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10th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285057095

Chapter 15.8, Problem 21E

Textbook Problem

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CONCEPT CHECK

Stokes’s Theorem Explain the benefit of Stokes’s Theorem when the boundary of the surface is a piecewise curve.

To determine

The benefits/importance of Stokes’s theorem if the boundary of the surface is a piece-wise curve.

When the boundary of the surface is a piecewise curve *C*, then the line integral, which actually has the fluid tendency that circulates around the boundary and has an exactly same measure of this rotational tendency which is taken over the entire surface (surface integral).

That is, the line integral can be calculated by calculation of the flux of the vector field through any surface bounded by the piecewise curve *C*.

For example:

It has to calculate the value of

*C*: triangle with vertices

For curl, first find the curl of vector field,

The surface *C: a* triangle with vertices

projected onto the *xy*-plane:

Partially differentiate *x*’.

Again, partially differentiate *y*’

Calculus

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Electrical Charge Let E = xi + yj + 2zk be an...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 - 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 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - 57095-15.7-1E-Question-Digital.docx Verifying the...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 1–6,...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 - 57095-15.7-17E-Question-Digital.docx Using the...Ch. 15.7 - Using the Divergence Theorem In Exercises 17 and...Ch. 15.7 - 57095-15.7-19E-Question-Digital.docx WRITING ABOUT...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 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...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 Stokes's Theorem In Exercises 9–18, 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 19 and 20, the...Ch. 15.8 - Motion of a Liquid In Exercises 19 and 20, the...Ch. 15.8 - 23. Proof Let C be a constant vector. Let S be an...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 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 In Exercises 21–26,...Ch. 15 - 57095-15-26RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...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 - 57095-15-35RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - 57095-15-37RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-38RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-39RE-Question-Digital.docx Work Find the...Ch. 15 - 57095-15-40RE-Question-Digital.docx Work A 20-ton...Ch. 15 - 57095-15-41RE-Question-Digital.docx Using the...Ch. 15 - 57095-15-42RE-Question-Digital.docx Using the...Ch. 15 - 43. Evaluating a Line Integral Evaluate the line...Ch. 15 - 57095-15-44RE-Question-Digital.docx Area and...Ch. 15 - 57095-15-45RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral In Exercises 45–50, use...Ch. 15 - 57095-15-47RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-48RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - 57095-15-53RE-Question-Digital.docx Investigation...Ch. 15 - 57095-15-54RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-55RE-Question-Digital.docx Approximating...Ch. 15 - 57095-15-56RE-Question-Digital.docx Mass A...Ch. 15 - Verifying the Divergence Theorem In Exercises 57...Ch. 15 - Verifying the Divergence Theorem In Exercises 57...Ch. 15 - 57095-15-59RE-Question-Digital.docx Verifying...Ch. 15 - 57095-15-60RE-Question-Digital.docx Verifying...Ch. 15 - 57095-15-61RE-Question-Digital.docx Proof Prove...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 - 57095-15-11PS-Question-Digital.docx Proof Let S be...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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