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Calculus: Early Transcendental Functions
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- Tilted disks Let S be the disk enclosed by the curveC: r(t) = ⟨cos φ cos t, sin t, sin φ cos t⟩ , for 0 ≤ t ≤ 2π, where 0 ≤ φ ≤ π/2 is a fixed angle. Use Stokes’ Theorem and a surface integral to find the circulation on C of the vector field F = ⟨ -y, x, 0⟩ as a function of φ. For what value of φ is the circulation a maximum?arrow_forward(a) Show that at every point on the curve r(u) = (e^u cos u, e^u sin u, e^u ) , the angle between the unit tangent vector and the z-axis is the same. Show that this is also true for the principal normal vector. (b) Give a parametric representation of the level surface e^(xyz) = 1. (c) Find the equation of the tangent plane to the surface z = 3 e x−y at the point (4, 4, 3).arrow_forwardGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forward
- 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 C has counterclockwise orientation and S has a consistent orientation. F = ⟨2z, -4x, 3y⟩; S is the cap of the sphere x2 + y2 + z2 = 169 above the plane z = 12 and C is the boundary of S.arrow_forwardCalculus Answer Calculate the work that a constant force field F does on a particle that moves uniformly once along the path of the curve x2 + y2 = 1.How much is the work now worth if we take F(x, y) = (αx, αy), where α is any positive constant?arrow_forwardTilted disks Let S be the disk enclosed by the curveC: r(t) = ⟨cos φ cos t, sin t, sin φ cos t⟩ , for 0 ≤ t ≤ 2π, where 0 ≤ φ ≤ π/2 is a fixed angle. Consider the vector field F = a x r, where a = ⟨a1, a2, a3⟩ is aconstant nonzero vector and r = ⟨x, y, z⟩. Show that the circulationis a maximum when a points in the direction of the normal to S.arrow_forward
- 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 C has counterclockwise orientation and S has a consistent orientation. F = ⟨y, -x, 10⟩; S is the upper half of the sphere x2 + y2 + z2 = 1 and C is the circle x2 + y2 = 1 in the xy-plane.arrow_forwardInterpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forwardCalculus Answer, please don't use italics, as I do not understand it Calculate the work that a constant force field F does on a particle that moves uniformly once along the path of the curve x2 + y2 = 1.How much is the work now worth if we take F(x, y) = (αx, αy), where α is any positive constant?arrow_forward
- Using a Function (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f (x, y) = c at P, (c) find the tangent line to the level curve f (x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 + 4y2, c = 40, P(2, −1)arrow_forwardVerifying 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 C has counterclockwise orientation and S has a consistent orientation. F = ⟨y - z, z - x, x - y⟩; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane z = √7 and C is the boundary of S.arrow_forwardFlux across hemispheres and paraboloids Let S be the hemispherex2 + y2 + z2 = a2, for z ≥ 0, and let T be the paraboloid z = a - (x2 + y2)/a, for z ≥ 0, where a > 0. Assume the surfaces have outward normal vectors.a. Verify that S and T have the same base (x2 + y2 ≤ a2) and thesame high point (0, 0, a).b. Which surface has the greater area?c. Show that the flux of the radial field F = ⟨x, y, z⟩ across S is 2πa3.d. Show that the flux of the radial field F = ⟨x, y, z⟩ across T is 3πa3/2.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning