Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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#### Concept explainers

Textbook Question

Chapter 14.7, Problem 5E

**Verifying Stokes’ Theorem** *Verify that the line integral and the surface integral of Stokes*’

*Theorem are equal for the following*vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation.

**5.** **F** = 〈*y*, – *x*, 10〉; *S* is the upper half of the sphere *x*^{2} + *y*^{2} + *z*^{2} = 1 and *C* is the circle *x*^{2} + *y*^{2} = 1 in the *xy*-plane.

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=
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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...

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...Ch. 14.1 - Matching vector Fields with graphs Match vector...Ch. 14.1 - Prob. 17ECh. 14.1 - Prob. 18ECh. 14.1 - Prob. 19ECh. 14.1 - Prob. 20ECh. 14.1 - Three-dimensional vector fields Sketch a few...Ch. 14.1 - Prob. 22ECh. 14.1 - Three-dimensional vector fields Sketch a few...Ch. 14.1 - Three-dimensional vector fields Sketch a few...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Gradient fields Find the gradient field F = for...Ch. 14.1 - Equipotential curves Consider the following...Ch. 14.1 - Equipotential curves Consider the following...Ch. 14.1 - Equipotential curves Consider the following...Ch. 14.1 - Equipotential curves Consider the following...Ch. 14.1 - Explain why or why not Determine whether the...Ch. 14.1 - Prob. 42ECh. 14.1 - Prob. 43ECh. 14.1 - Design your own vector field Specify the component...Ch. 14.1 - Design your own vector field Specify the component...Ch. 14.1 - Design your own vector field Specify the component...Ch. 14.1 - Design your own vector field Specify the component...Ch. 14.1 - Electric field due to a point charge The electric...Ch. 14.1 - Electric field due to a line of charge The...Ch. 14.1 - Gravitational force due to a mass The...Ch. 14.1 - Flow curves in the plane Let...Ch. 14.1 - Flow curves in the plane Let...Ch. 14.1 - Flow curves in the plane Let...Ch. 14.1 - Flow curves in the plane Let...Ch. 14.1 - Flow curves in the plane Let...Ch. 14.1 - Prob. 56ECh. 14.1 - Prob. 57ECh. 14.1 - Prob. 58ECh. 14.1 - Prob. 59ECh. 14.1 - Prob. 60ECh. 14.1 - Cartesian-to-polar vector field Write the vector...Ch. 14.2 - How does a line integral differ from the...Ch. 14.2 - Prob. 2ECh. 14.2 - If a curve C is given by r(t) = t, t2, what is...Ch. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Scalar line integrals with arc length as parameter...Ch. 14.2 - Prob. 12ECh. 14.2 - Scalar line integrals with arc length as parameter...Ch. 14.2 - Prob. 14ECh. 14.2 - Scalar line integrals in the plane a.Find a...Ch. 14.2 - Prob. 16ECh. 14.2 - Scalar line integrals in the plane a.Find a...Ch. 14.2 - Scalar line integrals in the plane a.Find a...Ch. 14.2 - Scalar line integrals in the plane a.Find a...Ch. 14.2 - Prob. 20ECh. 14.2 - Average values Find the average value of the...Ch. 14.2 - Average values Find the average value of the...Ch. 14.2 - Prob. 23ECh. 14.2 - Average values Find the average value of the...Ch. 14.2 - Scalar line integrals in 3Convert the line...Ch. 14.2 - Scalar line integrals in 3Convert the line...Ch. 14.2 - Prob. 27ECh. 14.2 - Scalar line integrals in 3Convert the line...Ch. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Length of curves Use a scalar line integral to...Ch. 14.2 - Length of curves Use a scalar line integral to...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Line integrals of vector fields in the plane Given...Ch. 14.2 - Work integrals Given the force field F, find the...Ch. 14.2 - Work integrals Given the force field F, find the...Ch. 14.2 - Work integrals Given the force field F, find the...Ch. 14.2 - Work integrals Given the force field F, find the...Ch. 14.2 - Work integrals in 3 Given the force field F, find...Ch. 14.2 - Work integrals in 3 Given the force field F, find...Ch. 14.2 - Work integrals in 3 Given the force field F, find...Ch. 14.2 - Work integrals in 3 Given the force field F, find...Ch. 14.2 - Circulation Consider the following vector fields F...Ch. 14.2 - Prob. 48ECh. 14.2 - Flux Consider the vector fields and curves in...Ch. 14.2 - Prob. 50ECh. 14.2 - Explain why or why not Determine whether the...Ch. 14.2 - Flying into a headwind An airplane flies in the...Ch. 14.2 - Prob. 53ECh. 14.2 - Prob. 54ECh. 14.2 - Changing orientation Let f(x, y) = x and let C be...Ch. 14.2 - Prob. 56ECh. 14.2 - Zero circulation fields 57.Consider the vector...Ch. 14.2 - Zero flux fields 58.For what values of a and d...Ch. 14.2 - Zero flux fields 59.Consider the vector field F =...Ch. 14.2 - Work in a rotation field Consider the rotation...Ch. 14.2 - Prob. 61ECh. 14.2 - Mass and density A thin wire represented by the...Ch. 14.2 - Mass and density A thin wire represented by the...Ch. 14.2 - Heat flux in a plate A square plate R = {(x, y): 0...Ch. 14.2 - Prob. 65ECh. 14.2 - Flux across curves in a vector field Consider the...Ch. 14.2 - Looking ahead: Area from line integrals The area...Ch. 14.2 - Looking ahead: Area from line integrals The area...Ch. 14.3 - What does it mean for a function to have an...Ch. 14.3 - What are local maximum and minimum values of a...Ch. 14.3 - What conditions must be met to ensure that a...Ch. 14.3 - How do you determine whether a vector field in 3...Ch. 14.3 - Briefly describe how to find a potential function ...Ch. 14.3 - If F is a conservative vector field on a region R,...Ch. 14.3 - If F is a conservative vector field on a region R,...Ch. 14.3 - Give three equivalent properties of conservative...Ch. 14.3 - How do you determine the absolute maximum and...Ch. 14.3 - Explain how a function can have an absolute...Ch. 14.3 - Testing for conservative vector fields Determine...Ch. 14.3 - Testing for conservative vector fields Determine...Ch. 14.3 - Testing for conservative vector fields Determine...Ch. 14.3 - Testing for conservative vector fields Determine...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Designing a function Sketch a graph of a function...Ch. 14.3 - Prob. 20ECh. 14.3 - Designing a function Sketch a graph of a function...Ch. 14.3 - Designing a function Sketch a graph of a function...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Finding potential functions Determine whether the...Ch. 14.3 - Evaluating line integrals Evaluate the line...Ch. 14.3 - Prob. 28ECh. 14.3 - Evaluating line integrals Evaluate the line...Ch. 14.3 - Evaluating line integrals Evaluate the line...Ch. 14.3 - Evaluating line integrals Evaluate the line...Ch. 14.3 - Evaluating line integrals Evaluate the line...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Line integrals of vector fields on closed curves...Ch. 14.3 - Explain why or why not Determine whether the...Ch. 14.3 - Line integrals Evaluate each line integral using a...Ch. 14.3 - Prob. 41ECh. 14.3 - Line integrals Evaluate each line integral using a...Ch. 14.3 - Line integrals Evaluate each line integral using a...Ch. 14.3 - Prob. 44ECh. 14.3 - Work in force fields Find the work required to...Ch. 14.3 - Work in force fields Find the work required to...Ch. 14.3 - Work in force fields Find the work required to...Ch. 14.3 - Work in force fields Find the work required to...Ch. 14.3 - Prob. 49ECh. 14.3 - Prob. 50ECh. 14.3 - Prob. 51ECh. 14.3 - Conservation of energy Suppose an object with mass...Ch. 14.3 - Gravitational potential The gravitational force...Ch. 14.3 - Radial Fields in 3 are conservative Prove that the...Ch. 14.3 - 55.Rotation fields are usually not conservative...Ch. 14.3 - Linear and quadratic vector fields a.For what...Ch. 14.3 - Prob. 57ECh. 14.3 - Prob. 58ECh. 14.3 - Prob. 59ECh. 14.3 - Alternative construction of potential functions...Ch. 14.3 - Prob. 61ECh. 14.4 - Explain why the two forms of Greens Theorem are...Ch. 14.4 - Referring to both forms of Greens Theorem, match...Ch. 14.4 - Prob. 3ECh. 14.4 - Prob. 4ECh. 14.4 - Prob. 5ECh. 14.4 - Why does a two-dimensional vector field with zero...Ch. 14.4 - Why does a two-dimensional vector field with zero...Ch. 14.4 - Sketch a two-dimensional vector field that has...Ch. 14.4 - Sketch a two-dimensional vector field that has...Ch. 14.4 - Discuss one of the parallels between a...Ch. 14.4 - Greens Theorem, circulation form Consider the...Ch. 14.4 - Prob. 12ECh. 14.4 - Greens Theorem, circulation form Consider the...Ch. 14.4 - Greens Theorem, circulation form Consider the...Ch. 14.4 - Greens Theorem, circulation form Consider the...Ch. 14.4 - Prob. 16ECh. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Area of regions Use a line integral on the...Ch. 14.4 - Greens Theorem, flux form Consider the following...Ch. 14.4 - Prob. 24ECh. 14.4 - Greens Theorem, flux form Consider the following...Ch. 14.4 - Greens Theorem, flux form Consider the following...Ch. 14.4 - Greens Theorem, flux form Consider the following...Ch. 14.4 - Greens Theorem, flux form Consider the following...Ch. 14.4 - Line integrals Use Greens Theorem to evaluate the...Ch. 14.4 - Line integrals Use Greens Theorem to evaluate the...Ch. 14.4 - Prob. 31ECh. 14.4 - Line integrals Use Greens Theorem to evaluate the...Ch. 14.4 - Prob. 33ECh. 14.4 - Line integrals Use Greens Theorem to evaluate the...Ch. 14.4 - General regions For the following vector fields,...Ch. 14.4 - General regions For the following vector fields,...Ch. 14.4 - General regions For the following vector fields,...Ch. 14.4 - General regions For the following vector fields,...Ch. 14.4 - Explain why or why not Determine whether the...Ch. 14.4 - Circulation and flux For the following vector...Ch. 14.4 - Circulation and flux For the following vector...Ch. 14.4 - Circulation and flux For the following vector...Ch. 14.4 - Prob. 43ECh. 14.4 - Special line integrals Prove the following...Ch. 14.4 - Special line integrals Prove the following...Ch. 14.4 - Prob. 46ECh. 14.4 - Area line integral Show that the value of...Ch. 14.4 - Area line integral In terms of the parameters a...Ch. 14.4 - Stream function Recall that if the vector field F...Ch. 14.4 - Stream function Recall that if the vector field F...Ch. 14.4 - Stream function Recall that if the vector field F...Ch. 14.4 - Stream function Recall that if the vector field F...Ch. 14.4 - Applications 5356. Ideal flow A two-dimensional...Ch. 14.4 - Applications 5356. Ideal flow A two-dimensional...Ch. 14.4 - Applications 5356. Ideal flow A two-dimensional...Ch. 14.4 - Applications 5356. Ideal flow A two-dimensional...Ch. 14.4 - Prob. 57ECh. 14.4 - Greens Theorem as a Fundamental Theorem of...Ch. 14.4 - Greens Theorem as a Fundamental Theorem of...Ch. 14.4 - Whats wrong? Consider the rotation field...Ch. 14.4 - Whats wrong? Consider the radial field...Ch. 14.4 - Prob. 62ECh. 14.4 - Flux integrals Assume the vector field F = (f, g)...Ch. 14.4 - Streamlines are tangent to the vector field Assume...Ch. 14.4 - Streamlines and equipotential lines Assume that on...Ch. 14.4 - Channel flow The flow in a long shallow channel is...Ch. 14.5 - Explain how to compute the divergence of the...Ch. 14.5 - Interpret the divergence of a vector field.Ch. 14.5 - What does it mean if the divergence of a vector...Ch. 14.5 - Explain how to compute the curl of the vector...Ch. 14.5 - Interpret the curl of a general rotation vector...Ch. 14.5 - What does it mean if the curl of a vector field is...Ch. 14.5 - What is the value of ( F)?Ch. 14.5 - What is the value of u?Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of vector fields Find the divergence of...Ch. 14.5 - Divergence of radial fields Calculate the...Ch. 14.5 - Prob. 18ECh. 14.5 - Prob. 19ECh. 14.5 - Prob. 20ECh. 14.5 - Divergence and flux from graphs Consider the...Ch. 14.5 - Divergence and flux from graphs Consider the...Ch. 14.5 - Curl of a rotational field Consider the following...Ch. 14.5 - Curl of a rotational field Consider the following...Ch. 14.5 - Curl of a rotational field Consider the following...Ch. 14.5 - Curl of a rotational field Consider the following...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Curl of a vector field Compute the curl of the...Ch. 14.5 - Prob. 35ECh. 14.5 - Prob. 36ECh. 14.5 - Prob. 37ECh. 14.5 - Prob. 38ECh. 14.5 - Explain why or why not Determine whether the...Ch. 14.5 - Another derivative combination Let F = (f, g, h)...Ch. 14.5 - Does it make sense? Are the following expressions...Ch. 14.5 - Zero divergence of the rotation field Show that...Ch. 14.5 - General rotation fields a.Let a = (0, 1, 0), r =...Ch. 14.5 - Prob. 44ECh. 14.5 - Curl of the rotation field For the general...Ch. 14.5 - Inward to outward Find the exact points on the...Ch. 14.5 - Maximum divergence Within the cube {(x, y, z): |x|...Ch. 14.5 - Prob. 48ECh. 14.5 - Zero component of the curl For what vectors n is...Ch. 14.5 - Prob. 50ECh. 14.5 - Find a vector Field Find a vector field F with the...Ch. 14.5 - Prob. 52ECh. 14.5 - Paddle wheel in a vector field Let F = z, 0, 0 and...Ch. 14.5 - Angular speed Consider the rotational velocity...Ch. 14.5 - Angular speed Consider the rotational velocity...Ch. 14.5 - Heat flux Suppose a solid object in 3 has a...Ch. 14.5 - Heat flux Suppose a solid object in 3 has a...Ch. 14.5 - Prob. 58ECh. 14.5 - Gravitational potential The potential function for...Ch. 14.5 - Electric potential The potential function for the...Ch. 14.5 - Navier-Stokes equation The Navier-Stokes equation...Ch. 14.5 - Stream function and vorticity The rotation of a...Ch. 14.5 - Amperes Law One of Maxwells equations for...Ch. 14.5 - Prob. 64ECh. 14.5 - Properties of div and curl Prove the following...Ch. 14.5 - Prob. 66ECh. 14.5 - Identities Prove the following identities. Assume...Ch. 14.5 - Identities Prove the following identities. Assume...Ch. 14.5 - Prob. 69ECh. 14.5 - Prob. 70ECh. 14.5 - Prob. 71ECh. 14.5 - Prob. 72ECh. 14.5 - Prob. 73ECh. 14.5 - Gradients and radial fields Prove that for a real...Ch. 14.5 - Prob. 75ECh. 14.6 - Give a parametric description for a cylinder with...Ch. 14.6 - Prob. 2ECh. 14.6 - Give a parametric description for a sphere with...Ch. 14.6 - Prob. 4ECh. 14.6 - Explain how to compute the surface integral of a...Ch. 14.6 - Prob. 6ECh. 14.6 - Prob. 7ECh. 14.6 - Explain what it means for a surface to be...Ch. 14.6 - Describe the usual orientation of a closed surface...Ch. 14.6 - Why is the upward flux of a vertical vector field...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Parametric descriptions Give a parametric...Ch. 14.6 - Identify the surface Describe the surface with the...Ch. 14.6 - Identify the surface Describe the surface with the...Ch. 14.6 - Identify the surface Describe the surface with the...Ch. 14.6 - Identify the surface Describe the surface with the...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface area using a parametric description Find...Ch. 14.6 - Surface integrals using a parametric description...Ch. 14.6 - Surface integrals using a parametric description...Ch. 14.6 - Surface integrals using a parametric description...Ch. 14.6 - Surface integrals using a parametric description...Ch. 14.6 - Surface area using an explicit description Find...Ch. 14.6 - Surface area using an explicit description Find...Ch. 14.6 - Prob. 33ECh. 14.6 - Prob. 34ECh. 14.6 - Surface integrals using an explicit description...Ch. 14.6 - Prob. 36ECh. 14.6 - Surface integrals using an explicit description...Ch. 14.6 - Surface integrals using an explicit description...Ch. 14.6 - Prob. 39ECh. 14.6 - Average values 40.Find the average squared...Ch. 14.6 - Average values 41.Find the average value of the...Ch. 14.6 - Average values 42.Find the average value of the...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Surface integrals of vector fields Find the flux...Ch. 14.6 - Explain why or why not Determine whether the...Ch. 14.6 - Miscellaneous surface integrals Evaluate the...Ch. 14.6 - Miscellaneous surface integrals Evaluate the...Ch. 14.6 - Miscellaneous surface integrals Evaluate the...Ch. 14.6 - Miscellaneous surface integrals Evaluate the...Ch. 14.6 - Cone and sphere The cone z2 = x2 + y2 for z 0,...Ch. 14.6 - Cylinder and sphere Consider the sphere x2 + y2 +...Ch. 14.6 - Flux on a tetrahedron Find the upward flux of the...Ch. 14.6 - Flux across a cone Consider the field F = x, y, z...Ch. 14.6 - Surface area formula for cones Find the general...Ch. 14.6 - Surface area formula for spherical cap A sphere of...Ch. 14.6 - Radial fields and spheres Consider the radial...Ch. 14.6 - Heat flux The heat flow vector field for...Ch. 14.6 - Heat flux The heat flow vector field for...Ch. 14.6 - Prob. 63ECh. 14.6 - Flux across a cylinder Let S be the cylinder x2 +...Ch. 14.6 - Flux across concentric spheres Consider the radial...Ch. 14.6 - Mass and center of mass Let S be a surface that...Ch. 14.6 - Mass and center of mass Let S be a surface that...Ch. 14.6 - Mass and center of mass Let S be a surface that...Ch. 14.6 - Mass and center of mass Let S be a surface that...Ch. 14.6 - Outward normal to a sphere Show that...Ch. 14.6 - Special case of surface integrals of scalar-valued...Ch. 14.6 - Surfaces of revolution Suppose y = f(x) is a...Ch. 14.6 - Rain on roofs Let z = s(x, y) define a surface...Ch. 14.6 - Surface area of a torus a.Show that a torus with...Ch. 14.6 - Prob. 75ECh. 14.7 - Explain the meaning of the integral S(F)ndS in...Ch. 14.7 - Explain the meaning of the integral S(F)ndS in...Ch. 14.7 - Explain the meaning of Stokes Theorem.Ch. 14.7 - Why does a conservative vector field produce zero...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Verifying Stokes Theorem Verify that the line...Ch. 14.7 - Stokes Theorem for evaluating line integrals...Ch. 14.7 - Stokes Theorem for evaluating line integrals...Ch. 14.7 - Stokes Theorem for evaluating line integrals...Ch. 14.7 - Stokes Theorem for evaluating line integrals...Ch. 14.7 - 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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Evaluate the integrals in Exercises 1–14.
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University Calculus: Early Transcendentals (4th Edition)

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*arrow_forward*Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (−1, 1, 1), (2, 1, −5), (2, 3, −11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1,-5) from (2, 1, 5) to (2, 3, -11) from (2, 3, -11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁ (t) = S₂(t) = S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) te [3, 4) UE [-1, 2], VE [1, 3]*arrow_forward*Evaluate Curlv•ñ, where v = 2xyi + (x² – 2y)j + xzk and i is a unit vector normal to the surface shown in the figure: (i) (ii) Z 1 surface z=1 surface y=1 1 1 1*arrow_forward* - 6. Use Stokes theorem to evaluate §. F·dr, where F = (-3y² , 4z, 6x) and C is the triangle in the plane z = ½ y with vertices (2,0, 0), (0, 2, 1) and (0, 0, 0) with a counterclockwise orientation looking down the positive z-axis.
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