Concept explainers
Stokes’ Theorem for evaluating line
13. F = 〈x2 – z2, y, 2xz〉; C is the boundary of the plane z = 4 – x – y in the first octant.
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- Using Stokes’ Theorem to evaluate a line integral Evaluate the lineintegral ∮C F ⋅ dr, where F = z i - z j + (x2 - y2) k and C consists of the three line segments that bound the plane z = 8 - 4x - 2y in the first octant, oriented as shown.arrow_forwardUse Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise. 3 In(3 + y) dx - -dy, where C is the triangle with vertices (0,0), (6, 0), and (0, 12) ху 3+y ху dy = 3 In(3 + y) dx - 3+ yarrow_forwardUse Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. 2x + 3y dx + e -3y dy, where C is the triangle with vertices (0, 0), (1, 0), (1, 1).arrow_forward
- Verify Stokes's Theorem for F = z²î+ x²j + y²k and S is the surface z2 = x2 + y2, y 2 0, and 0arrow_forwardEvaluate the surface integral. (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 sus4, 0svs 2. Need Help? Watch Itarrow_forwardUse Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate ∫C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + 3xzj + exyk, C is the circle x2 + y2 = 4, z = 6.arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨4x, -8z, 4y⟩; S is the part of the paraboloidz = 1 - 2x2 - 3y2 that lies within the paraboloid z = 2x2 + y2 .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_forward3- Use Stokes theorem to compute the line integral c F · dr, where F = [e-3y, e², e-2ª] and C is the boundary of the surface S: z = 2x2,0 < x < 2,0 < y < 1.arrow_forward5. Use Stokes' Theorem (and only Stokes' Theorem) to evaluate F dr, where F(r, y, z) be clear, if you want to evaluate this and use Stokes' Theorem then you must be calculating the surface integral of the curl of F of a certain surface S.) (3y,-2x, 3y) and C is the curve given by a +y? = 9, z = 2. (So to %3Darrow_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_forwardUse Stoke’s Theorem to calculate the line integral ∮C(z−y)dx +(x−z)dy +(y−x)dz. The curve C is the triangle with the vertices A(2,0,0), B(0,2,0), D(0,0,2) (Figure 3).arrow_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