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Chapter: 15 -Problem: 27 >> Compute the curl of the following vector fields. Transcribed Image Text: F = (x² - y², xy, z)
Answer Preview: ANSWER; To compute the curl of the vector field F = x…

, Chapter: 15 -Problem: 2 >> Interpret the volume integral in the Divergence Theorem.
Answer Preview: The volume integral in the Divergence Theorem represents the flux or flow of a vector field across t…

, Chapter: 15 -Problem: 11 >> Find the divergence of the following vector fields. Transcribed Image Text: F (12x, -6y, -6z) =
Answer Preview: To find the divergence of the vector field F = <12x, -6y, -6z>, we need to compute the part…

, Chapter: 15 -Problem: 15 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (x, y, z) 1 + x² + y²
Answer Preview: To find the divergence of the vector field F = (x, y, z)/(1 + x^2 + y^2), we need to compute the div…

, Chapter: 15 -Problem: 5 >> 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. Transcribed Image Text: F = (y, -x, 10); S is the upper half of th
Answer Preview: To verify the equality of the line integral and the surface integral using Stokes' Theorem, we need to evaluate both integrals and compare their results. Given: The vector field F = (y, -x, 10) The su…

, Chapter: 15 -Problem: 25 >> Evaluate the line integral ?c F • dr for the following vector fields F and curves C in two ways. a. By parameterizing C b. By using the Fundamental Theorem for line integrals, if possible Transcribed Image Text: F = V(xyz); C: r(t) = (cos t, sin t, t/?), for 0 ? t ? T
Answer Preview: a. To evaluate the line integral c F dr by parameterizing C, we need to substitute the parametric eq…

, Chapter: 15 -Problem: 9 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (2x, 4y, -3z)
Answer Preview: ANSWER The divergence of a vector field F = is given by: div(F…

, Chapter: 15 -Problem: 9 >> Describe the usual orientation of a closed surface such as a sphere.
Answer Preview: The usual orientation of a closed surface, such as a sphere, is defined by an outward-pointing norma…

, Chapter: 15 -Problem: 22 >> Consider the following vector fields, the circle C, and two points P and Q. a. Without computing the divergence, does the graph suggest that the divergence is positive or negative at P and Q? Justify your answer. b. Compute the divergence and confirm your conjecture in part (a). c. On what part of C is the flux outward? Inward? d. Is the net outward flux across C positive or negative?
Answer Preview: a. Based on the graph, we can make an inference about the divergence at points P and Q. The divergen…

, Chapter: 15 -Problem: 21 >> Consider the following vector fields, the circle C, and two points P and Q. a. Without computing the divergence, does the graph suggest that the divergence is positive or negative at P and Q? Justify your answer. b. Compute the divergence and confirm your conjecture in part (a). c. On what part of C is the flux outward? Inward? d. Is the net outward flux across C positive or negative?
Answer Preview: ANSWERS a. The graph suggests that the divergence is positive at P and negative a…

, Chapter: 15 -Problem: 12 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (x²yz, -xy²z, -xyz²)
Answer Preview: To find the divergence of the vector field F = , we need to compute the divergence operat…

, Chapter: 15 -Problem: 5 >> Explain how to compute the surface integral of a scalar-valued function f over a sphere using a parametric description of the sphere.
Answer Preview: To compute the surface integral of a scalar-valued function f over a sphere using a parametric descr…

, Chapter: 15 -Problem: 7 >> Suppose div F = 0 in a region enclosed by two concentric spheres. What is the relationship between the outward fluxes across the two spheres?
Answer Preview: By the divergence theorem, the outward flux across the …

, Chapter: 15 -Problem: 30 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: F = (x, 2y, 3z); D is the region between the cylinders x² + y² = 1 and x² + y² = 4, for 0 ? z ? 8.
Answer Preview: ANSWER; To compute the net outward flux of the vector field F = (x, 2y, 3z) across the boundary of t…

, Chapter: 15 -Problem: 21 >> For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. Transcribed Image Text: V = (0, 0, y)
Answer Preview: The given velocity field is \( \mathbf{v} = \langle 0, 0, y angle \). To compute the curl, we can use the formula: \[ \text{curl}(\mathbf{v}) …

, Chapter: 15 -Problem: 1 >> Give a parametric description for a cylinder with radius a and height h, including the intervals for the parameters.
Answer Preview: A parametric description for a cylinder with radius a and height h can be g…

, Chapter: 15 -Problem: 23 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x, y, z); S is the surface of the paraboloid z = 4x² - y², for z? 0, plus its base in the xy-plane.
Answer Preview: ANSWER To compute the net outward flux of the vector field F(x, y, …

, Chapter: 15 -Problem: 3 >> R2 Find the vector field F = ?? for the following potential functions. Sketch a few level curves of ? and sketch the general appearance of F in relation to the level curves. Transcribed Image Text: p(x, y) = x² + 4y2, for x = 5, y ? 5
Answer Preview: To find the vector field F = for the potential function (x, y) = x^2 + 4y^2, we need to compute the …

, Chapter: 15 -Problem: 4 >> Explain how to compute the surface integral of a scalar-valued function f over a cone using an explicit description of the cone.
Answer Preview: To compute the surface integral of a scalar-valued function f over a cone, we can use an explicit description of the cone and the formula for surface …

, Chapter: 15 -Problem: 18 >> Evaluate the line integral in Stokes’ Theorem to evaluate the surface integral ??s (? × F) • n ds. Assume that n points in the positive z-direction. Transcribed Image Text: F = r/r; S is the paraboloid x = 9 - y²z², for 0 ? x ? 9 (excluding its base), where r = (x, y, z).
Answer Preview: To evaluate the surface integral s ( F) n ds using Stokes' Th…

, Chapter: 15 -Problem: 21 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (y 2x, x³ y, y²z); S is the sphere {(x, y, z): x² + y² + z² = 4}.
Answer Preview: To compute the net outward flux of the vector field F across the surface S using the Divergence Theo…

, Chapter: 15 -Problem: 17 >> Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 15.8. Data from in Theorem 15.8 Transcribed Image Text: F= = (x, y, z) x² + y² + +2.² N || r |r|²
Answer Preview: ANSWER; To calculate the divergence of the radial field F = (x, y, z)/(x^2 + y^2 + z^2), we need to …

, Chapter: 15 -Problem: 24 >> Consider the following vector fields, where r = (x, y, z).a. Compute the curl of the field and verify that it has the same direction as the axis of rotation.b. Compute the magnitude of the curl of the field. Transcribed Image Text: F = (1,-1,0) x r
Answer Preview: ANSWER; To compute the curl of the vector field F = 1, -1, 0 * r, where r = x, y, z, we can use the …

, Chapter: 15 -Problem: 10 >> 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. Transcribed Image Text: F = (-y, -x - z, y - x); S is the part of
Answer Preview: ANSWER To verify that the line integral and the surface integral of Stokes' Theorem are equal for the given vector field F, surface S, and closed curv…

, Chapter: 15 -Problem: 10 >> Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. Transcribed Image Text: F = (x, y, z); D = {(x, y, z): [x] ? 1, |y| ? 1, |z| ? 1}
Answer Preview: To evaluate both integrals of the Divergence Theorem for the vector field F = (-x, -y, -z) and the r…

, Chapter: 15 -Problem: 19 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x, 2y, z); S is the boundary of the tetrahedron in the first octant formed by the plane x + y + z = 1.
Answer Preview: To compute the net outward flux of the field F = (x, 2y, z) across the surface S, we can apply the Divergence Theorem, also known as Gauss's theorem. …

, Chapter: 15 -Problem: 13 >> R3 Given the following force fields, find the work required to move an object on the given curve. Transcribed Image Text: F = (-y, z, x) on the path consisting of the line segment from (0, 0, 0) to (0, 1, 0) followed by the line segment from (0, 1, 0) to (0, 1, 4)
Answer Preview: ANSWER; To find the work required to move an object along a curve with a given force field, we can u…

, Chapter: 15 -Problem: 8 >> 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. Transcribed Image Text: F = (22,-4x, 3y); S is the cap of the sphe
Answer Preview: To verify the equality of the line integral and the surface integral of Stokes' Theorem for the give…

, Chapter: 15 -Problem: 12 >> Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. Transcribed Image Text: F = (x², y², z²); D = {(x, y, z): x ? 1, y ? 2, z] ? 3}
Answer Preview: To evaluate both integrals of the Divergence Theorem for the given vector field F = x^2, y^2, z^2 and region D = {(x, y, z) | |x| 1, |y| 2, |z| 3}, we'll first calculate the surface integral over the …

, Chapter: 15 -Problem: 9 >> Evaluate the following line integrals. Transcribed Image Text: [ye ye * ds; C is the path r(t) = (t, 3t, -6t), for 0 ? t ? In 8.
Answer Preview: previous step takes into account the correct orientation based …

, Chapter: 15 -Problem: 28 >> Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. Transcribed Image Text: F = (3x²y, x³ + 2yz², 2y²z)
Answer Preview: ANSWER; To use Stokes' Theorem to find the circulation of the vector field F around a closed curve C, we need to calculate the surface integral of the …

, Chapter: 15 -Problem: 9 >> 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. Transcribed Image Text: F = (yz, z x² + y² + z² = boundary of S. =
Answer Preview: To verify the equality of the line integral and the surface integral using Stokes' Theorem, we need to evaluate both integrals and compare their results. Given: The vector field F = (y - z, z - x, x - …

, Chapter: 15 -Problem: 24 >> For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. Transcribed Image Text: V = (0, -z, y)
Answer Preview: ANSWER To compute the curl of the velocity field v = 0, -z, y, we can use the curl formula: curl(v) …

, Chapter: 15 -Problem: 4 >> Explain how to compute the curl of the vector field Transcribed Image Text: F = (f, g, h).
Answer Preview: To compute the curl of a vector field F = , follow these steps: 1. Compute the partial derivatives o…

, Chapter: 15 -Problem: 17 >> Describe the surface with the given parametric representation. Transcribed Image Text: r(u, v) = (u, v, 2u + 3v - 1), for 1 ? u ? 3,2 ? y ? 4
Answer Preview: The given parametric representation of the surface is: r(u, v) = u, v, 2u + 3v - 1, for 1 u 3 and 2 …

, Chapter: 15 -Problem: 20 >> Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 15.8. Data from in Theorem 15.8 Transcribed Image Text: F = (x, y, z) (x² + y² + z²) = r|r|²
Answer Preview: ANSWER; To calculate the divergence of the radial field F = x, y, z(x^2 + y^2 + z^2) = r * |r|^2, we …

, Chapter: 15 -Problem: 22 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (y + z?x + 2, x + y); S consists of the faces of the cube {(x, y, z): x ? 1, [y] ? 1, |z| ? 1}.
Answer Preview: To compute the net outward flux of the vector field F = (y + z, x + z, x + y) across the surfaces of …

, Chapter: 15 -Problem: 26 >> Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. Transcribed Image Text: F = (2x, -2y, 2z.)
Answer Preview: To find the circulation of the vector field F = <2x, -2y, 2z> around a simple closed smooth curve C …

, Chapter: 15 -Problem: 4 >> R2 Find the vector field F = ?? for the following potential functions. Sketch a few level curves of ? and sketch the general appearance of F in relation to the level curves. Transcribed Image Text: p(x, y) = (x² - y²)/2, for x ? 2, y ? 2
Answer Preview: ANSWER; To find the vector field F = for the given potential function (x, y) = (x^2 - y^2)/2, we need to compute the gradient of . The gradient of , d…

, Chapter: 15 -Problem: 16 >> Evaluate the line integral ?C F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (2xy sin z, x² sin z, x² y cos z); C is the boundary of the plane z = 8 - 2x - 4y in the first oc
Answer Preview: ANSWER; To evaluate the line integral C F dr using Stokes' Theorem, we need to find an appropriate surface S whose boundary is C. The given curve C is the boundary of the plane z = 8 - 2x - 4y in the …

, Chapter: 15 -Problem: 14 >> R3 Given the following force fields, find the work required to move an object on the given curve. Transcribed Image Text: F = (x, y, z) (x² + y² + z²)³/2 1?t?2 on the path r(t) = (1²,31², -1²), for
Answer Preview: To find the work required to move an object on the given curve using the force field f = (x, y, z)/(…

, Chapter: 15 -Problem: 31 >> Use either form of Green’s Theorem to evaluate the following line integrals. Transcribed Image Text: $(x³ vertices (±1, ±1) with counterclockwise orientation. + xy) dy + (2y² - 2x²y) dx; C is the square with
Answer Preview: ANSWER; To evaluate the line integral using Green's Theorem, we need to calculate the double integral of the curl of the vector field over the region …

, Chapter: 15 -Problem: 28 >> Compute the curl of the following vector fields. Transcribed Image Text: (0,z² - y², -yz) F = (0,
Answer Preview: ANSWER To compute the curl of the vector field F = 0, z^2 - y^2, …

, Chapter: 15 -Problem: 13 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (x² - y², y² - z², z² - x²)
Answer Preview: ANSWER; To find the divergence of the vector field F = x^2 - y^2, y^2 - z^2, z^2 - x^2, we nee…

, Chapter: 15 -Problem: 1 >> Explain how to compute the divergence of the vector field Transcribed Image Text: F = (f, g, h).
Answer Preview: A NSWER; To compute the divergence of a vector field F = (f, g, h), where f, g, and h are functions …

, Chapter: 15 -Problem: 19 >> Evaluate the line integral in Stokes’ Theorem to evaluate the surface integral ??s (? × F) • n ds. Assume that n points in the positive z-direction. Transcribed Image Text: F = (2y, -2, its base) x² + y² + z² xyz); S is the cap of the sphere (excluding = 25, for 3 ? x ? 5.
Answer Preview: ANSWER; To evaluate the surface integral s ( F) n ds using Stokes' Theorem, we need to find the cur…

, Chapter: 15 -Problem: 30 >> Compute the curl of the following vector fields. Transcribed Image Text: F = r = (x, y, z)
Answer Preview: To compute the curl of the vector field F = r = , where r = represents the position vecto…

, Chapter: 15 -Problem: 26 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: F = r r = (x, y, z) ?x² + y² + z²; D is the region between the spheres of radius 1 and 2 centered at the origin.
Answer Preview: ANSWER; To compute the net outward flux of the vector field F across the boundary of the region D, we can use the Divergence Theorem. The Divergence T…

, Chapter: 15 -Problem: 18 >> Describe the surface with the given parametric representation. Transcribed Image Text: r(u, v) = (u, u + v, 2-u-v), for 0 ? u ? 2,0 ? y ? 2
Answer Preview: The surface described by the parametric representation r(u,v) = , where 0 u 2 and 0 v 2, is a triang…

, Chapter: 15 -Problem: 6 >> 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. Transcribed Image Text: F = (0, -x, y); x² + y² + z2²: = S is the
Answer Preview: To verify the equality of the line integral and surface integral of Stokes' Theorem for the given vector field, surface, and closed curve, we'll fol…

, Chapter: 15 -Problem: 27 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: (x, y, z) ?x² + y² + z² spheres of radius 1 and 2 centered at the origin. ;D is the region between the
Answer Preview: ANSWER; To compute the net outward flux of the vector field F across the boundary of the region D, w…

, Chapter: 15 -Problem: 20 >> Describe the surface with the given parametric representation. Transcribed Image Text: r(u, v) = (v, 6 cos u, 6 sin u), for 0 ? u ? 2?, 0 ? y ? 2
Answer Preview: The surface described by the parametric representation r(u,v) = , where 0 u 2 and 0 v 2, is a …

, Chapter: 15 -Problem: 8 >> If div F > 0 in a region enclosed by a small cube, is the net flux of the field into or out of the cube?
Answer Preview: If the divergence of the vector field F is positive (div F > 0) in a region enclosed by a smal…

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, Chapter: 15 -Problem: 11 >> Evaluate the line integral ?c F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (2y, -2, x); C is the circle x² + y² = 12 in the plane z = 0.
Answer Preview: ANSWER; To evaluate the line integral c F dr using Stokes' Theorem, we need to find an appropriate s…

, Chapter: 15 -Problem: 31 >> Compute the curl of the following vector fields. Transcribed Image Text: F (x, y, z) 2 (x² + y² + z²)³/2 r 3 r³
Answer Preview: ANSWER; To compute the curl of the vector field, let's start by expressing the vector field in term…

, Chapter: 15 -Problem: 12 >> Evaluate the line integral ?C F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (y, xz, -y); C is the ellipse x² + y²/4 = 1 in the plane z = 1.
Answer Preview: To evaluate the line integral C F dr using Stokes' Theorem, we need to find an appropriate surface S whose boundary is C. In this case, C is the ellipse defined by the equation x^2 + y^2 / 4 = 1 in t…

, Chapter: 15 -Problem: 22 >> For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. Transcribed Image Text: v = (1-z², 0, 0)
Answer Preview: Given velocity field is v = 1 - z^2, 0, 0. The curl of v is given by the formula: curl(…

, Chapter: 15 -Problem: 5 >> Interpret the curl of a general rotation vector field.
Answer Preview: ANSWER The curl of a general rotation vector field measures the local rotational effects or vorticit…

