Verifying the Divergence Theorem In Exercises 3–8, verify the Divergence Theorem by evaluating
as a surface
S: surface bounded by
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Multivariable Calculus
- Channel flow The flow in a long shallow channel is modeled by the velocity field F = ⟨0, 1 - x2⟩, where R = {(x, y): | x | ≤ 1 and | y | < 5}.a. Sketch R and several streamlines of F.b. Evaluate the curl of F on the lines x = 0, x = 1/4, x = 1/2, and x = 1.c. Compute the circulation on the boundary of the region R.d. How do you explain the fact that the curl of F is nonzero atpoints of R, but the circulation is zero?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 line integral using Green's Theorem and check the answer by evaluating it directly. ∮C6 y2dx+3 x2dy∮C6 y2dx+3 x2dy, where CC is the square with vertices (0,0)(0,0), (3,0)(3,0), (3,3)(3,3), and (0,3)(0,3) oriented counterclockwise.arrow_forward
- 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_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 = ⟨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.arrow_forwardContinuity 4. Let f(r, y) be a function defined on a disk D not containing the origin which is given by f(r, y) = x² + y? Show that 0 < f(x, y) < 2 for (r, y) € D.arrow_forward
- Application of Green's theorem Assume that u and v are continuously differentiable functions. Using Green's theorem, prove that SS'S D Ux Vx |u₁|dA= udv, C Wy Vy where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forward5. Show that the function h(r, y) defined by if a + y < 1 h(r, y) = 0; if a + y 2 1 is discontinuous.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 = ⟨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.arrow_forward
- Stokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.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_forwardUse Stokes’ Theorem to evaluate ∫ F*dr where C is oriented counter-clockwise as viewed from above. F(x,y,z) = yi-zj+x2k C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1) Note: The triangle is a portion of the plane x+y+z=1arrow_forward
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