CONCEPT CHECK
Tangent VectorConsider a point
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Multivariable Calculus
- Flux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.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_forwardHow do you define and calculate the work done by a variable force directed along a portion of the x-axis? How do you calculate the work it takes to pump a liquid from a tank? Give examples.arrow_forward
- Flux 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_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_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_forward
- 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. 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_forwardVerifying Stokes’ Theorem Confirm that Stokes’ Theorem holds forthe vector field F = ⟨z - y, x, -x⟩, where S is the hemisphere x2 + y2 + z2 = 4, for z ≥ 0, and C is the circle x2 + y2 = 4 oriented counterclockwise.arrow_forwardBendixson’s criterion The streamlines of a planar fluid floware the smooth curves traced by the fluid’s individual particles.The vectors F = M(x, y)i + N(x, y)j of the flow’s velocity fieldare the tangent vectors of the streamlines. Show that if the flowtakes place over a simply connected region R (no holes or missingpoints) and that if Mx + Ny 0 throughout R, then none of thestreamlines in R is closed. In other words, no particle of fluid everhas a closed trajectory in R. The criterion Mx + Ny ≠ 0 is calledBendixson’s criterion for the nonexistence of closed trajectories.arrow_forward
- 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_forwardUse Stokes' Theorem to find the work done on a particle moves along the line segments from the origin to the points (2,0,0) (2,4,3) , (0,4,3). and back to the origin. Note that this (counterclockwise) path is a rectangle on the plane z = 3/4 y. The motion is under the influence of the force field F = z2 i+ 2xy j + 4y2 karrow_forwardHeat flux in a plate A square plate R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T(x, y) = 100 - 50x - 25y.a. Sketch two level curves of the temperature in the plate.b. Find the gradient of the temperature ∇T(x, y).c. Assume the flow of heat is given by the vector field F = -∇T(x, y). Compute F.d. Find the outward heat flux across the boundary {(x, y): x = 1, 0 ≤ y ≤ 1}.e. Find the outward heat flux across the boundary {(x, y): 0 ≤ x ≤ 1, y = 1}.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