, Chapter: 15 -Problem: 23 >> Consider the following vector fields, where r = (x, y, z). a. Compute the curl of the field and verify that it has the same direction as the axis of rotation. b. Compute the magnitude of the curl of the field. Transcribed Image Text: F = (1, 0, 0) X r
Answer Preview: a. To compute the curl of the vector field F = <1, 0, 0>*r, where r = (x, y, z), we can use the form…

, Chapter: 15 -Problem: 16 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (yz sin x, xz cos y, xy cos z)
Answer Preview: ANSWER; To find the divergence of the vector field F = (yz sin x, xz cos y, xy cos z), we …

, Chapter: 15 -Problem: 20 >> Evaluate the line integral in Stokes’ Theorem to evaluate the surface integral ??s (? × F) • n ds. Assume that n points in the positive z-direction. Transcribed Image Text: F = (x + y, y + z, z + x); S is the tilted disk enclosed by r(t) = (cos t, 2 sin t, V3 cos t).
Answer Preview: To evaluate the surface integral s ( F) n ds using Stokes' Theorem, we need to follow a few steps: …

, Chapter: 15 -Problem: 24 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x, y, z); S is the surface of the cone z² = x² + y², for 0 ? z ? 4, plus its top surface in the plane z = 4.
Answer Preview: To compute the net outward flux of the field F = (x, y, z) across the given surfaces, we can use the …

, Chapter: 15 -Problem: 5 >> R3 Find the vector field F = ?? for the following potential functions. Transcribed Image Text: p(x, y, z) = 1/[r], where r = (x, y, z)
Answer Preview: To find the vector field F = , where (x, y, z) = 1/|r|, and r = (x, y, z) is t…

, Chapter: 15 -Problem: 29 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: F = (x²,-y², z²); D is the region in the first octant between the planes z = 4 - x - y and z = 2 - x - y.
Answer Preview: ANSWER; To compute the net outward flux of the vector field F = (x^2, y^2, z^2) across the boundary of the region D, we can apply the Divergence Theor…

, Chapter: 15 -Problem: 15 >> Evaluate the line integral ?C F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (y²,-z², x); C is the circle r(t) = (3 cos t, 4 cos t, 5 sin t), for 0 ? t ? 2.
Answer Preview: To evaluate the line integral C F dr using Stokes' Theorem, we need to find an appropriate surface …

, Chapter: 15 -Problem: 26 >> Evaluate the line integral ?c F • dr for the following vector fields F and curves C in two ways. a. By parameterizing C b. By using the Fundamental Theorem for line integrals, if possible Transcribed Image Text: F = (x,y); C is the square with vertices (±1, ±1) with counterclockwise orientation.
Answer Preview: a. By parameterizing C: To evaluate the line integral c F dr using parameterization, we need to find a parameterization for the curve C. In this case, C is the square with vertices (1, 1) with counter…

, Chapter: 15 -Problem: 24 >> Evaluate the line integral ?c F • dr for the following vector fields F and curves C in two ways. a. By parameterizing C b. By using the Fundamental Theorem for line integrals, if possible Transcribed Image Text: F = V(x²y); C: r(t) = (9- t², t), for 0 ? t ? 3
Answer Preview: a. To evaluate the line integral c F dr by parameterizing the curve C, we need to calculate the dot …

, Chapter: 15 -Problem: 29 >> Compute the curl of the following vector fields. Transcribed Image Text: F = (x²z², 1, 2xz)
Answer Preview: To compute the curl of the vector field F = , we can apply the curl operator ( ) to F. Th…

, Chapter: 15 -Problem: 6 >> R3 Find the vector field F = ?? for the following potential functions. Transcribed Image Text: p(x, y, z) = -e-x2-y2-2 2
Answer Preview: To find the vector field F = for the given potential function (x, y, z) …

, Chapter: 15 -Problem: 13 >> Evaluate the line integral ?C F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (x²z², y, 2xz); C is the boundary of the plane z = 4 - x - y in the first octant.
Answer Preview: To evaluate the line integral C F dr using Stokes' Theorem, we need to find an appropriate surface S whose boundary is C. In this case, C is the boundary of the plane z = 4 - x - y in the first octan…

, Chapter: 15 -Problem: 19 >> Describe the surface with the given parametric representation. Transcribed Image Text: r(u, v) = (v cos u, v sin u, 4v), for 0 ? u ? ?,0 ? v ? 3
Answer Preview: ANSWER; The given parametric representation describes a surface in three-dimensional space. Let's break it down to understand the surface more clearly…

, Chapter: 15 -Problem: 1 >> Explain the meaning of the surface integral in the Divergence Theorem.
Answer Preview: The Divergence Theorem connects the flux of a vector field across a closed surface to the vector fie…

, Chapter: 15 -Problem: 9 >> Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. Transcribed Image Text: F = (2x, 3y, 4z); D = {(x, y, z): x² + y² + z² ? 4}
Answer Preview: ANSWER; To evaluate the integrals of the Divergence Theorem for the given vector field F = (2x, 3y, …

, Chapter: 15 -Problem: 29 >> Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. Transcribed Image Text: F = (y²z³, 2xyz³, 3xy²z²)
Answer Preview: ANSWER; To find the circulation of the vector field F = (y^2z^3, 2xyz^3, 3xyz^2) around any simple closed smooth curve C using Stokes' Theorem, we nee…

, Chapter: 15 -Problem: 7 >> 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. Transcribed Image Text: F = (x, y, z); S is the paraboloid z = 8 -
Answer Preview: To verify the equality of the line integral and the surface integral of Stokes' Theorem, we need to evaluate both integrals and compare their results. Let's start by calculating the line integral: Lin…

, Chapter: 15 -Problem: 17 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x,-2y, 3z); S is the sphere {(x, y, z): x² + y² + z² = 6}.
Answer Preview: ANSWER; To compute the net outward flux of the vector field F = (x, -2y, 3z) across the given surfac…

, Chapter: 15 -Problem: 18 >> Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 15.8. Data from in Theorem 15.8 Transcribed Image Text: F = (x, y, z) (x² + y² + z²)3/2 2 r/3
Answer Preview: ANSWER; To calculate the divergence of the radial field F = (x, y, z) / (x + y + z)^(3/2) * 2r / r^3…

, Chapter: 15 -Problem: 14 >> Evaluate the line integral ?C F • dr by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. Transcribed Image Text: F = (x² - y², z² - x², y²z²); C is the boundary of the square x ? 1, y ? 1 in the plane z = 0.
Answer Preview: ANSWER To evaluate the line integral C F dr using Stokes' Theorem, we need to find an appropriate s…

, Chapter: 15 -Problem: 17 >> Evaluate the line integral in Stokes’ Theorem to evaluate the surface integral ??s (? × F) • n ds. Assume that n points in the positive z-direction. Transcribed Image Text: F = (x, y, z); S is the upper half of the ellipsoid x²/4 + y²/9 + z² = 1.
Answer Preview: To evaluate the line integral in Stokes' Theorem and subsequently the surface integral, we need to follow these steps: Step 1: Determine the boundary …

, Chapter: 15 -Problem: 19 >> Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 15.8. Data from in Theorem 15.8 Transcribed Image Text: F (x, y, z) (x² + y² + z²)² r
Answer Preview: ANSWER; To calculate the divergence of the given radial field, we need to find the dot product of th…

, Chapter: 15 -Problem: 18 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x², 2xz, y²); S is the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1.
Answer Preview: To compute the net outward flux of the vector field F = (x, 2xz, y) across the surface S, we can use …

, Chapter: 15 -Problem: 2 >> Give a parametric description for a cone with radius a and height h, including the intervals for the parameters.
Answer Preview: A parametric description for a cone with radius a and height h can be give…

, Chapter: 15 -Problem: 29 >> Use either form of Green’s Theorem to evaluate the following line integrals. Transcribed Image Text: fxy² dx dx + x²y dy; C is the triangle with vertices (0, 0), (2, 0), and (0, 2) with counterclockwise orientation.
Answer Preview: ANSWER; To evaluate the line integral using Green's Theorem, we need to compute the double integral …

, Chapter: 15 -Problem: 23 >> For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. Transcribed Image Text: v = (-2z, 0, 1)
Answer Preview: To compute the curl of the given velocity field V = (-2z, 0, 1), we can use the formula: curl(V) = (…

, Chapter: 15 -Problem: 8 >> Evaluate the following line integrals. Transcribed Image Text: ?(x² - 2xy + y²) ds; C is the upper half of the circle r(t) = (5 cos t, 5 sin t), for 0 ? t ?, with counterclockwise orientation.
Answer Preview: ANSWER To evaluate the line integral C (x^2 - 2xy + y^2) ds, where C is the upper half of the circ…

, Chapter: 15 -Problem: 10 >> Why is the upward flux of a vertical vector field F = (0, 0, 1) across a surface equal to the area of the projection of the surface in the xy-plane?
Answer Preview: The upward flux of a vertical vector field F = (0, 0, 1) across a surface is equal to the area of th…

, Chapter: 15 -Problem: 3 >> Give a parametric description for a sphere with radius a, including the intervals for the parameters.
Answer Preview: To parametrically describe a sphere with radius "a," we c…

, Chapter: 15 -Problem: 25 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: F = (zx, xy, 2y z); D is the region between the spheres of radius 2 and 4 centered at the origin.
Answer Preview: To compute the net outward flux of the vector field F = (z - x, x - y, 2yz) across the boundary of the region D, which is the region between the spher…

, Chapter: 15 -Problem: 27 >> Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. Transcribed Image Text: F V (xsin ye²) =
Answer Preview: Stokes' Theorem states that the circulation of a vector field F around a simple closed curve C is e…

, Chapter: 15 -Problem: 28 >> Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. Transcribed Image Text: F = (zy, xz, 2y - x); D is the region between two cubes: {(x, y, z): 1 ? x ? 3,1 ? y ? 3, 1 ? |z| ? 3}.
Answer Preview: ANSWER; To compute the net outward flux of the vector field F = (zy, xz, 2yx) across the boundary of …

, Chapter: 15 -Problem: 25 >> Consider the following vector fields, where r = (x, y, z). a. Compute the curl of the field and verify that it has the same direction as the axis of rotation. b. Compute the magnitude of the curl of the field. Transcribed Image Text: F = (1,-1, 1) Xr
Answer Preview: a. To compute the curl of the vector field F = <1, -1, 1>*r, where r = (x, y, z), we can use the for…

, Chapter: 15 -Problem: 11 >> Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. Transcribed Image Text: F = (zy, x, x); D {(x, y, z): x²/4 + y²/8 + z²/12 ? 1}
Answer Preview: To evaluate both integrals of the Divergence Theorem for the vector field F = (z - y, x, -x) and the …

, Chapter: 15 -Problem: 7 >> Explain how to compute a surface integral ??S F • n dS over a sphere using a parametric description of the sphere and a given orientation.
Answer Preview: To compute a surface integral S F n dS over a sphere using a parametric description and a given orientation, you can follow these steps: 1. Parameteri…

, Chapter: 15 -Problem: 20 >> Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. Transcribed Image Text: F = (x², y², z²); S is the sphere {(x, y, z): x² + y² + z² = 25}.
Answer Preview: ANSWER. To compute the net outward flux of the vector field F = x^2, y^2, z^2 across the sphere S gi…

, Chapter: 15 -Problem: 6 >> Explain how to compute a surface integral ??S F • n dS over a cone using an explicit description and a given orientation of the cone.
Answer Preview: To compute a surface integral over a cone, we need an explicit description of the cone and the orientation of its surface. Let's assume we have a cone …

, Chapter: 15 -Problem: 10 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (-2y, 3x, z)
Answer Preview: To find the divergence of the vector field F = (-2y, 3…

, Chapter: 15 -Problem: 10 >> Evaluate the following line integrals. Transcribed Image Text: (xz - y²) ds; C is the line segment from (0, 1, 2) to (-3, 7,-1).
Answer Preview: ANSWER; To evaluate the line integral (xzy^2) ds over the line segment C from (0, 1, 2) to (-3, 7, -…

, Chapter: 15 -Problem: 30 >> Use either form of Green’s Theorem to evaluate the following line integrals. Transcribed Image Text: (-3y + x³/2) dx + (x - y2/3) dy; C is the boundary of the C half disk {(x, y): x² + y² ? 2, y ? 0} with counterclockwise orientation.
Answer Preview: ANSWER To evaluate the line integral C (-3y + x^(3/2)) dx + (x - y^(2/3)) dy over the boundary C of the half disk {(x, y) | x^2 + y^2 2, y 0} with cou…

, Chapter: 15 -Problem: 14 >> Find the divergence of the following vector fields. Transcribed Image Text: F = (ex+yey+², e-z+x)
Answer Preview: To find the divergence of the vector field F = , we need …

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