Evaluate F.N dS, where F(x, y, z) = 9x'i + 9xyj + 3x'y°k and S is the surface consisting of all faces of the tetrahedron bounded by plane x + y + z = 1 and the coordinate planes, with %3D outward unit normal vector N.

Q: Evaluate the surface integral. y dS, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

A: We have to evaluate the surface integral ∬SydS Given vector equation is r→u,v=ucosv,usinv,v…

Q: Calculate F . dS where F = (e sin y, eº cos y, yz²) and S is the surface of the box bounded by the…

A: Here, F=<exsiny,excosy,yz2>

Q: Q3// A surface S is defined by 2x2 + z = 1 and the bounded by the planes y = 0,y 2, x 0,z = 0 in the…

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Q: Use stokes theorem to evaluate f.dr where f = (4z+y, 3x, 7y) and C is the curve of intersection of…

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Q: 4. Let S be the open surface, x² + y² + z² = 4; z > 0. Evaluate the following surface integral,…

A: Given that S is the open surface x2+y2+z2=4, z≥0 To evaluate ∬SF·ndS Here, F=x,0,0 Let us…

Q: Calculate the surface integralff, ds, where S is the hemisphere x² + y² + z² = 1, z ≥ 0. The…

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Q: Find the circulation of F(x, y, z) = (-y sin 2, x sin z, xy cosz +12) around the circle cut from the…

A: Given that,F(x, y, z)=(-ysinz, xsinz, xycosz+12) around the circle cut from the sphere x2+y2+z2=5 by…

Q: a) Evaluate // (-xi- yj+2z k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y² between…

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Q: Let S be the disk enclosed by the curve C: r(t) = (cos q cos t, sint, sin p cos t), for 0sts2a,…

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Q: What is the value of F.ds, where F = 4xz,- y° i, + yz, ? Here, s is the surface bounded by x = 0, x…

A: Here function is given find f.ds as below

Q: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

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Q: Find parametric equations for the normal line to the surface z=e2x< + 9y< at the point (0,0, 1). O x…

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Q: Compute the surface area of the surface segment parametrised by x(u, v) = u² cos v, y(u, v) = u² sin…

A: Given, xu,v=u2cosv,yu,v=u2sinv,zu,v=u2where 1≤u≤3 and 0≤v≤2π To find the surface area.

Q: 6. Let F(x,y,z) = (x+y,–2y + z,8z – x) and let U be the sphere where x2 + y? + z2 with…

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Q: Given F(x, y, z) = x^2i + 2xyj + 3xzk and a region, being cut from the 1st octant by the sphere x^2…

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Q: Evaluate Curlv-ñ, where v = 2xyi + (x-2y)j+xzk and ñ is a unit vector normal to the surface shown in…

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Q: Compute ,F - dS for F = (4e*, 2z, 2x), S being a surface with parametrization D(r, s) = (rs,r +…

A: The integral of a vector valued function can be calculated as follows, using the parametrization:…

Q: Q3 Evaluate Curlv-ñ , where v=2xyi +(x -2y)j+xzk and ñ is a unit vector normal to the surface shown…

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Q: The vector tell f (xxz) = (z=+,2oxy"6) igven. Calkulate, using the stores theorem, the surface…

A: Given that  Fx,y,z=z2−1, z+xy3,6 To evaluate the integral ∫∫S∇×FndS using the stokes theorem.

Q: What is the value of J]F ds, ds, where F = 4xzi,- y² % + yzi, ? Here, s is the surface bounded by x…

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Q: Use Stokes' Theorem to evaluate SSs curlF · dS, where F(x, y, z) = 7xyi + 3yzj+4z²k and S is the…

A: Consider the provided question, Use Stokes' Theorem to evaluate ∫∫Scurl F·dS,where F(x, y, z)…

Q: Q3// A surface S is defined by 2x2 + z 1 and the bounded by the planes y = 0,y 2, x = 0,z = 0 in the…

A: A surface S is defined by 2x2+z=1 which is bounded by the planes y=0, y=2, x=0, z=0 in the first…

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Evaluate Curlv.ñ , where v = 2xyi +(x² – 2y)j + xzk and ñ is a unit vector normal to the surface…

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Q: Evaluate F.N dS, where F(x, y, z) = 9x²i + 9xyj + 3x'y'k and S is the surface consisting of all…

