Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0sus3,0svs2n. Then we have O A. S is a cylinder of radius 2 and the normal vector is N= [0, 0, -4u]. O B. S is a disc of radius 6 and the normal vector is N= [4u, 0, 0]. O C. S is a cylinder of radius 2 and the normal vector is N= [4u, 0, 0]. O D. S is a disc of radius 6 and the normal vector is N= [0, 0, -4u].

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: Find the surface integral F. dS of the vector field F(x, y, z) = (-x, -4, yz®) along the part of the…

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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: (b) Let D be the disk of radius a in R², and let C denote it's boundary. Let N be the unit normal…

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Q: 2. Show that the xy plane W = (x, y,0) in Rº is generated by: (i)u = and v = (ii)u : and v = 3

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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: Let S {(x1, x2, x3) € R³ | x3 = 5-x²-x², x3 ≥ 1}, a portion of a circular paraboloid. Endow S with…

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Q: Let F = (y², x, z²)and S be the part of the paraboloid z = x² + y² with z < 1 and outward facing…

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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: Let S be the surface defined by the vector function R(u, v) = (v + 4, 4u - v, uv²). Set up an…

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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: 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: 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: 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: 03. a. If F = (y² + z² − x²)i + (z² + x² − y²)j + (x² + y² − z²)k, then evaluate, SSV × F·n dA…

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

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Q: Find the flux of F over the surface S of equation z2 = x2 + y2 , 0 ≤ z ≤ 3 and parameterized by x =…

A: Let E be a three-dimensional solid bounded by a piecewise smooth closed surface S that has…

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: Let S be the portion of the ellipsoid x² + 4y² + z² = 4 where z ≥ √3x. Suppose that S is oriented so…

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Q: F = 2xi+2yj+3zk and σ is the portion of the cone z=sqrt(x^2+y^2) between the planes z=3 and z=5…

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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: 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: Let Q be the solid region bounded by the surface S of equation given by z2 = x2 + y2 , 0 ≤ z ≤ 3,…

A: Find outward flux of the vector field G(x, y, z) = x2 i+ y2 j+ z2 k with Q the solid region bounded…

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: Find the flux of F acros S, F N dS IS where N is the upward unit normal vector to S. F(x, y, z) = xi…

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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: Let S be the upper half of the unit sphere X+y+zwhere Osos2n and 0<0Sa/2. See if you get Curl…

A: Given S is an upper unit sphere x2+y2+z2=1 with an outward pointing normal vector. Suppose…

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: Evaluate the flux, in the specified direction, of the vector field across the surface. F(x, y, z) =…

A: To calculate the flux, we shall use the standard formula of flux in terms of the vector form of the…

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: Given that, F(x,y,z)=xy2i+x2yj+yzk Using the stoke's theorem, ∮F.dr=∬S(curl f.n)dS Now let's find…

Q: 3. Let F(x, y, z) = 3zi – 2y5 – (x + y) k. For each of the surfaces in Parts (a) (e), say whether…

A: We have to find the flux of F through the surface is positive, negative or zero.

Q: Use Stokes theorem to evaluate: ∫∫Sc  n⋅(∇×F)dS, where F = 5y i − 5x j + 3⋅x^4⋅y^7 k, n is the…

A: Use Stokes theorem to evaluate:   ∫∫Sc n⋅(∇×F)dS,   where F = 5y i − 5x j + 3⋅x^4⋅y^7 k, n is the…

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: Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0 <us3, 0svs2n. Then…

A: Parametrization

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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