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 - 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.

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 - 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.

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Additional Topics In Trigonometry

Section: Chapter Questions

Problem 10PS

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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 = ⟨x, y, z⟩; S is the paraboloid z = 8 - x2 - y2, for0 ≤ z ≤ 8, and C is the circle x2 + y2 = 8 in the xy-plane.
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.
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 = ⟨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.
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 = ⟨0, -x, y⟩; S is the upper half of the sphere x2 + y2 + z2 = 4 and C is the circle x2 + y2 = 4 in the xy-plane.
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 - 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.
Flux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.
Verifying Stokes’ Theorem Confirm that Stokes’ Theorem holds forthe vector field F = ⟨z - y, x, -x⟩, where S is the hemisphere x2 + y2 + z2 = 4, for z ≥ 0, and C is the circle x2 + y2 = 4 oriented counterclockwise.
Radial fields and zero circulation Consider the radial vectorfields F = r/ | r | p, where p is a real number and r = ⟨x, y, z⟩ .Let C be any circle in the xy-plane centered at the origin.a. Evaluate a line integral to show that the field has zero circulation on C.b. For what values of p does Stokes’ Theorem apply? For those values of p, use the surface integral in Stokes’ Theorem to show that the field has zero circulation on C.
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.
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