S₁- X 5. Let S be the capped cylindrical surface shown in Figure 8.2.14. S is the union of two surfaces, S₁ and S₂, where S₁ is the set of (x, y, z) with x² + y² = 1,0 ≤z ≤ 1, and S₂ is the set of (x, y, z) with x² + y² + (z - 1)² = 1, z≥ 1. Set F(x, y, z) = (2x + z²y + x)i + (z³yx + y)j + z¹x²k. Compute (V x F). dS. (HINT: Stokes' theorem holds for this surface.) noolled in oil A Figure 8.2.14 The capped cylinder is the union of S₁ and S₂.

Q: Let l,,l2,lz be the lines X(t) = (1,1, 2) + t(1,2,3), X(s)= (9,8,11) + s(2,1,1) and X (r) = (0,0,0)…

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Q: c) The surface S is the portion of the plane x+y+z=1 in the first quadrant of the Oxyz Cartesian…

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Q: 3. The point (1, k, 4) lies on the plane [x, y, z] = [1, 2, 0] + s[3, 0, 1] + t[2, −1, 1]. Find the…

A: Given point (1,k,4) is on plane.plane  is

Q: 31. Use sine and cosine to parametrize the intersection of the surfaces x² + y² = 1 and z = 4x²…

A: Given x2+y2=1 & z=4x2 x2+y2=1 is the equation of circle. That is x2+y2=12,  Therefore, Radius is…

Q: 3) A vector is given as A = x(z+2)â,-()x+1)â, +(y+2)â. Prove the Stoke's theorem for the geometry…

A: From the below figure, ∮cA.dt=∫ABA.ds+∫BCA.ds+∫CDA.ds+∫DAA.ds Here ax=i, ay=j, ak=k Since…

Q: 3. (.J 6) For two vectors u = [1,0,0]" and v = [1,1, 1]", find a third vector with unit length such…

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Q: 5 me cap of the sphere x² + y² + z² =25, for 2 ≤z≤5

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Q: Let S be the portion of the cone with equation z² = x² + y² which lies between the planes z = 1 and…

A: Given that S is the portion of the cone with the equation z2=x2+y2 which lies between the planes z=1…

Q: 9. Let W be the pyramid with top vertex (0, 0, 1), and base vertices at (0, 0, 0), (1,0,0), (0,…

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Q: 9. Let E be the tetrahedron with vertices: (0,0,0), (3,2,0), (0,3,0), (0, 0, 2). E is bounded by…

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Q: (C) (D) 6 32. Let AABC - ADEF and their areas be, respectively 64 cm? and 121, cm2. If EE = 15.4 cm,…

A: Here a given set of questions. The solution to each of the given problem is as follows.

Q: 12. Find the vector with length 5 and direction angles a = B = 120°, y = 135°. = 60°,

A: 12. Given that a vector with length 5 and direction angles α=60o, β=120o, γ=1350.

Q: 2 The values of a and b such that the surfaces ax - byz = (a + 2)x and 4x2y + z3 = 4 cut…

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Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), (3, 2,0), (0,3,0), (0, 0,2). E is bounded by…

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Q: (a) Find two points on the line x = -1+3t, y=t + 2, z= 2t – 7. (b) ( Find The point at which the…

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Q: The remainder upon the division of x98 +4 by x - 2 in z, is: 6. None

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Q: Prove that the z-axis is a nodal line with unodes a -2) and (0, 0,2/3) for the surface 3 2xy +…

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Q: Find the points on the sphere with center at the point P-(3,3,3) of radius 3 where the function f(x,…

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Q: 2. Which of the following describes a different plane than the other three? (a) x + 2y + 3z = 6 (b)…

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Q: . Evaluate || (2- 1) do, where S is the part of the cone z = Vr + y² which lies under the plane z =…

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Q: 4. a) Find the point on the unit sphere x² + y² + z² = 1 that is closest to (1,2,3). b) Find the…

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Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), E is bounded by four planes: z = 0, x=0, 2 3…

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Q: projection of a point P on a plane is the point P' such that PP' is a line segment that is endicular…

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Q: 3. Let H = 2xi – 3xy j+xz² k. Calculate ſ, Ĥ •d where S is the surface of the cube with corners at…

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Q: Find the (x, y, z) coordinates of the points P where the line I(t) = (x (t),y(t),z(t)) = (2 + 3t, 1…

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Q: Let F = (z, 0, y) and let S be the oriented surface parametrized by Þ(u, v) = (u² – v, u, v²) for 0…

A: From the given information: