Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S,
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Calculus
- Using 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_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_forward
- a. Outward flux and area Show that the outward flux of theposition vector field F = xi + yj across any closed curve towhich Green’s Theorem applies is twice the area of the regionenclosed by the curve.b. Let n be the outward unit normal vector to a closed curve towhich Green’s Theorem applies. Show that it is not possiblefor F = x i + y j to be orthogonal to n at every point of C.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_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
- 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_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)-1/24pi b)-1/8pi c)1/24pi d)-1/48piarrow_forwardEvaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + (z-y) j + x k S is the surface of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1).arrow_forward
- Evaluate the surface integral F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + (z − y) j + x k S is the surface of the tetrahedron with vertices (0, 0, 0), (3, 0, 0), (0, 3, 0), and (0, 0, 3)arrow_forwardLine 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_forward
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