Question

1. Verify Stokes' Theorem for the given vector field and surface, oriented with an upward-pointing normal: F = ⟨ey - z, 0, 0⟩, F = ⟨ey - z, 0, 0⟩, the square with vertices (4, 0, 3), (4, 4, 3), (0, 4, 3), and (0, 0, 3). ∫CF ⋅ ds = ∫CF ⋅ ds = ∬Scurl(F) ⋅ dS = 2. Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1, 0, 1), (0, 1, 0), and (0, 0, 1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral: n = (n = ( , , )) (the unit normal vector) ∇ × F = (∇ × F = ( , , )) The value of the integral is 3. Use Stokes' theorem to evaluate ∬S(∇ × F) ⋅ dS where F(x, y, z) = -4yz i + 4xz j + 7(x^2 + y^2) k and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 = 1, oriented upward. 4. Use Stokes' Theorem to evaluate ∬M(∇ × F) ⋅ dS where M is the hemisphere x^2 + y^2 + z^2 = 25, x ≥ 0, with the normal in the direction of the positive x direction, and F = ⟨x^6, 0, y^2⟩. Begin by writing down the "standard" parametrization of ∂M as a function of the angle θ (denoted by "t" in your answer) x = x = , y = y = , z = z = . ∫∂M F ⋅ ds = ∫2π0 f(θ) dθ ∫∂M F ⋅ ds = ∫02π f(θ) dθ, where f(θ) = f(θ) = (use "t" for theta). The value of the integral is 5. Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 1, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 1, z ≥ 1. For the vector field F = (zx + z^2y + 9y, z^3yx + 5x, z^4x^2), compute ∬M(∇ × F) ⋅ dS in any way you like. ∬M(∇ × F) ⋅ dS = 6. Use the divergence theorem to find the outward flux of the vector field F(x, y, z) = 2x^2i + 5y^2j + 2z^2k across the boundary of the rectangular prism: 0 ≤ x ≤ 4, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3 7. Evaluate ∬∂W F ⋅ dS where F = (x^2 + y, z^2, ey - z) and W is the solid rectangular box whose sides are bounded by the coordinate planes, and the planes x = 2, y = 7, z = 5. 8. Evaluate ∬M F ⋅ dS where F = (3xy^2, 3x^2y, z^3) and M is the surface of the sphere of radius 1 centered at the origin. 9. Verify the Divergence Theorem for the vector field and region: F = ⟨6x, 3z, 3y⟩ and the region x^2 + y^2 ≤ 1, 0 ≤ z ≤ 4. ∬S F ⋅ dS = ∬S F ⋅ dS = ∭R div(F) dV =

          1. Verify Stokes' Theorem for the given vector field and surface, oriented with an upward-pointing normal:
    F = ⟨ey - z, 0, 0⟩, F = ⟨ey - z, 0, 0⟩, the square with vertices (4, 0, 3), (4, 4, 3), (0, 4, 3), and (0, 0, 3).
    ∫CF ⋅ ds = ∫CF ⋅ ds = 
    ∬Scurl(F) ⋅ dS = 
    2. Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1, 0, 1), (0, 1, 0), and (0, 0, 1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:
    n = (n = ( , , )) (the unit normal vector)
    ∇ × F = (∇ × F = ( , , ))
    The value of the integral is
    3. Use Stokes' theorem to evaluate ∬S(∇ × F) ⋅ dS where F(x, y, z) = -4yz i + 4xz j + 7(x^2 + y^2) k and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 = 1, oriented upward.
    4. Use Stokes' Theorem to evaluate ∬M(∇ × F) ⋅ dS where M is the hemisphere x^2 + y^2 + z^2 = 25, x ≥ 0, with the normal in the direction of the positive x direction, and F = ⟨x^6, 0, y^2⟩. Begin by writing down the "standard" parametrization of ∂M as a function of the angle θ (denoted by "t" in your answer) x = x = , y = y = , z = z = . ∫∂M F ⋅ ds = ∫2π0 f(θ) dθ ∫∂M F ⋅ ds = ∫02π f(θ) dθ, where f(θ) = f(θ) = (use "t" for theta). The value of the integral is
    5. Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 1, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 1, z ≥ 1. For the vector field F = (zx + z^2y + 9y, z^3yx + 5x, z^4x^2), compute ∬M(∇ × F) ⋅ dS in any way you like. ∬M(∇ × F) ⋅ dS =
    6. Use the divergence theorem to find the outward flux of the vector field F(x, y, z) = 2x^2i + 5y^2j + 2z^2k across the boundary of the rectangular prism: 0 ≤ x ≤ 4, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3
    7. Evaluate ∬∂W F ⋅ dS where F = (x^2 + y, z^2, ey - z) and W is the solid rectangular box whose sides are bounded by the coordinate planes, and the planes x = 2, y = 7, z = 5.
    8. Evaluate ∬M F ⋅ dS where F = (3xy^2, 3x^2y, z^3) and M is the surface of the sphere of radius 1 centered at the origin.
    9. Verify the Divergence Theorem for the vector field and region: F = ⟨6x, 3z, 3y⟩ and the region x^2 + y^2 ≤ 1, 0 ≤ z ≤ 4. ∬S F ⋅ dS = ∬S F ⋅ dS = ∭R div(F) dV =

Added by Julio H.

Question

Please give Ace some feedback