Verifying Stokes's Theorem In Exercises 3-6, verify Stokes’s Theorem by evaluating
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- Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y, -x, 10⟩; S is the upper half of the sphere x2 + y2 + z2 = 1 and C is the circle x2 + y2 = 1 in the xy-plane.arrow_forwardwww Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = zi + 5xj+yk across the surface S. r(r,0) =r cos 0i +r sin 0j+ (4-r) k, 0srs2,0 s0s 2x in the direction away from the origin.arrow_forwardEvaluate F · dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. (2z + 4y) dx + (4x – 3z) dy + (2x – 3y) dz (a) C: line segment from (0, 0, 0) to (1, 1, 1) (b) C: line segment from (0, 0, 0) to (0, 0, 1) to (1, 1, 1) (c) C: line segment from (0, 0, 0) to (1, 0, 0) to (1, 1, 0) to (1, 1, 1)arrow_forward
- Work by a constant force Evaluate a line integral to show thatthe work done in moving an object from point A to point B in thepresence of a constant force F = ⟨a, b, c⟩ is F ⋅ AB.arrow_forwardCalculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the surface of part of the cone √x² + y2 that lies between the two planes z = 1 and z = 8 with an upward-pointing unit normal, vector using a line integral. F = (yz, -xz, z³) (Use symbolic notation and fractions where needed.) curl(F) = flux of curl(F) = [arrow_forwardF(x, y, z) = xi + yj + xyk, r(t) = cos t i + sin tj + tk, 0<1arrow_forwardSurface integral of a vector field? Let T be the upper surface of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 4. Calculate the integral of the image below, where S is the face of T that is in the xy plane.arrow_forwardUse Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate S curl F · dS. F(x, y, z) = zeyi + x cos(y)j + xz sin(y)k, S is the hemisphere x2 + y2 + z2 = 9, y ≥ 0, oriented in the direction of the positive y-axis. F(x, y, z) = zeyi + x cos(y)j + xz sin(y)k, S is the hemisphere x2 + y2 + z2 = 9, y ≥ 0, oriented in the direction of the positive y-axis.arrow_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨2z, -4x, 3y⟩; S is the cap of the sphere x2 + y2 + z2 = 169 above the plane z = 12 and C is the boundary of S.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 = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y - z, z - x, x - y⟩; S is the part of the plane z = 6 - ythat lies in the cylinder x2 + y2 = 16 and C is the boundary of S.arrow_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y - z, z - x, x - y⟩; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane z = √7 and C is the boundary of S.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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