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Verifying Stokes’ Theorem Verify that the line
9. F = 〈y – z, z – x, x – y〉; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane
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Chapter 14 Solutions
Calculus: Early Transcendentals (2nd Edition)
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- M2arrow_forwardsketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. Surface: x2+y2=4; z=x2 Parameter: x=2sin(t)arrow_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_forward
- 2. A cartesian equation for the surface is? 3. Draw the graph and the tangent planearrow_forwardStreamlines and equipotential lines Assume that on ℝ2, the vectorfield F = ⟨ƒ, g⟩ has a potential function φ such that ƒ = φxand g = φy, and it has a stream function ψ such that ƒ = ψy andg = -ψx. Show that the equipotential curves (level curves of φ)and the streamlines (level curves of ψ) are everywhere orthogonal.arrow_forwardBendixson’s criterion The streamlines of a planar fluid floware the smooth curves traced by the fluid’s individual particles.The vectors F = M(x, y)i + N(x, y)j of the flow’s velocity fieldare the tangent vectors of the streamlines. Show that if the flowtakes place over a simply connected region R (no holes or missingpoints) and that if Mx + Ny 0 throughout R, then none of thestreamlines in R is closed. In other words, no particle of fluid everhas a closed trajectory in R. The criterion Mx + Ny ≠ 0 is calledBendixson’s criterion for the nonexistence of closed trajectories.arrow_forward
- V = 6y – 16x + 9z + V¢ Find the curl of vector V in Cartesian coordinates, o is Continuously differentiable. Find: V x Varrow_forwardplease indicate whether parallel perpendicular or neitherarrow_forwardTop The base of the closed cubelike surface shown here is the unit square in the xy-plane. The four sides lie in the planes x=0, x= 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F=xi - 2yj + (z + 3)k and suppose the outward flux of F through Side A is 1 and through Side B is - 3. Can you conclude anything about the outward flux through the top? Give reasons for your answer. Side A Choose correct answer below, and, if necessary, fill in the answer box to complete your choice. A. Yes, the outward flux through the top can be calculated exactly, and it is (Type an integer or a fraction.) B. Yes, the outward flux through the top is positive, but it cannot be calculated exactly. C. Yes, the outward flux through the top is negative, but it cannot be calculated exactly. D. No, the outward flux through the top cannot be determined without its identity. Give reasons for your answer. O A. By the Divergence Theorem, the outward flux is 0. Side A and…arrow_forward
- 2. A cartesian equation for the surface is? 3. Draw the graph and the tangent planearrow_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, -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_forward2. A cartesian equation for the surface is? 3. Draw the grapharrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
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