0,y0) and fy(x0,y0)fy(x0,y0) do not exist
differentiable
a function f(x,y)f(x,y) is differentiable at (x0,y0)(x0,y0) if f(x,y)f(x,y) can be expressed in the form f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),
where the error term E(x,y)E(x,y) satisfies lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0
directional derivative
the derivative of a function in the direction of a given unit vector
discriminant
the discriminant of the function f(x,y)f(x,y) is given by the formula D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2
function of two variables
a function z=f(x,y)z=f(x,y) that maps each ordered pair (x,y)(x,y) in a subset DD of 22 to a unique real number zz
generalized chain rule
the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables
gradient
the gradient of the function f(x,y)f(x,y) is defined to be f(x,y)=(f/x)i+(f/y)j,f(x,y)=(f/x)i+(f/y)j, which can be generalized to a function of any number of independent variables
graph of a function of two variables
a set of ordered triples (x,y,z)(x,y,z) that satisfies the equation z=f(x,y)z=f(x,y) plotted in three-dimensional Cartesian space
higher-order partial derivatives
second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives
interior point
a point P0P0 of RR is a boundary point if there is a δδ disk centered around P0P0 contained completely in RR
intermediate variable
given a composition of functions (e.g., f(x(t),y(t))),f(x(t),y(t))), the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function f(x(t),y(t)),f(x(t),y(t)), the variables xandyxandy are examples of intermediate variables
Lagrange multiplier
the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable λλ
level curve of a function of two variables
the set of points satisfying the equation f(x,y)=cf(x,y)=c for some real number cc in the range of ff
level surface of a function of three variables
the set of points satisfying the equation f(x,y,z)=cf(x,y,z)=c for some real number cc in the range of ff
linear approximation
given a function f(x,y)f(x,y) and a tangent plane to the function at a point (x0,y0),(x0,y0), we can approximate f(x,y)f(x,y) for points near (x0,y0)(x0,y0) using the tangent plane formula
method of Lagrange multipliers
a method of solving an optimization problem subject to one or more constraints
mixed partial derivatives
second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables
objective function
the function that is to be maximized or minimized in an optimization problem
open set
a set SS that contains none of its boundary points
optimization problem
calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers
partial derivative
a derivative of a function of more than one independent variable in which all the variables but one are held constant
partial differential equation
an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives
region
an open, connected, nonempty subset of 22
saddle point
given the function z=f(x,y),z=f(x,y), the point (x0,y0,f(x0,y0))(x0,y0,f(x0,y0)) is a saddle point if both fx(x0,y0)=0fx(x0,y0)=0 and fy(x0,y0)=0,fy(x0,y0)=0, but ff does not have a local extremum at (x0,y0)(x0,y0)
surface
the graph of a function of two variables, z=f(x,y)z=f(x,y)
tangent plane
given a function f(x,y)f(x,y) that is differentiable at a point (x0,y0),(x0,y0), the equation of the tangent plane to the surface z=f(x,y)z=f(x,y) is given by z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)
total differential
the total differential of the function f(x,y)f(x,y) at (x0,y0)(x0,y0) is given by the formula dz=fx(x0,y0)dx+fy(x0,y0)dydz=fx(x0,y0)dx+fy(x0,y0)dy
tree diagram
illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for
vertical trace
the set of ordered triples (c,y,z)(c,y,z) that solves the equation f(c,y)=zf(c,y)=z for a given constant x=cx=c or the set of ordered triples (x,d,z)(x,d,z) that solves the equation f(x,d)=zf(x,d)=z for a given constant y=dy=d
δδ ball
all points in 33 lying at a distance of less than δδ from (x0,y0,z0)(x0,y0,z0)
δδ disk
an open disk of radius δδ centered at point (a,b)(a,b)
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