Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Chapter 14.3, Problem 18E

Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ¡2 and ¡3 respectively, that do not include the origin.

18.     F = 〈 x , y 〉 x 2 + y 2 on   R *

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Chapter 14 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3ECh. 14.1 - Prob. 4ECh. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...
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Stokes Theorem for evaluating line integrals...Ch. 14.7 - Stokes Theorem for evaluating line integrals...Ch. 14.7 - Stokes Theorem for evaluating surface integrals...Ch. 14.7 - Stokes Theorem for evaluating surface integrals...Ch. 14.7 - Stokes Theorem for evaluating surface integrals...Ch. 14.7 - Stokes Theorem for evaluating surface integrals...Ch. 14.7 - Interpreting and graphing the curl For the...Ch. 14.7 - Interpreting and graphing the curl For the...Ch. 14.7 - Interpreting and graphing the curl For the...Ch. 14.7 - Interpreting and graphing the curl For the...Ch. 14.7 - Explain why or why not Determine whether the...Ch. 14.7 - Conservative fields Use Stokes Theorem to find the...Ch. 14.7 - Conservative fields Use Stokes Theorem to find the...Ch. 14.7 - Conservative fields Use Stokes Theorem to find the...Ch. 14.7 - Conservative fields Use Stokes Theorem to find the...Ch. 14.7 - Tilted disks Let S be the disk enclosed by the...Ch. 14.7 - Tilted disks Let S be the disk enclosed by the...Ch. 14.7 - Tilted disks Let S be the disk enclosed by the...Ch. 14.7 - Tilted disks Let S be the disk enclosed by the...Ch. 14.7 - Prob. 34ECh. 14.7 - Circulation in a plane A circle C in the plane x +...Ch. 14.7 - No integrals Let F = (2z, z, 2y + x) and let S be...Ch. 14.7 - Compound surface and boundary Begin with the...Ch. 14.7 - Ampres Law The French physicist AndrMarie Ampre...Ch. 14.7 - Maximum surface integral Let S be the paraboloid z...Ch. 14.7 - Area of a region in a plane Let R be a region in a...Ch. 14.7 - Choosing a more convenient surface The goal is to...Ch. 14.7 - Radial fields and zero circulation Consider the...Ch. 14.7 - Zero curl Consider the vector field...Ch. 14.7 - Average circulation Let S be a small circular disk...Ch. 14.7 - Proof of Stokes Theorem Confirm the following step...Ch. 14.7 - Stokes Theorem on closed surfaces Prove that if F...Ch. 14.7 - Rotated Greens Theorem Use Stokes Theorem to write...Ch. 14.8 - Review Questions 1.Explain the meaning of the...Ch. 14.8 - Interpret the volume integral in the Divergence...Ch. 14.8 - Explain the meaning of the Divergence Theorem.Ch. 14.8 - What is the net outward flux of the rotation field...Ch. 14.8 - What is the net outward flux of the radial field F...Ch. 14.8 - What is the divergence of an inverse square vector...Ch. 14.8 - Suppose div F = 0 in a region enclosed by two...Ch. 14.8 - If div F 0 in a region enclosed by a small cube,...Ch. 14.8 - Verifying the Divergence Theorem Evaluate both...Ch. 14.8 - F = x, y, z; D = {(x, y, z): |x| 1, |y| 1, |z| ...Ch. 14.8 - Basic Skills 912.Verifying the Divergence Theorem...Ch. 14.8 - F = x2, y2, z2; D = {(x, y, z): |x| 1, |y| 2,...Ch. 14.8 - Rotation fields 13.Find the net outward flux of...Ch. 14.8 - Rotation fields 14.Find the net outward flux of...Ch. 14.8 - Find the net outward flux of the field F = bz cy,...Ch. 14.8 - Rotation fields 16.Find the net outward flux of F...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - F = x, 2y, z; S is the boundary of the tetrahedron...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - F = y 2x, x3 y, y2 z; S is the sphere {(x, y,...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - Computing flux Use the Divergence Theorem to...Ch. 14.8 - Divergence Theorem for more general regions Use...Ch. 14.8 - Divergence Theorem for more general regions Use...Ch. 14.8 - Divergence Theorem for more general regions Use...Ch. 14.8 - Divergence Theorem for more general regions Use...Ch. 14.8 - F = x2, y2, z2); D is the region in the first...Ch. 14.8 - Divergence Theorem for more general regions Use...Ch. 14.8 - Explain why or why not Determine whether the...Ch. 14.8 - Flux across a sphere Consider the radial field F =...Ch. 14.8 - Flux integrals Compute the outward flux of the...Ch. 14.8 - Flux integrals Compute the outward flux of the...Ch. 14.8 - Flux integrals Compute the outward flux of the...Ch. 14.8 - Prob. 36ECh. 14.8 - Singular radial field Consider the radial field...Ch. 14.8 - Logarithmic potential Consider the potential...Ch. 14.8 - Gauss Law for electric fields The electric field...Ch. 14.8 - Gauss Law