Differentiation of Vector-Valued FunctionsIn Exercises 3–10, find
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Multivariable Calculus
- Gravitational 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_forwardUsing Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t) (b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)] (d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)] (f) d dt [u(2t)] 26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t k * only d ,e, f *arrow_forwardFlux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?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 = 65, P(3, 2)arrow_forwardCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ 2π. F = ⟨y - x, y⟩arrow_forwardUsing Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t) (b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)] (d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)] (f) d dt [u(2t)] 26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t karrow_forward
- Subject differential geometry Let X(u,v)=(vcosu,vsinu,u) be the coordinate patch of a surface of M. A) find a normal and tangent vector field of M on patch X B) q=(1,0,1) is the point on this patch?why? C) find the tangent plane of the TpM at the point p=(0,0,0) of Marrow_forwardEvaluating line integrals Evaluate the line integral ∫C F ⋅ drfor 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 = ∇(x2y); C: r(t) = ⟨9 - t2, t⟩ , for 0 ≤ t ≤ 3arrow_forwardTrajectories on circles and spheres Determine whether the following trajectories lie on either a circle in ℝ2 or a sphere in ℝ3 centered at the origin. If so, find the radius of the circle or sphere, and show that the position vector and the velocity vector are everywhere orthogonal. r(t) = ⟨sin t + √3 cos t, √3 sin t - cos t⟩ , for 0 ≤ t ≤ πarrow_forward
- Flux across the boundary of an annulus Find the outward flux of the vector field F = ⟨xy2, x2y⟩ across the boundary of the annulusR = {(x, y): 1 ≤ x2 + y2 ≤ 4}, which, when expressed in polar coordinates, is the set {(r, θ): 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}arrow_forwardSurface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨e-y, 2z, xy⟩ across the curved sides of the surfaceS = {(x, y, z): z = cos y, | y | ≤ π, 0 ≤ x ≤ 4}; normal vectorspoint upward.arrow_forwardFlux 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_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning