 16.59PE: Flux Across a Plane Curve or SurfaceCylindrical can Use the Diverge...
 16.60PE: Flux Across a Plane Curve or SurfaceHemisphere Find the flux of F =...
 16.1AAE: Finding Areas with Green’s TheoremUse the Green’s Theorem area form...
 16.1PE: Evaluating Line IntegralsThe accompanying figure shows two polygona...
 16.1QGY: What are line integrals? How are they evaluated? Give examples.
 16.2AAE: Finding Areas with Green’s TheoremUse the Green’s Theorem area form...
 16.2PE: Evaluating Line IntegralsThe accompanying figure shows three polygo...
 16.2QGY: How can you use line integrals to find the centers of mass of sprin...
 16.3AAE: Finding Areas with Green’s TheoremUse the Green’s Theorem area form...
 16.3PE: Evaluating Line IntegralsIntegrate over the circle
 16.3QGY: What is a vector field? A gradient field? Give examples.
 16.4AAE: Finding Areas with Green’s TheoremUse the Green’s Theorem area form...
 16.4PE: Evaluating Line IntegralsIntegrate over the involute curve
 16.4QGY: What is the flow of a vector field along a curve? What is the work ...
 16.5AAE: Theory and Applicationsa. Give an example of a vector field F (x, y...
 16.5PE: Evaluating Line IntegralsEvaluate the integrals in Exercises
 16.5QGY: What is the Fundamental Theorem of line integrals? Explain how it r...
 16.6AAE: Theory and ApplicationsFind all points (a, b, c) on the sphere x2 +...
 16.6PE: Evaluating Line IntegralsEvaluate the integrals in Exercises
 16.6QGY: Specify three properties that are special about conservative fields...
 16.7AAE: Theory and ApplicationsFind the mass of a spherical shell of radius...
 16.7PE: Evaluating Line IntegralsIntegrate around the circle cut from the s...
 16.7QGY: What is special about path independent fields?
 16.8AAE: Theory and ApplicationsFind the mass of a helicoid Reference: Exerc...
 16.8PE: Evaluating Line IntegralsIntegrate F = 3x2yi + (x3 + 1)j + 9z2k aro...
 16.8QGY: What is a potential function? Show by example how to find a potenti...
 16.9AAE: Theory and ApplicationsAmong all rectangular regions find the one f...
 16.9PE: Evaluating Line IntegralsEvaluate the integrals in Exercise. C is t...
 16.9QGY: What is a differential form? What does it mean for such a form to b...
 16.10AAE: Theory and ApplicationsFind an equation for the plane through the o...
 16.10PE: Evaluating Line IntegralsEvaluate the integrals in Exercise. C is t...
 16.10QGY: What is Green's Theorem? Discuss how the two forms of Green's Theor...
 16.11AAE: Theory and ApplicationsA string lies along the circle x2 + y2 = 4 f...
 16.11PE: Finding and Evaluating Surface IntegralsArea of an elliptical regio...
 16.11QGY: How do you calculate the area of a parametrized surface in space? O...
 16.12AAE: Theory and ApplicationsA thin sheet lies along the portion of the p...
 16.12PE: Finding and Evaluating Surface IntegralsArea of a parabolic cap Fin...
 16.12QGY: How do you integrate a function over a parametrized surface in spac...
 16.13AAE: Theory and ApplicationsArchimedes’ principle If an object such as a...
 16.13PE: Finding and Evaluating Surface IntegralsArea of a spherical cap Fin...
 16.13QGY: What is an oriented surface? What is the surface integral of a vect...
 16.14AAE: Theory and ApplicationsFluid force on a curved surface A cone in th...
 16.14PE: Finding and Evaluating Surface Integralsa. Hemisphere cut by cylind...
 16.14QGY: What is the curl of a vector field? How can you interpret it?
 16.15AAE: Theory and ApplicationsFaraday’s Law If E(t, x, y, z) and B(t, x, y...
 16.15PE: Finding and Evaluating Surface IntegralsArea of a triangle Find the...
 16.15QGY: What is Stokes' Theorem? Explain how it generalizes Green's Theorem...
 16.16AAE: Theory and ApplicationsLet be the gravitational force field defined...
 16.16PE: Finding and Evaluating Surface IntegralsParabolic cylinder cut by p...
 16.16QGY: What is the Divergence Theorem? How can you interpret it?
 16.17AAE: Theory and ApplicationsIf ƒ(x, y, z) and g(x, y, z) are continuousl...
 16.17PE: Finding and Evaluating Surface IntegralsCircular cylinder cut by pl...
 16.17QGY: How does the Divergence Theorem generalize Green’s Theorem?
 16.18AAE: Theory and ApplicationsSuppose that over a region D enclosed by the...
 16.18PE: Finding and Evaluating Surface IntegralsArea of Wyoming The state o...
 16.18QGY: How can Green’s Theorem, Stokes’ Theorem, and the Divergence Theore...
 16.19AAE: Theory and ApplicationsProve or disprove that if
 16.19PE: Parametrized SurfacesFind parametrizations for the surfaces in Exer...
 16.20AAE: Theory and ApplicationsLet S be an oriented surface parametrized by...
 16.20PE: Parametrized SurfacesFind parametrizations for the surfaces in Exer...
 16.21AAE: Theory and ApplicationsShow that the volume V of a region D in spac...
 16.21PE: Parametrized SurfacesFind parametrizations for the surfaces in Exer...
 16.22PE: Parametrized SurfacesFind parametrizations for the surfaces in Exer...
 16.23PE: Parametrized SurfacesPortion of paraboloid The portion of the parab...
 16.24PE: Parametrized SurfacesPortion of hemisphere The portion of the hemis...
 16.25PE: Parametrized SurfacesSurface area Find the area of the surface
 16.26PE: Parametrized SurfacesSurface integral Integrate ƒ(x, y, z) = xy  z...
 16.27PE: Parametrized SurfacesArea of a helicoid Find the surface area of th...
 16.28PE: Parametrized SurfacesSurface integral Evaluate the integral where S...
 16.29PE: Conservative FieldsWhich of the fields in Exercise are conservative...
 16.30PE: Conservative FieldsWhich of the fields in Exercise are conservative...
 16.31PE: Conservative FieldsWhich of the fields in Exercise are conservative...
 16.32PE: Conservative FieldsWhich of the fields in Exercise are conservative...
 16.33PE: Conservative FieldsFind potential functions for the fields in Exerc...
 16.34PE: Conservative FieldsFind potential functions for the fields in Exerc...
 16.35PE: Work and CirculationIn Exercise, find the work done by each field a...
 16.36PE: Work and CirculationIn Exercise, find the work done by each field a...
 16.37PE: Work and CirculationFinding work in two ways Find the work done by ...
 16.38PE: Work and CirculationFlow along different paths Find the flow of the...
 16.39PE: Work and CirculationIn Exercise, use the surface integral in Stokes...
 16.40PE: Work and CirculationIn Exercise, use the surface integral in Stokes...
 16.41PE: Masses and MomentsWire with different densities Find the mass of a ...
 16.42PE: Masses and MomentsWire with variable density Find the center of mas...
 16.43PE: Masses and MomentsWire with variable density Find the center of mas...
 16.44PE: Masses and MomentsCenter of mass of an arch A slender metal arch li...
 16.45PE: Masses and MomentsWire with constant density A wire of constant den...
 16.46PE: Masses and MomentsHelical wire with constant density Find the mass ...
 16.47PE: Masses and MomentsInertia and center of mass of a shell Find Iz and...
 16.48PE: Masses and MomentsMoment of inertia of a cube Find the moment of in...
 16.49PE: Flux Across a Plane Curve or SurfaceUse Green’s Theorem to find the...
 16.50PE: Flux Across a Plane Curve or SurfaceUse Green’s Theorem to find the...
 16.51PE: Flux Across a Plane Curve or SurfaceZero line integral Show that fo...
 16.52PE: Flux Across a Plane Curve or Surfacea. Outward flux and area Show t...
 16.53PE: Flux Across a Plane Curve or SurfaceIn Exercises , find the outward...
 16.54PE: Flux Across a Plane Curve or SurfaceIn Exercises , find the outward...
 16.55PE: Flux Across a Plane Curve or SurfaceIn Exercises , find the outward...
 16.56PE: Flux Across a Plane Curve or SurfaceIn Exercises , find the outward...
 16.57PE: Flux Across a Plane Curve or SurfaceHemisphere, cylinder, and plane...
 16.58PE: Flux Across a Plane Curve or SurfaceCylinder and planes Find the ou...
Solutions for Chapter 16: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 16
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 16 includes 99 full stepbystep solutions. This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13th. Thomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077. Since 99 problems in chapter 16 have been answered, more than 44538 students have viewed full stepbystep solutions from this chapter.

Absolute maximum
A value ƒ(c) is an absolute maximum value of ƒ if ƒ(c) ? ƒ(x) for all x in the domain of ƒ.

Convenience sample
A sample that sacrifices randomness for convenience

Distance (on a number line)
The distance between real numbers a and b, or a  b

Equation
A statement of equality between two expressions.

First quartile
See Quartile.

Multiplication property of inequality
If u < v and c > 0, then uc < vc. If u < and c < 0, then uc > vc

Natural logarithmic function
The inverse of the exponential function y = ex, denoted by y = ln x.

Nonsingular matrix
A square matrix with nonzero determinant

Octants
The eight regions of space determined by the coordinate planes.

Polar coordinates
The numbers (r, ?) that determine a point’s location in a polar coordinate system. The number r is the directed distance and ? is the directed angle

Quadratic equation in x
An equation that can be written in the form ax 2 + bx + c = 01a ? 02

Reflection across the xaxis
x, y and (x,y) are reflections of each other across the xaxis.

Solve a triangle
To find one or more unknown sides or angles of a triangle

Standard form of a polynomial function
ƒ(x) = an x n + an1x n1 + Á + a1x + a0

Sum of functions
(ƒ + g)(x) = ƒ(x) + g(x)

Terminal point
See Arrow.

Triangular form
A special form for a system of linear equations that facilitates finding the solution.

Velocity
A vector that specifies the motion of an object in terms of its speed and direction.

Venn diagram
A visualization of the relationships among events within a sample space.

Zero factor property
If ab = 0 , then either a = 0 or b = 0.