Consider the surface s shawn in the figure with the following parameters. r(u,v) = (ucosv, usin v, V7 -uz!) with Osu<2 %3D The differential of the surfaces 2u dA 2 -u A) ds is ? 4u? в) dA 4 - u? C) dA 2u D) dA

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

Q: 4. Find the surface area of the paraboloid S defined by the equation: (r, y, z) = (u cos v, usin v,…

A: We will find surface area by using formula.

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 %3D O x- y…

A: The solution is given in the next step.

Q: Verify that the curve r(t) lies on the surface. Give the name of the surface. r(t) = (/3t cos t, V3t…

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Q: Find ervstion for the tangen plare to the Surface an Siven by x2y?3 =12 at the point Z-2,1,3)

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Q: Which of the following is equivalent to the surface integral (x+y-z) dS, where S is the surface S…

A: Given ∬(x+y+z)dSr(u,v)=2u,5+u+3v,1+2u+4v

Q: If A = 2yz i - (x +3y - 2)j + (x²+z)k, evaluate |(V×A) •n dS over the surface of intersection of the…

A: Given:  A=2yzi-x+3y-2j+x2+zk, With the cylinder x2+y2=a2,x2+z2=a2 Now, calculating ∇×A.…

Q: 1. 3 Consider the parametric surface 7(u, v) = (2uv, u? - v, u² +v²), where u +v < 2. Evaluate the…

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Q: Express the surface z = 6 cos 0 sin o, y = 7 sin 0 sin ø, z = 2 cos ¢ as a level surface f(z, y, 2)…

A: Given x= 6 cos θ sinϕ  , y = sinθ sinϕ  , z= 2cosϕ

Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: (b) A = 2pza, + 3z sin ø a, – 4p cos p a̟ and S is the surface of the wedge 0 0 < ¢ < 45°, 0 <z < 5

A: Let us solve the b part step by step with given A value

Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: Evaluate f, A.n ds, where A = zi +j- 3y²zk and S is the surface of cylinder x? + y? = 16 included in…

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Q: 39. Find the surface area of the portion of the hyperbolic paraboloid r(u, v) = (u+ v) i + (u – v)…

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Q: (5) Use Stokes' Theorem to find the line integral |(* + 2y°)i + (y + z°) +(= + 2r*)k) • dř where C…

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Q: 5) Consider the surface S shown in the graph, whose parametrization is given by by: r(u, v) = (v²,…

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Q: Let G(x, y, z)= (1+9xy)² and consider a surface S given by the parametrization r(u, v)=[u, u³, v], 0…

A: In this question, we find the surface area integral for a given parameterized surface in terms of u…

Q: Q#5. Evaluate the surface integral ff(x,y,z)dS When f(x, y, z)= 3xyz and is the portion of the plane…

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Q: Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent…

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Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Let G(x, y, z)= (1+9xy)2 and consider a surface S given by the parametrization r(u, v)= [u, u, v],…

A: Step:-1 Given that Gx, y, z =1+9xy12  And   r u, v = u, u3, v0≤u≤1,  -1≤v≤1 So,  Gx, y, z =1+9xy12…

Q: Consider the normal line to the surface z = x2 + y² (paraboloid) at the point P(1, 1, 2). Find the…

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Q: Consider the equation of the surface S given by: ln(x2yz3) + xz − x2y = 0 An equation for the…

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Q: Consider the surface z = x In y – x²y. (a) Show that Q = (1,1, –1) lies on this surface. (b) Find…

A: Consider the surface: z=xlny-x2y…

Q: Consider the surface S(u, v) = -dS.

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Q: Find a parametric representation for the part of the cylinder y2 + z² = 36 that lies between the…

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Q: Use Stokes' theorem to evaluate frotF. ds. F(x, y, z) = xyi + e²j + xy²k; S consists of the four…

A: To find- Use Stokes' theorem to evaluate ∫SrotF·dS, Fx, y, z = xyi + e2j + xy2k; S consists of the…

Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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Q: (4) Use Stokes' Theorem to evaluate 11. Vx (yz, xz, xy) dS where S is the surface S that is the part…

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Q: 2)Use Stokes' Theorem to evaluate [F.dr where E2xyi, y² j‡ 2xzk and Sis the first octant с surface…

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Q: Q4 (a) Calculate the surface integral: [] (² + (y² + 2yz) dS where o is the first octant of the…

A: To evaluate the surface integral Given plane 2x+y+2z=6

Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: t A be a surface parametrized by R(u, v) = (u sin v, u cos v , u – v), u E R and v E [0, 2n). Solve…

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Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: Find all points on the surface xy - z2 = 1 which are closest to that origin. Let x,y,z be the…

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Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: 1. Consider the ellipsoid a?a? + B*y? +v°z? = 7? (a) Parametrize the surface S = r(u, v) by x = and…

A: Given  α=1, β=6, γ=13The ellipsoid α2x2+β2y2+γ2z2=γ2and parametrize the surface S=ru,v…

Q: Given a surface 3xy - In (3z - 2) = -5, where z = f(x,y). Find zx at y = 3, z = 1. Leave your answer…

A: This question is based on Partial differentiation of function

Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: Consider the surface S shown in the graph, whose parameterization is given by: r(u, v) = (v², 2u²,…

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Q: Which of the following is the equation of the tangent plane of the z = 3x^2+x-y^2+y surface at the…

A: Equation of tangent plane problem.....

Q: Consider the surface S: z = 4x2 + 2y2 . An equation of the plane tangent to S at the point P (−1, 2,…

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Q: Evaluate the surface integral. (x + y + z) ds, S is the parallelogram with parametric equations x =…

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Q: 2) Find the area of the surface r(u,v)=u cos vi+u sin vj+2vk where 0su<3 and 0<vs 2n .

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Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

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Q: b) Use Stokes' Theorem to evaluate fF•dr for F(x,y,z) =-yi+ xz j+ zk where C is the curve of…

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Q: Consider the spherical surface E with center at C= ( 4, 3, 2 ) and radius 4 . Let π be the tangent…

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Q: (a) The part of the surface z = 1+ 3x + 2y² that lies above the triangle with vertices (0,0), (0,…

A: a) Given part of surface is  z=1+3x+2y2, which lies above the triangle with vertices (0, 0), (0, 1),…

Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Q: A) dS = v36u4 + 16u² + 36v² dA B) dS = V16u4 + 36u² + 36v² dA C) dS = v6u? – 4u – 6v dA D) dS =…

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