Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0

Q: Consider the solid Q shown in the figure 2.x + 3y +3z = 6 +x = 2 Let S be the boundary of the solid…

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Q: Use the divergence (Gauss) theorem to find the value of (image 1) where F(x, y, z) = (4x3, ex2 − y,…

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Q: Let S be the surface with parametrization F(u, v) = (V1+ u² cos v, /1+ u² sin v, u), 0< u< 1,0 < v <…

A: Given the surface with the parametrization r→u,v=1+u2cosv, 1+u2sinv,u, 0≤u≤1, 0≤v≤2π. (a)…

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

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Q: Consider the positively oriented surface S₂ defined by the vector function R(u, v) = (u sin v-3, u…

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Q: Evaluate the flux integral ∫∫scurl(F).dS, where F(x,y,z) = [xey^2z3 + 2xyze(x^2)+z, x + z2e(x^2)+z,…

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Q: Consider the positively oriented surface S₂ defined by the vector function R(u, v) = (u sin v-3, u…

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Q: Consider the surface S given by: 4x2 + 4z2 = (1+ y)?, Osys2.Compute: (a) a normal vector at each…

A: Given:-                 Consider the surface S given by: 4x 2 + 4z 2 = (1 + y) 2,0≤y≤2  Compute: (a)…

Q: Let S be the surface defined by the vector function R(u, v) = (u sin v, u cos v, u – v), where u e R…

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Q: Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P…

A: For part 1 we are given two spheres of different sizes i.e different radius which are tangent to…

Q: 9. Use Stokes' Theorem to evaluate F. dr where F(z, y, 2) = (2 – y) i + (x + 2)j – (x + y) k and C…

A: The given problem is to evalute the contour integral using stokes theorem, where surface C is the…

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: Let r(u, v)= [2ucos(v), 2usin(v), 2] be a parametrization of a surface S, where 0su33,0svS2 T. Then…

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Q: 11. If F= (y? +z? - x²) î + (z² + x² – y²)j+(x² + y² – z²) k, evaluate fcurl F-î ds integrated over…

A: Given :-  If  F→ = y2 + z2 - x2 i +  z2 + x2 - y2 j +  x2 + y2 - z2 k To Determine :-  ∫ curl F→. n^…

Q: Let F = 2x3i + 2y3j + 2z3k and let S be the surface consisting of the hemisphere x2 + y2 + z2 = 1, z…

A: Given, F=2x3i+2y3j+2z3k and  S be the surface consisting of the hemisphere x2+y2+z2=1, z≥0 and the…

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: 6. Use the Normal Vector to a Smooth Parametric Surface to find the equation of the tangent plane to…

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Q: Find the flux of the following vector field across the given surface with the specified orientation.…

A: Let r be the vector on the surface of  the cylinder y=6x2 for 0 ≤ x ≤ 4​, 0 ≤ z ≤ 2​.Let's…

Q: Find a vector normal to the surface x2 + y2 - z² = 18 at P = (3, 5, 4). %3D

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Q: Consider the ellipsoid 4x2 + 36y2 + 225 z2 = 225 (a) Parametrize the surface S = r(u, v) by x = a…

A: We will answer the first two questions as we don't answer multiple questions at a time. Please…

Q: Verity Gauss' theorem for the vector feld f (x,y, z)= (y,x,2) on the surface of the sechere with…

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Q: Verify Stokes' theorem for the vector ficld v=x'i+x' yj. C is the boundary of the rectangle whose…

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Q: Let S be the flat, shaded surface oriented as shown by the normal vector i in the diagram above.…

A: The flux of the vector field is ∫CF·ndS=∫∫RdivFdA by Green's theorem. As ∫CF·ndS=0, then ∫∫RdivFdA=0…

Q: In the space referred to a direct orthonormal system ( 0;i, j, k), given the points: A (1,1, 2), B…

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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: 3. firmdian .a. A. vector.onthogonal.tr.xv.and.xw. .b.. A. vecter.ortheanal.br. and. Co.…

A: Use the following formulae, to obtain the solution. The vectors u→ and v→ are said to be orthogonal…

Q: Let T be the triangular region with vertices (1, 0,0), (0, 1,0), and (0, 0, 1) oriented with…

A: given Velocity field V=4i+4j

Q: 1. Consider the ellipsoid (a) Parametrize the surface S = r(u,v) by z = a sin(u) sin(v), y = bcos(u)…

A: Using the parameters on the equation of ellipsoid:…

Q: Let S be the closed surface consisting of the part of the cone x2 + y – z2 = 0 that lies between the…

A: The vector field is Fx,y,z=-x i-y j+z3 k and the surface is the part of the cone z2=x2+y2 between…

Q: (b) Let D be the disk of radius a in R?, and let C denote it's boundary. Let N be the unit normal…

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Q: for the given vector field F and the oriented surface S. In other words, find the F(x, y, z) = zexi…

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Q: 6) Let S be a surface parameterized by: r(u, v) = (u³ + v*)î + (u³ – v³)ĵ+ (2uv)k con u € [0,1], v E…

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Q: Which of the following is a parametrization of the line through P = (3, 10, 8) perpendicular to the…

A: Use Parametrization to solve

Q: Evaluate an area of a surface where the vector that is normal to the surface is defined as; d. Q =…

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Q: Find a parametric representation for the surface: The plane through the origin that contains the…

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Q: Use Stokes's theorem to evaluate F.dr, where F(r, y, z) = ry i + x°yj+yz k, where C is a triangular…

A: Stroke's theorem:  ∫Fdr=∬ (Curl F) n^dS Curl F=ijkxy2x2yyzy2x2y=iz n^=∇s∥s∥=i^+k^2

Q: Use Gauss’ theorem to evaluate ∫∫σF.ndS, if σ is the closed surface of the hemisphere z=√25-x²-y²…

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Q: Let r(u, v)= [2ucos(v), 2usin(v), 2] be a parametrization of a surface S, where 0 sus1,0svs2n. Then…

A: Parametrization

Q: Verify Stokes' Theorem for the vector field F(x,y,z) = z^2i - 2xj + y^3k taking the surface S as the…

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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: 4. Let S is the surface z = 5- - y, 1s z in R. Let S be oriented with the normal vector pointing…

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Q: Find the surface integral F. dS of the vector field F(x, y, 2) = (-æ, -y, yz®) along the part of the…

A: The complete solutions are given below

Q: Let F (x, y, z) = (y + z2, xy + ez^3, xy) be a vector field, and let S be the closed surface between…

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Q: Let o be a surface z = a – Vx2 + y² that lies above plane z=0 , oriented by an upward unit normal…

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Q: Find the flux of the vector field F(z, y, z) (2,y,3) over the surface enclosed by the paraboloid, F…

A: The objective is to find the surface integral ∫∫SF· dS where Fx,y,z=x,y,3 over the surface enclosed…

Q: Find the flux of the vector field F = (y,- z, z) across the part of the plane z = 4 + 3z + 2y above…

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Q: 2. Let S denote the piece of parabolic surface x2 + y? = z, 0 < z < 4. For the vector field F(x, y,…

A: According to the given information, Let S denote the piece of parabolic surface as:

Q: Let vector Ñ be normal to plane x – 5y – 2z = 3, and let Ē = (3, – 2, 1). Find the following. 1. The…

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