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Circulation and flux For the following
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Calculus: Early Transcendentals (3rd Edition)
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- Circulation and flux For the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = ⟨-y, x⟩; R is the annulus {(r, θ); 1 ≤ r ≤ 3, 0 ≤ θ ≤ π}.arrow_forwardHeat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. T(x, y, z) = 100e-x2 - y2 - z2; S is the sphere x2 + y2 + z2 = a2.arrow_forwardCirculation and flux For the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = ⟨x, y⟩; R is the half-annulus {(r, θ); 1 ≤ r ≤ 2, 0 ≤ θ ≤ π}.arrow_forward
- (1 point) Evaluate the line integral foF. dr, where F(x, y, z) = -5xi – yi + 3zk and C is given by the vector function r(t) = (sin t, cos t, t), 0arrow_forwardCirculation and flux For the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = ⟨2x + y, x - 4y⟩; R is the quarter-annulus {(r, θ); 1 ≤ r ≤ 4, 0 ≤ θ ≤ π/2}.arrow_forwardHeat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. T(x, y, z) = -ln (x2 + y2 + z2); S is the sphere x2 + y2 + z2 = a2.arrow_forwardHeat flux Suppose a solid object in ℝ3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is∇ ⋅ F = -k∇⋅ ∇T = -k∇2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.arrow_forward38. Motion along a circle Show that the vector-valued function r(t) = (2i + 2j + k) %3D + cos t V2 j) + sin t V2 j + V3 V3 V3 describes the motion of a particle moving in the circle of radius 1 centered at the point (2, 2, 1) and lying in the plane x + y – 2z = 2.arrow_forwardHeat flux Suppose a solid object in ℝ3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is∇ ⋅ F = -k∇⋅ ∇T = -k∇2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. T(x, y, z) = 100e-x2 + y2 + z2arrow_forwardarrow_back_iosarrow_forward_ios
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