Use the Divergence Theorem to find the flux of across the surface of the solid bounded above by z = and below by the xy-plane with outward orientation. Hint: Use spherical coordinates. NOTE: Enter the exact answer. Φ = 384 T. 10 X F(x, y, z) = (x³ — e³)i + (y³ + sin z)j + (z³ − xy)k √4x² - y²

Q: What is the surface area of the three dimensional solid formed by rotating the following parametric…

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Q: O) Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = xyi + j + xyk and C is the curve of…

A: (c) The curve, C can be easily parametrized using the following parametrization: x=cos t, y=sin t…

Q: Find the spherical coordinate expression for the function F(x, y, z). F(x, y, z) = x^²√√x² + y² + z²…

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Q: Verify Stokes' theorem for F = (x+y-4) i+3xy j+ )k, over the surface of the hemisphere x2 + y2 + z2…

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Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above. Jc…

A: Use Stokes' Theorem to evaluate CF · dr where C is oriented counterclockwise as viewed from above.…

Q: :f (x, y, 2) = x + y; E = {(x, y, z) |1 0, y > 0}

A: In this question given that a function and it is in the cartetion coordinate. We can convert in…

Q: Use a parametrization to find the flux F.n do across the surface in the given direction. F= - xi -…

A: The electric flux through a surface depends only on amount of charge enclosed by the surface. It…

Q: 2. Change Vx? + y? dz dy dx into cylindricalcoordinates, then evaluate. J-1 Jo

A: Given I=∫01∫-1-x21-x2∫x2+y22-x2-y2x2+y2 dz dy dx

Q: i. Find b × c in terms of λ . ii. Show that a · (b × c) is independent of λ .  iii. Find the…

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Q: 10. Use Stokes' Theorem to evaluate ||(V×F)· dS where F= (z,2x, y) and Sis the hemisphere of x' + y?…

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Q: (2² + y²)dV, where D is the solid that lies Use spherical coordinates: Evaluate above the xy-plane…

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Q: Describe the bifurcation for the following dy - Ay - y %3D dt

A: The saddle node-bifurcation: Let y˙=f(α,t) is the system of equation, the to find the saddle-node…

Q: Match the parametrizations to the surfaces shown in the figures. Þ(u, v) = v 4² u cos(v) (u, U) = |…

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Q: eff curl(F). ds. F(x, y, z) = x² sin(z)i + y²j + xyk, S is the part of the paraboloid z = 9 - x² -…

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Q: Let C be the circle centered at the origin with radius 4 oriented clockwise, and let F(x, y) = (2x +…

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Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: Use Stokes' theorem frotF. ds to evaluate F(x, y, z) = (x + tan¯¹yz)i + y²zj + zk, S is the part of…

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Q: Evaluate II| 6x(x2 + y2) dv, where E is the solid in the first octant that lies beneath the…

A: Since the integral lies in first octant so 0≤θ≤π/2 and z≥0.  Converting into cylindrical coordinates…