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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

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
Evaluate the following line integrals. √(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.
Evaluate the following line integrals. [ye ye * ds; C is the path r(t) = (t, 3t, -6t), for 0 ≤ t ≤ In 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
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.
Find the divergence of the following vector fields. F = (2x, 4y, -3z)
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
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
Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. F = (2x, 3y, 4z); D = {(x, y, z): x² + y² + z² ≤ 4}
Find the divergence of the following vector fields. F = (-2y, 3x, z)
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.
Evaluate the following line integrals. (xz - y²) ds; C is the line segment from (0, 1, 2) to (-3, 7,-1).
Suppose div F = 0 in a region enclosed by two concentric spheres. What is the relationship between the outward fluxes across the two spheres?
Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. F = (x, y, z); D = {(x, y, z): [x] ≤ 1, |y| ≤ 1, |z| ≤ 1}
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?
Describe the usual orientation of a closed surface such as a sphere.
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?
Find the divergence of the following vector fields. F (12x, -6y, -6z) =
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. F = (2y, -2, x); C
Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. F = (zy, x, x); D {(x, y, z): x²/4 + y²/8 + z²/12 ≤ 1}
R3 Given the following force fields, find the work required to move an object on the given curve. F = (x, y, z) (x² + y² + z²)³/2 1≤t≤2 on the path r(t) = (1²,31², -1²), for
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
Matching vector fields Match vector fields a–f with the graphs A–F. Let r = (x, y). a. F = (x, y) c. F = r/r e. F = (e, e*) b. F = (-2y, 2x) d. F = (y - x,x) F = f. (sin 7x, sin #y)
Compute the curl of the following vector fields. F = (x² - y², xy, z)
Explain how to compute the divergence of the vector field F = (f, g, h).
R2 Find the vector field F = ∇φ for the following potential functions. Sketch a few level curves of φ and sketch the general appearance of F in relation to the level curves. p(x, y) = x² + 4y2,
R2 Find the vector field F = ∇φ for the following potential functions. Sketch a few level curves of φ and sketch the general appearance of F in relation to the level curves. p(x, y) = (x² -
Explain how to compute the curl of the vector field F = (f, g, h).
R3 Find the vector field F = ∇φ for the following potential functions. p(x, y, z) = 1/[r], where r = (x, y, z)
Explain the meaning of the surface integral in the Divergence Theorem.
Give a parametric description for a cylinder with radius a and height h, including the intervals for the parameters.
Interpret the volume integral in the Divergence 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
Give a parametric description for a cone with radius a and height h, including the intervals for the parameters.
Give a parametric description for a sphere with radius a, including the intervals for the parameters.
Explain how to compute the surface integral of a scalar-valued function f over a cone using an explicit description of the cone.
Interpret the curl of a general rotation vector field.
Explain how to compute the surface integral of a scalar-valued function f over a sphere using a parametric description of the sphere.
R3 Find the vector field F = ∇φ for the following potential functions. p(x, y, z) = -e-x2-y2-2 2
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
Find the divergence of the following vector fields. F = (x²yz, -xy²z, -xyz²)
Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. F = (x², y², z²); D = {(x, y, z): x ≤ 1, y ≤ 2, z] ≤ 3}
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. F = (y, xz, -y); C
R3 Given the following force fields, find the work required to move an object on the given curve. F = (-y, z, x) on the path consisting of the line segment from (0, 0, 0) to (0, 1, 0) followed by the
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. F = (x²z², y,
Find the divergence of the following vector fields. F = (x² - y², y² - z², z² - x²)
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
Find the divergence of the following vector fields. F = (x, y, z) 1 + x² + y²
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. F = (y²,-z², x);
Find the divergence of the following vector fields. F = (ex+yey+², e-z+x)
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. F = (2xy sin z, x²
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. F = (x² - y², z²
Find the divergence of the following vector fields. F = (yz sin x, xz cos y, xy cos z)
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 F
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x,-2y, 3z); S is the sphere {(x, y, z): x² + y² + z² = 6}.
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. F = r/r; S is the paraboloid x = 9 -
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. F = (x, y, z); S is the upper half of
Describe the surface with the given parametric representation. r(u, v) = (u, v, 2u + 3v - 1), for 1 ≤ u ≤ 3,2 ≤ y ≤ 4
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x², 2xz, y²); S is the surface of the cube cut from the first octant by the
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 F (x,
Describe the surface with the given parametric representation. r(u, v) = (u, u + v, 2-u-v), for 0 ≤ u ≤ 2,0 ≤ y ≤ 2
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 F =
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. F = (2y, -2, its base) x² + y² +
Describe the surface with the given parametric representation. r(u, v) = (v cos u, v sin u, 4v), for 0 ≤ u ≤ π,0 ≤ v ≤ 3
Describe the surface with the given parametric representation. r(u, v) = (v, 6 cos u, 6 sin u), for 0 ≤ u ≤ 2π, 0 ≤ y ≤ 2
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x, 2y, z); S is the boundary of the tetrahedron in the first octant formed by the
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. F = (x + y, y + z, z + x); S is the
For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. V = (0, 0, y)
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
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x², y², z²); S is the sphere {(x, y, z): x² + y² + z² = 25}.
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
For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. v = (1-z², 0, 0)
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (y + z₂x + 2, x + y); S consists of the faces of the cube {(x, y, z): x ≤ 1,
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (y 2x, x³ y, y²z); S is the sphere {(x, y, z): x² + y² + z² = 4}.
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
Evaluate the line integral ∫c F • dr for the following vector fields F and curves C in two ways.a. By parameterizing Cb. By using the Fundamental Theorem for line integrals, if possible F =
For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. v = (-2z, 0, 1)
Evaluate the line integral ∫c F • dr for the following vector fields F and curves C in two ways.a. By parameterizing Cb. By using the Fundamental Theorem for line integrals, if possible F =
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x, y, z); S is the surface of the paraboloid z = 4x² - y², for z≥ 0, plus its
For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. V = (0, -z, y)
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = (zx, xy, 2y z); D is the region between the spheres of radius
Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x, y, z); S is the surface of the cone z² = x² + y², for 0 ≤ z ≤ 4, plus
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
Evaluate the line integral ∫c F • dr for the following vector fields F and curves C in two ways.a. By parameterizing Cb. By using the Fundamental Theorem for line integrals, if possible F =
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. (x, y, z) √x² + y² + z² spheres of radius 1 and 2 centered
Compute the curl of the following vector fields. (0,z² - y², -yz) F = (0,
Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. F = (2x, -2y, 2z.)
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = r r = (x, y, z) √x² + y² + z²; D is the region
Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. F = (3x²y, x³ + 2yz², 2y²z)
Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. F V (xsin ye²) =
Use Stokes’ Theorem to find the circulation of the following vector fields around any simple closed smooth curve C. F = (y²z³, 2xyz³, 3xy²z²)
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = (x²,-y², z²); D is the region in the first octant between
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = (zy, xz, 2y - x); D is the region between two cubes: {(x, y,
Use either form of Green’s Theorem to evaluate the following line integrals. fxy² dx dx + x²y dy; C is the triangle with vertices (0, 0), (2, 0), and (0, 2) with counterclockwise orientation.
Compute the curl of the following vector fields. F = r = (x, y, z)
Compute the curl of the following vector fields. F = (x²z², 1, 2xz)
Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. F = (x, 2y, 3z); D is the region between the cylinders x² + y²
Compute the curl of the following vector fields. F (x, y, z) 2 (x² + y² + z²)³/2 r 3 r³
Use either form of Green’s Theorem to evaluate the following line integrals. (-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
Use either form of Green’s Theorem to evaluate the following line integrals. $(x³ vertices (±1, ±1) with counterclockwise orientation. + xy) dy + (2y² - 2x²y) dx; C is the square with

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