1: Evaluate the (outer) flux of F = (y, x, 1) over the closed surface S which consists of the xy, xz, yz planes on the first octant and the portion of the sphere x² + y? + z2 = 4 on the first octant. In other words, S is the surface in the first octant bounded by the sphere and three coordinate planes by %3D

Q: 3.4.8 Evaluate the surface integral ff, U(x, y, z) dS, where S is the surface of the paraboloid z =…

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Q: 1. A solid G is bounded by the paraboloid z = 4 – x² – y², the cylinder x² + y? = 4 and the plane z…

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Q: 18. Evaluate J (V x F) · N dS where F = yi + (x- 2xz) j- xy k and S is the surface of the sphere x…

A: Stokes theorem is used

Q: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

A: Parametrize the three curves separately and c is the sum of the three integrals.

Q: 1. Use Green's Theorem to find the counterclockwise circulation and outward flux for the given filed…

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Q: 1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate E.dr for F=(xz,-y,x²y), where C is the boundary of…

A: Given: F=&lt;xz,-y,x2y&gt; C surface, S is bounded by x=0, y=0 and 2 not include xy-plane Now,…

Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

A: We need to find the length of the curve BD. The curve BD is found by the intersection of the…

Q: a)Calculate the integral [D zdV on the D region bounded by the 6 = x2 + y2 paraboloid from the…

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Q: Suppose that a point (X,Y) is chosen at random from the region S on the xy-plane containing all…

A: Consider  two random variables x and y, suppose x is a sex of a person as well as  whether…

Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: 2). Find the flux of F(x, y, z) = (0, yz, z) outward through the surface S cut from the cylinder y +…

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Q: a) Let G be the wedge in the first octant that is cut from the cylindrical solid y² + z² ≤ 4 by the…

A: To Evaluate: ∫∫∫GxyzdV. Where G: y2+z2≤4, y=2x , x=0. As per policy, we are solving only thr first…

Q: Calculate S," (2x+y) dV where V is a closed space bounded by the cylinder z = 4 – x2 and the planes…

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Q: 2. (a) Let S be the part of hemisphere x² + y² + z² = 4 in the first octant. Compute faz²ds. (b)…

A: Given: x2+y2+z2=4 To find: Solution

Q: If S is the solid bounded by the paraboloid z = 3x2 + 3y and the plane z = 4 (with constant…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate F.dr for E = (xz,-y,x²y), where C is the boundary…

A: Applying stokes theorem we solve it

Q: Let S be the surface of z = 6 –- x² – 4y² with z > 2 Find the flux of F = [1 – 4y, 1 + x, 2z]on S

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Q: 2. A solid G is bounded by the paraboloid z = 4-²-y², the cylinder x² + y² = 4 and the plane z = 5.…

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Q: 1. The part of the sphere x? + y² + z² = a² that lies within the cylinder x² + y² : = ax and above…

A: To Determine :-  The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax…

Q: If E is the solid shape bounded by x 20,y 20,z 20 and the plane 6x + 4y + 3z = 12 evaluate %3D (y +…

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Q: 2. Set up the triple integral f[Sp I+y+zdV, where D is the tetrahedron with vertices at (0,0,0),…

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Q: Let S be part of the outward-oriented hyperboloid of one sheet x² + y² = z² = 1 that lies within the…

A: Given x2+y2-z2=1 and x2+y2+z2=3

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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Q: 1: Evaluate the (outer) flux of F = (y, x, 1) over the closed surface S which consists of the ry,…

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Q: Evaluate SSS z²dV, where R is the region given R by above the plane z=O and between the spheres 2. +…

A: A scalar function is a rule that is used to associate a number, or quantity with a particular point…

Q: 3.4.8 Show that the sphere x² + y² + z² = 1, z ≥ 0. JJ,z²dS=, where S is the surface of

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Q: 13. F(x, y, z) =x°zi+ xz°j + y In(x + 1) k, S is the surface of the solid bounded by the planes x +…

A: The solution are next step

Q: Compute the flux of F(r, y, z) = [1,–2, z] through the surface S, which is part of the graph z =…

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Q: 3. Evaluate AP G if G is a solid in the first octant bounded by the plane y + z = 2 and the surface…

A: The given problem is to evalute the triple integral over the given surface of first octant bounded…

Q: Use Green's Theorem to calculate S,x²y³dx + (xy – y²)dy where C is the boundary of the region lying…

A: Greens Theorem:

Q: Evaluate x dV , where E is the E region in the first octant bounded by the sphere x² + y² + z² = 1…

A: given region is in the first octant. using spherical coordinates x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ and…

Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

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Q: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: 6. Compute the flux of F = [0, 0, z²] through the spherical surface S given as the upper hemisphere…

A: To find- Compute the flux of F→ = 0, 0, z2 through the spherical surface S given as the upper…

Q: 1) Evaluate the triple integral SS, f(x,y,z)dv : B) f(x,y,z) = 3y² – 2z , Q is the tetrahedron…

A: given f(x,y,z) = 3y2-2z, Q=3x+2y-z=6 The region bounded by the equation Q is a tetrahedron cutting…

Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

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Q: c, y, z) = x²zi + xz'j+ y ln(x + 1) k, s the surface of the solid bounded by the 2z = 4, y = 3, x =…

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Q: 1. Evaluate the surface integral [[' xz² ds ; o is the part of the cone z = x² + y² that lies…

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Q: (a) Given that F(x,y,z) = (3x² + 3y²)i – 2xyj + yzk, evaluate S, F·dr for which C is the boundary of…

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Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…

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Q: Let S be the piece of the surface z = 9-r that lies inside the cylinder r2 +y² = 9, and let F = (x,…

A: Given, S be the piece of the surface z=9-x2 that lies inside the cylinder x2+y2=9, and let F=x,y2,z.…

Q: Evaluate || 4.nds , where A=zi+xj – 3y°zk and S is the surface of the cylinder x² + y =4 included in…

A: Let ∅=x2+y2-4  Then the normal to the surface S is ∇∅=2xi+2yj The unit normal is…

Q: a) Evaluate the surface integral ſ 2x²y dS over the surface y² + z² = 1 betwee x = -1 and x = 5.

A: a We have to evaluate the surface integral ∬σ2x2ydSover the surface y2+z2=1 between x=-1 and x=5.…

Q: calculate ∫ ∫ ∫Ty dv, being the T region inside the paraboloid z= 1/(x2+y2)the sphere x2+y2+z2=5

A: We need to calculate the triple integral ∫ ∫ ∫Ty dv, where T being the region inside the paraboloid…

Q: . Find the solution of the following Cauchy problem ", = F(x, y) given u = f(x). du - = g(x) on the…

A: It is given that…

Q: Calculate the flux of F across G. F(x, y, z) = 2xi + 2yj + 2zk; G is the surface of the sphere x2 +…

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Q: 1. Evaluate the surface integral SL xz² ds ; o is the part of the cone z = Vx2 + y² that lies…

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Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

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