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Q: Let S be the surface parametrized by R (u, v) =< u, uv, uv

A: It is given that, the parametrized surface is R(u,v)=u, uv, uv2, 0≤u≤2, 0≤v≤3.  We have to calculate…

Q: Calculate F- dS where F = (e" sin y, e* cos y, yz²) S is the surface of the box bounded by the…

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Q: Evaluate the surface integral. y ds, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

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Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

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Q: Find a parametrization of the curve. The intersection of the surfaces z = x^2 − y^2 and z = x^2 + xy…

A: Given : The surfaces z = x2 - y2 and z = x2 + xy - 1.   To find the parameterization of the curve.

Q: Compute F - dS for F = (4e*, 2z, 2x), S being a surface with parametrization (r, s) = (rs, r + 4s,…

A: Given F=4ez,2z,2x, S being a surface with parametrization φ(r,s)=rs,r+4s,r, 0≤r≤1, 0≤s≤3, oriented…

Q: = xyz exists over a surface S c ed by the plane x-0,y%3D0,z%3D0, z=y V dS over the curve surface.

A: Given:  V= xyz x2+y2=9 To find: ∫s V ds  For solving the convergence x2+y2=9 (mod 1s) we use Chinese…

Q: Calculate F - dS where F = (e" sin y, e" cos y, yz²) S is the surface of the box bounded by the…

A: Given F¯={exsiny,excosy,yz2)

Q: Find a unit normal vector to the surface f(x, y, z) = 0 at the point P(4, – 5, – 5) for the fur 5x…

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Q: Find the surface area of the parametric surface F (и, v) — (2 сos(2и) + vcos(u), 2 sin(2u) +…

A: Here, we need to find the surface area of the parametric surface                               r→(u,…

Q: Use spherical coordinates to evaluate the triple integral ∫∫∫ √(x^2 + y^2 + z^2) dv where G is the…

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Q: Calculate F. dS where F = (e" sin y, e" cos y, yz²) S is the surface of the box bounded by the…

A: We have to find double integral of F. ds

Q: Find the flux integral ∫∫scurl(F).dS, where F(x,y,z) = [-y + 2cos(πy)e2x + z2, x2cos(zπ/2) -…

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Q: 4. Obtain the equation of the tangent plane to the surface with parametric representation r(u, v) =…

A: Note: We'll answer the first question since the exact one wasn't specified. Please submit a new…

Q: 6. Let F (x,y, z) = (x+ y,–2y + z, 8z – x) and let U be the sphere where x2 + y? + z² = 25 oriented…

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Q: Find the surface normal vector to the surface z 4-(x +y²) at the point (1, 1,v2) by (a) Vf (b) N =…

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Q: Evaluate || F:ñ dS where S is the surface bounded by z = a - Vx² + y² and the xy-plane including the…

A: The vector field is F→=xπ, yπ, 3z-2π. The surface is bounded by z=a-x2+y2 and xy-plane. The function…

Q: Let F(r, y) = 4xî + 3yî and let i be the outward unit normal vector to the positively oriented…

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Q: Use Stokes' theorem to calculate , F.ds, where F = (z, x, y) and C is the boundary %3D of a triangle…

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Q: Use the polar coordinates to evaluate volumes : 2) Below z = V1- x2 – y2 inside x² + y2 = above the…

A: Using the polar coordinates we have to evaluate the volumebelow z=1-x2-y2 inside x2+y2=14 above xy…

Q: 9. Let S be the surface parametrized by (u, v³, u + v) where 0 <u <1 and 0 < v<1. Calculate y dS.

A: The solution are next step

Q: Calculate the integral J, VX (y1).ri de if i is the outward normal unit vector of an ellipsoidal…

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Q: Q3 Evaluate Curlvoñ, where v = 2.xyi +(x² – 2 y)j + xzk and i is a unit vector normal to the surface…

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Q: Find a parametrization of the curve x^(2/3)+y^(2/3)=8^(2/3) and use it to compute the area of the…

A: Finding the parameters:

Q: Use Stokes' Theorem to calculate fe F-dĩ, where F = zyi+3=j+5zk, and C is the closed path around a…

A: Here given that F→=xy i^+3z j^+5x k^ and curve C in xz-plane. To find: ∫C F→·dr→ by Stokes' theorem.