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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
component
a scalar that describes either the vertical or horizontal direction of a vector
coordinate plane
a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the xy-plane, xz-plane, or the yz-plane
cross product
u×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)k,u×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)k, where u=u1,u2,u3u=u1,u2,u3 and v=v1,v2,v3v=v1,v2,v3
cylinder
a set of lines parallel to a given line passing through a given curve
cylindrical coordinate system
a way to describe a location in space with an ordered triple (r,θ,z),(r,θ,z), where (r,θ)(r,θ) represents the polar coordinates of the point’s projection in the xy-plane, and zz represents the point’s projection onto the z-axis
determinant
a real number associated with a square matrix
direction angles
the angles formed by a nonzero vector and the coordinate axes
direction cosines
the cosines of the angles formed by a nonzero vector and the coordinate axes
direction vector
a vector parallel to a line that is used to describe the direction, or orientation, of the line in space
dot product or scalar product
u·v=u1v1+u2v2+u3v3u·v=u1v1+u2v2+u3v3 where u=u1,u2,u3u=u1,u2,u3 and v=v1,v2,v3v=v1,v2,v3
ellipsoid
a three-dimensional surface described by an equation of the form x2a2+y2b2+z2c2=1;x2a2+y2b2+z2c2=1; all traces of this surface are ellipses
elliptic cone
a three-dimensional surface described by an equation of the form x2a2+y2b2z2c2=0;x2a2+y2b2z2c2=0; traces of this surface include ellipses and intersecting lines
elliptic paraboloid
a three-dimensional surface described by an equation of the form z=x2a2+y2b2;z=x2a2+y2b2; traces of this surface include ellipses and parabolas
equivalent vectors
vectors that have the same magnitude and the same direction
general form of the equation of a plane
an equation in the form ax+by+cz+d=0,ax+by+cz+d=0, where n=a,b,cn=a,b,c is a normal vector of the plane, P=(x0,y0,z0)P=(x0,y0,z0) is a point on the plane, and d=ax0by0cz0d=ax0by0cz0
hyperboloid of one sheet
a three-dimensional surface described by an equation of the form x2a2+y2b2z2c2=1;x2a2+y2b2z2c2=1; traces of this surface include ellipses and hyperbolas
hyperboloid of two sheets
a three-dimensional surface described by an equation of the form z2c2x2a2y2b2=1;z2c2x2a2y2b2=1; traces of this surface include ellipses and hyperbolas
initial point
the starting point of a vector
magnitude
the length of a vector
normal vector
a vector perpendicular to a plane
normalization
using scalar multiplication to find a unit vector with a given direction
octants
the eight regions of space created by the coordinate planes
orthogonal vectors
vectors that form a right angle when placed in standard position
parallelepiped
a three-dimensional prism with six faces that are parallelograms
parallelogram method
a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram
parametric equations of a line
the set of equations x=x0+ta,x=x0+ta, y=y0+tb,y=y0+tb, and z=z0+tcz=z0+tc describing the line with direction vector v=a,b,cv=a,b,c passing through point (x0,y0,z0)(x0,y0,z0)
quadric surfaces
surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)
right-hand rule
a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the z-axis in such a way that the fingers curl from the positive x-axis to the positive y-axis, the thumb points in the direction of the positive z-axis
rulings
parallel lines that make up a cylindrical surface
scalar
a real number
scalar equation of a plane
the equation a(xx0)+b(yy0)+c(zz0)=0a(xx0)+b(yy0)+c(zz0)=0 used to describe a plane containing point P=(x0,y0,z0)P=(x0,y0,z0) with normal vector n=a,b,cn=a,b,c or its alternate form ax+by+cz+d=0,ax+by+cz+d=0, where d=ax0by0cz0d=ax0by0cz0
scalar multiplication
a vector operation that defines the product of a scalar and a vector
scalar projection
the magnitude of the vector projection of a vector
skew lines
two lines that are not parallel but do not intersect
sphere
the set of all points equidistant from a given point known as the center
spherical coordinate system
a way to describe a location in space with an ordered triple (ρ,θ,φ),(ρ,θ,φ), where ρρ is the distance between PP and the origin (ρ0),(ρ0), θθ is the same angle used to describe the location in cylindrical coordinates, and φφ is the angle formed by the positive z-axis and line segment OP,OP, where OO is the origin and 0φπ0φπ
standard equation of a sphere
(xa)2+(yb)2+(zc)2=r2(xa)2+(yb)2+(zc)2=r2 describes a sphere with center (a,b,c)(a,b,c) and radius rr
standard unit vectors
unit vectors along the coordinate axes: i=1,0,j=0,1i=1,0,j=0,1
standard-position vector
a vector with initial point (0,0)(0,0)
symmetric equations of aa line
the equations xx0a=yy0b=zz0cxx0a=yy0b=zz0c describing the line with direction vector v=a,b,cv=a,b,c passing through point (x0,y0,z0)(x0,y0,z0)
terminal point
the endpoint of a vector
three-dimensional rectangular coordinate system
a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple (x,y,z)(x,y,z) that plots its location relative to the defining axes
torque
the effect of a force that causes an object to rotate
trace
the intersection of a three-dimensional surface with a coordinate plane
triangle inequality
the length of any side of a triangle is less than the sum of the lengths of the other two sides
triangle method
a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector
triple scalar product
the dot product of a vector with the cross product of two other vectors: u·(v×w)u·(v×w)
unit vector
a vector with margnitude 11
vector
a mathematical object that has both magnitude and direction
vector addition
a vector operation that defines the sum of two vectors
vector difference
the vector difference vwvw is defined as v+(w)=v+(−1)wv+(w)=v+(−1)w
vector equation of a line
the equation r=r0+tvr=r0+tv used to describe a line with direction vector v=a,b,cv=a,b,c passing through point P=(x0,y0,z0),P=(x0,y0,z0), where r0=x0,y0,z0,r0=x0,y0,z0, is the position vector of point PP
vector equation of a plane
the equation n·PQ=0,n·PQ=0, where PP is a given point in the plane, QQ is any point in the plane, and nn is a normal vector of the plane
vector product
the cross product of two vectors
vector projection
the component of a vector that follows a given direction
vector sum
the sum of two vectors, vv and w,w, can be constructed graphically by placing the initial point of ww at the terminal point of v;v; then the vector sum v+wv+w is the vector with an initial point that coincides with the initial point of v,v, and with a terminal point that coincides with the terminal point of ww
work done by a force
work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector F and the displacement of an object by a vector s, then the work done by the force is the dot product of F and s.
zero vector
the vector with both initial point and terminal point (0,0)(0,0)
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