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Chapter 14 Solutions
Calculus: Early Transcendentals, 2nd Edition
- (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 = ⟨-y, x⟩; R is the annulus {(r, θ); 1 ≤ r ≤ 3, 0 ≤ θ ≤ π}.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 = ⟨2x + y, x - 4y⟩; R is the quarter-annulus {(r, θ); 1 ≤ r ≤ 4, 0 ≤ θ ≤ π/2}.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_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_forwardFind div F and curl F if F(x,y, z) = x°i – 2j + yzk. div F= curl F=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_forwardFind a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector VF(x, y, z).] Surface Point z = x3 (3, -6, 27)arrow_forwardEvaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =arrow_forwardStokes' Theorem (1.50) Given F = x²yi – yj. Find (a) V x F (b) Ss F- da over a rectangle bounded by the lines x = 0, x = b, y = 0, and y = c. (c) fc ▼ x F. dr around the rectangle of part (b).arrow_forwardEvaluate the circulation of G = xyi+zj+7yk around a square of side 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Prevs So F.dr-arrow_forwardc) Verify Stokes's Theorem for F = (x²+y²)i-2xyj takes around the rectangle bounded by the lines x=2, x=-2, y=0 and y=4arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning
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