for gravitation The gravitational force...Ch. 14.8 - Heat transfer Fouriers Law of heat transfer (or...Ch. 14.8 - Heat transfer Fouriers Law of heat transfer (or...Ch. 14.8 - Heat transfer Fouriers Law of heat transfer (or...Ch. 14.8 - Heat transfer Fouriers Law of heat transfer (or...Ch. 14.8 - Heat transfer Fouriers Law of heat transfer (or...Ch. 14.8 - Inverse square fields are special Let F be a...Ch. 14.8 - A beautiful flux integral Consider the potential...Ch. 14.8 - Integration by parts (Gauss' Formula) Recall the...Ch. 14.8 - Prob. 49ECh. 14.8 - Prob. 50ECh. 14.8 - Greens Second Identity Prose Greens Second...Ch. 14.8 - Prob. 52ECh. 14.8 - Prob. 53ECh. 14.8 - Prob. 54ECh. 14 - Explain why or why not Determine whether the...Ch. 14 - Matching vector fields Match vector fields a-f...Ch. 14 - Gradient fields in 2 Find the vector field F = ...Ch. 14 - Gradient fields in 2 Find the vector field F = ...Ch. 14 - Gradient fields in 3 Find the vector field F = ...Ch. 14 - Gradient fields in 3 Find the vector field F = ...Ch. 14 - Normal component Let C be the circle of radius 2...Ch. 14 - Line integrals Evaluate the following line...Ch. 14 - Prob. 9RECh. 14 - Line integrals Evaluate the following line...Ch. 14 - Two parameterizations Verify that C(x2y+3z)ds has...Ch. 14 - Work integral Find the work done in moving an...Ch. 14 - Work integrals in R3 Given the following force...Ch. 14 - Work integrals in 3 Given the following force...Ch. 14 - Circulation and flux Find the circulation and the...Ch. 14 - Circulation and flux Find the circulation and the...Ch. 14 - Circulation and flux Find the circulation and the...Ch. 14 - Circulation and flux Find the circulation and the...Ch. 14 - Flux in channel flow Consider the flow of water in...Ch. 14 - Conservative vector fields and potentials...Ch. 14 - Conservative vector fields and potentials...Ch. 14 - Conservative vector fields and potentials...Ch. 14 - Conservative vector fields and potentials...Ch. 14 - Evaluating line integrals Evaluate the line...Ch. 14 - Evaluating line integrals Evaluate the line...Ch. 14 - Evaluating line integrals Evaluate the line...Ch. 14 - Evaluating line integrals Evaluate the line...Ch. 14 - Radial fields in R2 are conservative Prove that...Ch. 14 - Greens Theorem for line integrals Use either form...Ch. 14 - Greens Theorem for line integrals Use either form...Ch. 14 - Greens Theorem for line integrals Use either form...Ch. 14 - Greens Theorem for line integrals Use either form...Ch. 14 - Areas of plane regions Find the area of the...Ch. 14 - Areas of plane regions Find the area of the...Ch. 14 - Circulation and flux Consider the following vector...Ch. 14 - Circulation and flux Consider the following vector...Ch. 14 - Parameters Let F = ax + by, cx + dy, where a, b,...Ch. 14 - Divergence and curl Compute the divergence and...Ch. 14 - Divergence and curl Compute the divergence and...Ch. 14 - Divergence and curl Compute the divergence and...Ch. 14 - Divergence and curl Compute the divergence and...Ch. 14 - Identities Prove that (1|r|4)=4r|r|6 and use the...Ch. 14 - Prob. 43RECh. 14 - Paddle wheel in a vector field Let F = 0, 2x, 0...Ch. 14 - Surface areas Use a surface integral to find the...Ch. 14 - Surface areas Use a surface integral to find the...Ch. 14 - Surface areas Use a surface integral to find the...Ch. 14 - Surface areas Use a surface integral to find the...Ch. 14 - Surface integrals Evaluate the following surface...Ch. 14 - Surface integrals Evaluate the following surface...Ch. 14 - Surface integrals Evaluate the following surface...Ch. 14 - Flux integrals Find the flux of the following...Ch. 14 - Flux integrals Find the flux of the following...Ch. 14 - Three methods Find the surface area of the...Ch. 14 - Flux across hemispheres and paraboloids Let S be...Ch. 14 - Surface area of an ellipsoid Consider the...Ch. 14 - Stokes Theorem for line integrals Evaluate the...Ch. 14 - Stokes Theorem for line integrals Evaluate the...Ch. 14 - Stokes Theorem for surface integrals Use Stokes...Ch. 14 - Stokes Theorem for surface integrals Use Stokes...Ch. 14 - Conservative fields Use Stokes Theorem to find the...Ch. 14 - Computing fluxes Use the Divergence Theorem to...Ch. 14 - Computing fluxes Use the Divergence Theorem to...Ch. 14 - Computing fluxes Use the Divergence Theorem to...Ch. 14 - General regions Use the Divergence Theorem to...Ch. 14 - General regions Use the Divergence Theorem to...Ch. 14 - Flux integrals Compute the outward flux of the...Ch. 14 - Stokes Theorem on a compound surface Consider the...

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