Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S,
where N is the upward unit normal
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Chapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Flux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_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
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- Find 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_forwardUsing 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_forwardFind the flux of the vector field F across the surface S in the indicated direction.F = 4x i + 4y j + z k; S is portion of the plane x + y + z = 4 for which 0 ≤ x ≤ 4 and 0< or equal y< or equal to3 direction is outward (away from origin) a) 174 b) - 78 c) 348 d) 150arrow_forward
- Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨y, x⟩ on the line segment from (1, 1) to (5, 10)arrow_forwardFind the flux of the vector field F across the surface S in the indicated direction.F = x 4y i - z k; S is portion of the cone z = 3 square root of x2+y2 between z = 0 and z = 3; direction is outward a) - 1 b) - 6π c) 2π d) - 2πarrow_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_forward
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