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11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 15.8, Problem 5E

Textbook Problem

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Verifying Stokes’s Theorem In Exercises 3-6, verify Stokes’s Theorem by evaluating

To determine

**To prove:** The Stoke’s Theorem by calculating

**Given:**

The given vector field is:

**Formula used:**

The curl F of a vector field is given as,

From the Stoke’s Theorem,

Unit normal vector,

**Proof:**

The line integral can be calculated by considering the path *C* that iscomposed of three paths *C _{1}*,

Now, apply the Stoke’s theorem.

It implies that,

In order to evaluate

Multivariable Calculus

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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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