Evaluating a Flux Integral In Exercises 31 and 32, find the flux of F over the closed surface. (Let N be the outward unit normal
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Chapter 15 Solutions
Calculus (MindTap Course List)
- Rain 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_forwarda. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of Sarrow_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_forward
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 6) a) 0 b) 252 c) 84 d) 42arrow_forwardUsing 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_forwardSurface integral of a vector field? Let T be the upper surface of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 4. Calculate the integral of the image below, where S is the face of T that is in the xy plane.arrow_forward
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = (x 2 + y 2) i + (x - y) j; C is the rectangle with vertices at (0, 0), (6, 0), (6, 9), and (0, 9) a)540 b)432 c)0 d)-432arrow_forwardUsing Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.arrow_forwardFind the flux of the vector field F across the surface S in the indicated direction.F = x 2y i - z k; S is portion of the cone z = 4 square root of x^2+y^2 between z = 0 and z = 1; direction is outward a)-1/24 pi b)-1/8 pi c)1/24 pi d)-1/48 piarrow_forward
- Computing 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 paraboloidz = 4 - x2 - y2, for z ≥ 0, plus its base in the xy-plane.arrow_forwardCirculation and flux Consider the following vector fields.a. Compute the circulation on the boundary of the region R (withcounterclockwise orientation).b. Compute the outward flux across the boundary of R. F = r/ | r | , where r = ⟨x, y⟩ and R is the half-annulus{(r, θ): 1 ≤ r ≤ 3, 0 ≤ θ ≤ π}arrow_forwardFind the flux of the vector field F(x, y, z) =< x^3, y^3, z^3 > across the surface σ,where σ is the surface of the solid G bounded below by z = 1 + x^2 + y^2, andabove by z = 2 and is oriented by outward normals. Sketch the surface σaccurately. Simplify your answerarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning