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11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 15.3, Problem 43E

Textbook Problem

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True or False? In Exercises 43-46, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If

To determine

Whether the provided statement, “If C_{1}, C_{2}, and C_{3} have the same initial and terminal points and

Even though

Multivariable Calculus

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Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line IntegralIn Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 17 and 18,...Ch. 15.2 - Evaluating a Line Integral In Exercises 17 and 18,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Mass In Exercises 23 and 24, find the total mass...Ch. 15.2 - Mass In Exercises 23 and 24, find the total mass...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field Using...Ch. 15.2 - Evaluating a Line Integral of a Vector Field Using...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 3742, find the work done by the...Ch. 15.2 - 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The figure shows a region R...Ch. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Think About It Let I=Cydxxdyx2+y2 Where C is a...Ch. 15.4 - Think About It For each given path, verify Greens...Ch. 15.4 - Proff (a) Let C be the line segment joining...Ch. 15.4 - Area Use the result of Exercise 49(b) to find the...Ch. 15.4 - Proof In Exercises 51 and 52, prove the identity,...Ch. 15.4 - Proof In Exercises 51 and 52, prove the identity,...Ch. 15.4 - Proof Let F=Mi+Nj, where M and N have continuous...Ch. 15.4 - Find the least possible area of a convex set in...Ch. 15.5 - CONCEPT CHECK Parametric Surface Explain how a...Ch. 15.5 - CONCEPT CHECK Surface Area A surface S is...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 38, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding a Tangent Plane In Exercises 3336, find an...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 3742, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - Representing a Cone Parametrically Show that the...Ch. 15.5 - HOW DO YOU SEE IT? The figures below are graphs of...Ch. 15.5 - Astroidal Sphere An equation of an astroidal...Ch. 15.5 - Different Views of a Surface Use a computer...Ch. 15.5 - Investigation Use a computer algebra system to...Ch. 15.5 - Investigation Consider the function in Exercise...Ch. 15.5 - Surface Area The surface of the dome on a new...Ch. 15.5 - Hyperboloid Find a vector-valued function for the...Ch. 15.5 - Area Use a computer algebra system to graph one...Ch. 15.5 - Surface Area Lei f be a nonnegative function such...Ch. 15.5 - Open-Ended Project The parametric equations...Ch. 15.5 - Mobius Strip The surface shown in the figure is...Ch. 15.6 - CONCEPT CHECK Surface Integral Explain how to set...Ch. 15.6 - CONCEPT CHECK Surface Integral For what conditions...Ch. 15.6 - CONCEPT CHECK Orientation of a Surface Integral...Ch. 15.6 - CONCEPT CHECK Flux What is the physical...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Evaluating a Surface Integral In Exercises15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Flow Rate In Exercises 33 and 34, use a computer...Ch. 15.6 - Flow Rate In Exercises 33-34, use a computer...Ch. 15.6 - Gauss's Law In Exercises 35 and 36, evaluate SENdS...Ch. 15.6 - Gauss's Law In Exercises 35 and 36, evaluate SENdS...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - EXPLORING CONCEPTS Using Different Methods...Ch. 15.6 - HOW DO YOU SEE IT? Is the surface shown in the...Ch. 15.6 - Investigation (a) Use a computer algebra system to...Ch. 15.7 - CONCEPT CHECK Using Different Methods Suppose that...Ch. 15.7 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - EXPLORING CONCEPTS Closed Surface What is the...Ch. 15.7 - HOW DO YOU SEE IT? The graph of a vector field F...Ch. 15.7 - Volume (a) Use the Divergence Theorem to verify...Ch. 15.7 - Constant Vector Field For the constant vector...Ch. 15.7 - Volume For the vector field F(x,y,z)=xi+yj+zk,...Ch. 15.7 - Verifying an Identity For the vector field...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.8 - CONCEPT CHECK Stokess Theorem Explain the benefit...Ch. 15.8 - Curl What is the physical interpretation of curl?Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokes Theorem In Exercises 3-6, verify...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Motion of a Liquid In Exercises 17 and 18, the...Ch. 15.8 - Motion of a Liquid In Exercises 17 and 18, the...Ch. 15.8 - EXPLORING CONCEPTS Think About It Let K be a...Ch. 15.8 - HOW DO YOU SEE IT? Let S1 be the portion of the...Ch. 15.8 - Let G(x,y)=(yx2+4y2,xx2+4y2,0). Prove or disprove...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Mass In Exercises 33 and 34, find the total mass...Ch. 15 - Mass In Exercises 33 and 34, find the total mass...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Work In Exercises 39 and 40, find the work done by...Ch. 15 - Work In Exercises 39 and 40, find the work done by...Ch. 15 - Evaluating a Line Integral in Differential Form In...Ch. 15 - Evaluating a Line Integral in Differential Form...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Line Integral of a Conservative Vector Field In...Ch. 15 - Line Integral of a Conservative Vector Field In...Ch. 15 - Using the Fundamental Theorem of Line Integrals In...Ch. 15 - Using the Fundamental Theorem of Line Integrals In...Ch. 15 - Using the Fundamental Theorem of Line Integrals in...Ch. 15 - Using the Fundamental Theorem of Line Integrals in...Ch. 15 - Finding Work in a Conservative Force Field In...Ch. 15 - Finding Work in a Conservative Force Field In...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Work In Exercises 59 and 60, use Greens Theorem to...Ch. 15 - Work In Exercises 59 and 60, use Greens Theorem to...Ch. 15 - Area In Exercises 61 and 61, use a line integral...Ch. 15 - Area In Exercises 61 and 62, use a line integral...Ch. 15 - Sketching a Parametric Surface In Exercise 63 and...Ch. 15 - Sketching a Parametric Surface In Exercise 63 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Representing a Surface Parametrically In Exercises...Ch. 15 - Representing a Surface Parametrically In Exercises...Ch. 15 - Representing a Surface of Revolution...Ch. 15 - Representing a Surface of Revolution...Ch. 15 - Finding a Surface Area In Exercises 71 and 72,...Ch. 15 - Finding a Surface Area In Exercises 71 and 72,...Ch. 15 - Evaluating a Surface Integral In Exercises 73 and...Ch. 15 - Evaluating a Surface Integral In Exercises 73 and...Ch. 15 - Mass In Exercises 75 and 76, find the mass of the...Ch. 15 - Mass In Exercises 75 and 76, find the mass of the...Ch. 15 - Evaluating a Surface Integral In Exercises 77 and...Ch. 15 - Evaluating a Surface Integral In Exercises 77 and...Ch. 15 - Evaluating a Flux Integral In Exercises 79 and 80,...Ch. 15 - Evaluating a Flux IntegralIn Exercises 79 and 80,...Ch. 15 - Using the Divergence TheoremIn Exercises 81 and...Ch. 15 - Using the Divergence TheoremIn Exercises 81 and...Ch. 15 - Using Stokess Theorem In Exercises 83 and 84, use...Ch. 15 - Using Stokess Theorem In Exercises 83 and 84, use...Ch. 15 - Motion of a Liquid In Exercises 85 and 86, the...Ch. 15 - Motion of a Liquid In Exercises 85 and 86, the...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Moments of Inertia Consider a wire of density...Ch. 15 - Moments of Inertia Using the formulas from...Ch. 15 - Laplace's Equation Let F(x,y,z)=xi+yj+zk, and let...Ch. 15 - Greens Theorem Consider the line integral...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Work The force field F(x,y)=(x+y)i+(x2+1)j acts on...Ch. 15 - Work The force field F(x,y)=3x2y2i+2x3yj is shown...Ch. 15 - Area and Work How does the area of the ellipse...Ch. 15 - Verifying Identities (a) Let f and g be scalar...

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