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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
circulation
the tendency of a fluid to move in the direction of curve C. If C is a closed curve, then the circulation of F along C is line integral CF·Tds,CF·Tds, which we also denote CF·TdsCF·Tds
closed curve
a curve for which there exists a parameterization r(t),r(t), atb,atb, such that r(a)=r(b),r(a)=r(b), and the curve is traversed exactly once
closed curve
a curve that begins and ends at the same point
connected region
a region in which any two points can be connected by a path with a trace contained entirely inside the region
conservative field
a vector field for which there exists a scalar function ff such that f=Ff=F
curl
the curl of vector field F=P,Q,R,F=P,Q,R, denoted ×F,×F, is the “determinant” of the matrix |ijkxyzPQR||ijkxyzPQR| and is given by the expression (RyQz)i+(PzRx)j+(QxPy)k;(RyQz)i+(PzRx)j+(QxPy)k; it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point
divergence
the divergence of a vector field F=P,Q,R,F=P,Q,R, denoted ×F,×F, is Px+Qy+Rz;Px+Qy+Rz; it measures the “outflowing-ness” of a vector field
divergence theorem
a theorem used to transform a difficult flux integral into an easier triple integral and vice versa
flux
the rate of a fluid flowing across a curve in a vector field; the flux of vector field F across plane curve C is line integral CF·n(t)n(t)dsCF·n(t)n(t)ds
flux integral
another name for a surface integral of a vector field; the preferred term in physics and engineering
Fundamental Theorem for Line Integrals
the value of line integral Cf·drCf·dr depends only on the value of ff at the endpoints of C: Cf·dr=f(r(b))f(r(a))Cf·dr=f(r(b))f(r(a))
Gauss’ law
if S is a piecewise, smooth closed surface in a vacuum and Q is the total stationary charge inside of S, then the flux of electrostatic field E across S is Q/ε0Q/ε0
gradient field
a vector field FF for which there exists a scalar function ff such that f=F;f=F; in other words, a vector field that is the gradient of a function; such vector fields are also called conservative
Green’s theorem
relates the integral over a connected region to an integral over the boundary of the region
grid curves
curves on a surface that are parallel to grid lines in a coordinate plane
heat flow
a vector field proportional to the negative temperature gradient in an object
independence of path
a vector field F has path independence if C1F·dr=C2F·drC1F·dr=C2F·dr for any curves C1C1 and C2C2 in the domain of F with the same initial points and terminal points
inverse-square law
the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge
line integral
the integral of a function along a curve in a plane or in space
mass flux
the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area
orientation of a curve
the orientation of a curve C is a specified direction of C
orientation of a surface
if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations
parameter domain (parameter space)
the region of the uv plane over which the parameters u and v vary for parameterization r(u,v)=x(u,v),y(u,v),z(u,v)r(u,v)=x(u,v),y(u,v),z(u,v)
parameterized surface (parametric surface)
a surface given by a description of the form r(u,v)=x(u,v),y(u,v),z(u,v),r(u,v)=x(u,v),y(u,v),z(u,v), where the parameters u and v vary over a parameter domain in the uv-plane
piecewise smooth curve
an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves
potential function
a scalar function ff such that f=Ff=F
radial field
a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin
regular parameterization
parameterization r(u,v)=x(u,v),y(u,v),z(u,v)r(u,v)=x(u,v),y(u,v),z(u,v) such that ru×rvru×rv is not zero for point (u,v)(u,v) in the parameter domain
rotational field
a vector field in which the vector at point (x,y)(x,y) is tangent to a circle with radius r=x2+y2;r=x2+y2; in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin
scalar line integral
the scalar line integral of a function ff along a curve C with respect to arc length is the integral Cfds,Cfds, it is the integral of a scalar function ff along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral
simple curve
a curve that does not cross itself
simply connected region
a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region
Stokes’ theorem
relates the flux integral over a surface S to a line integral around the boundary C of the surface S
stream function
if F=P,QF=P,Q is a source-free vector field, then stream function g is a function such that P=gyP=gy and Q=gxQ=gx
surface area
the area of surface S given by the surface integral SdSSdS
surface independent
flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface
surface integral
an integral of a function over a surface
surface integral of a scalar-valued function
a surface integral in which the integrand is a scalar function
surface integral of a vector field
a surface integral in which the integrand is a vector field
unit vector field
a vector field in which the magnitude of every vector is 1
vector field
measured in 2,2, an assignment of a vector F(x,y)F(x,y) to each point (x,y)(x,y) of a subset DD of 2;2; in 3,3, an assignment of a vector F(x,y,z)F(x,y,z) to each point (x,y,z)(x,y,z) of a subset DD of 33
vector line integral
the vector line integral of vector field F along curve C is the integral of the dot product of F with unit tangent vector T of C with respect to arc length, CF·Tds;CF·Tds; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral
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