Divergence of a Cross Product In Exercises 73 and 74, find
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Chapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Using Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. IF = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 7) a) 112 b) 392 c) 0 d) 56arrow_forwardRain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.arrow_forward
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. F = xy i + x j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 4) a) 0 b) -80/3 c) 112/3 d) 64/3arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 6) a) 216 b) 72 c) 0 d) 36arrow_forwardGradient fields in ℝ3 Find the vector field F = ∇φ for thefollowing potential functions. φ(x, y, z) = 1/ | r | , where r = ⟨x, y, z⟩arrow_forward
- Find the linearizations of the functions in Exercises 57 and 58 at the given points. 57. ƒ(x, y, z) = xy + 2yz - 3xz at (1, 0, 0) and (1, 1, 0) 58. ƒ(x, y, z) = 22 cos x sin ( y + z) at (0, 0, pai/4) and (pai/4, pai/4, 0)arrow_forwardFlux 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_forwardComputing flux Use the Divergence Theorem to compute thenet outward flux of the following fields across the given surface S. F = ⟨x, y, z⟩; S is the surface of the cone z2 = x2 + y2, for0 ≤ z ≤ 4, plus its top surface in the plane z = 4.arrow_forward
- Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.F = (y-x) i + (z-y) j + (z-x) k ; D: the region cut from the solid cylinder x 2 + y 2 ≤ 49 by the planes z = 0 and z=2 a) 0 b) 98π c) -98π d) -98arrow_forwardSplitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨x, y⟩ on ℝ2arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning