Let 1 G (x, у, 2) 3 (1+9ху) and consider a surface given by the parametrization .3 r(u, v)= [u, u³, v] O

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: Calculate the surface integral / G(r) dA, where G = (12xz + 36)1/2, the surface S is given by the…

A:

Q: Consider the rormal line to the surface z= z + y? (parabokoid) at the point P(1,1, 2) Find the point…

A:

Q: Evaluate the surface integral. || x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A: Given: ∫∫sx2yzdS where S is the part of the plane z=1+2x+3y that lies above the rectangle…

Q: Find the surface area for the portion of the surface f(x,y)=1-x² + y that lies above the triangular…

A: Solution :

Q: Let L denote the intersection line between the planes 2x-y-3z=4 and 2x+y-z=2 A) Determine a…

A: Solution of the system of equation will give the equation of the line of intersection.

Q: Explain briefly how Vø relates to surfaces of constant o(x, y, x). Hence, use Vø to find points in…

A: For the surface of constants ϕx,y,z=k, the gradient vector ∇ϕx0,y0,z0 is orthogonal to the level…

Q: How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface…

A:

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: 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: 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: Integrate G(x, y, z)=x4 y ( y2 + z2 ) over the surface of the cylinder y2 +z2 = 25 that lies in the…

A:

Q: Evaluate the surface integral ff G (x, y, z) dS, C where G (x, y, z) = x+y+z, and C is the…

A:

Q: Use Stoke's theorem to evaluate the integral xydx +2x 'dy-&r 'zdz where C is the boundary of the…

A:

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

A:

Q: The parametric representation of the surface x² – 8x + z? = 0,0 < y < 5 is a) ř(u, v) = (4 cos u , 4…

A:

Q: Let F(z, y, 2) = (y, Y, z + z²), and let S'be the surface of a very small sphere centered at (-1,0,…

A: The sphere has the equation (x+1)2+y2+(z-1)2=r2, where r is the radius of sphere. If the sphere is…

Q: Verify Stokes's Theorem for F = z²î+ x²j + y²k and S is the surface z2 = x2 + y², y 2 0, and 0 <…

A:

Q: Let x = x(4, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface…

A: Let x=x(u,v) be a regular parametrized surface, A parallel surface to x is a parametrized surface…

Q: Let Y be the portion of the sphere x² + y² + (z − 2)² = 4 which lies below the plane z = 3 with…

A: Given  ∫Cz2+3yi→+2z-2xj→+z2k→.dr→ The portion of the sphere x2+y2+z-22=4 and plane z=3

Q: Evaluate the surface integral. J[x²³x² x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A:

Q: (6) Calculate the surface integral SSs F · ñdS where F = (ze", –3ze"V, xy) and S is the…

A: Given that,

Q: Let S be the negatively oriented surface defined by R(u, v) = (u², v², uv), ere (u, v) € [0, 1] ×…

A: This is a problem of vector calculus.

Q: Let R be the triangular region in the xy-plane with corner points (0,0). (1,0) and (0,1). and let…

A:

Q: Let S be the negatively oriented surface defined by R(u, v) = (u², -uv, v²), where (u, v) € [0.2] ×…

A: Given Ru,v=u2,-uv,v2 and Fx,y,z=y,-xz,y Now, Ruu,v=2u,-v,0Rvu,v=0,-u,2v So,…

Q: Verify Stoke's theorem for A=(2x-y)i-yz²j-y*zk where S is the upper half 5A. surface of the sphere *…

A: Consider the given vector and sphere equation. A→=2x−yi−yz2j−y2zk and x2+y2+z2=1 Since S is the…

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: Evaluate the surface integral [[ (a² + y²) dS where S is the surface with vector function r(u, v) =…

A:

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

A:

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: According to the company policy, we are supposed to answer the first question only. For the rest…

Q: What is the surface area of the portion of the surface z=x2+y that lays over the triangle on xy-…

A:

Q: 1- Calculate the surface integral [fs G(r)dA, where G = (1+½xz)} and 5 is given by the…

A:

Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate (z –…

A:

Q: Find a parametrization for the portion of the sphere x2 +y2 + z2 - 16 between the planes z =- 2/2…

A: we have been given a sphere x2+y2+z2=16 between the planes z=-22 and z=2

Q: Set up and evaluate the surface integral ||(4x+z+6)dS over the surface S ř (u,v)=(v, 4cos u, 4sin u)…

A:

Q: S is the surface with parametric equations x = Зи+ v, у — Зи— v, z = uv for (u, v) in the region D…

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

Q: 2. Find the exact area of the surface. a. z = 1+ 2x + 3y + 4y², 1 < x < 4, 0 < y < 1. b. z = 1+x +…

A:

Q: Verify Stoke's Theorem for A = 3yî – xzj + yzk, where S is the surface of the paraboloid 2z = x + y?…

A:

Q: Sketch or describe the surfaces in R³ determined by the equa- tions in Exercises 40–46. z = y? + 2

A:

Q: Evaluate the surface integral f, F. dS where F = 3xyi + 9x²j+3yzk and S is the surface z = xe", 0≤x≤…

A:

Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 <x< 3 %3D is a) F(u, v) = (v, 2 + 2…

A:

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

A:

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: 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: The parametric representation of the surface x2 -8x + z2 = 0,0 < y 5 5 %3D is a) (u, v) = (4 cos u ,…

A:

Q: Let G(x, y, z)= (1+9xy)² and consider a surface S given by the parametrizatio Osus2,0svs2. Then the…

A: Here we have,    and surface S by the parametrization We have to find,

Q: Verify Stoke's theorem for F = -2yzi + vj +3xk over the surface defined z = 5- x² -y? above the z=1…

A:

Q: Evaluate the surface integral ff G (x, y, z) dS, C where G (x, y, z) = x+y+z, and C is the…

A:

Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…

Q: Evaluate the surface integral. || x2yz ds, s is the part of the plane z = 1+ 2x + 3y that lies above…

A: