Match the parametrizations to the surfaces shown in the figures. Answer Bank u cos (v) Ф (и, v) —D| и sin (v) (2V1+u² cos (v) O (u, v) =| 2VI+u? sin (v) 3 cos (u) sin (v) O (u, v) = 2 sin (u) sin (v) cos (v) Ф (и, о) %3D u

Q: Consider the surface S which is the part of the paraboloid y = x ^2 + z^ 2 that lies inside the…

A: Given, the surface S: y=x2+z2 which lies inside the cylinder x2+z2=16.   a) Let x=vcosuz=vsinuThen…

Q: A torus of radius 2 (and cross-sectional radius 1) can be represented parametrically by the function…

A: The surface area (A) of torus is given by the formula mentioned below with given values as.

Q: find an equation for the plane that is tangent to the given surface z = ln (x + y2),  at the given…

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Q: 31. Use sine and cosine to parametrize the intersection of the surfaces x² + y² = 1 and z = 4x²…

A: Given x2+y2=1 & z=4x2 x2+y2=1 is the equation of circle. That is x2+y2=12,  Therefore, Radius is…

Q: 312. Evaluate surface integral zdA, where S is surface z = Vx2 + y², 0 < z< 2.

A: to evaluate the given surface integral first we parametrize  the surface

Q: I. Evaluate '9 + 3z°dS for the surface S defined by S r(u, v) = [6 cos(u) sin(v), 6 sin(u) sin(v), 3…

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Q: 1: Describe the solid bounded by the sphere ² +y² + z² = 9 and the pla 2 over the xy-plane using %3D…

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Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

Q: find a parametrization of the surface Circular cylinder band The portion of the cylinder y2 +(z -…

A: Given:The portion of the cylinder y2+(z-5)2=25 between the planes x=0 and x=10.

Q: Find the parametric equations for the curve of intersection between the cylindrical surface S: x +…

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Q: Find the surface area of the torus with parametric equations: r(u, v) = ((4+ cos v) cosu, (4+cos…

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Q: 15. Find the radius of curvature at 0 = ÷n of the parametric equations x = 3(0 + cos 0), y = 3(1 —…

A: Given two parametric equations, x=3(θ+cosθ),y=3(1-cosθ) To find the radius of curvature at θ=π3

Q: 103. Show that the parametrization of the ellipse by the angle 0 is ab cos 0 X = a² sin² e + b² cos²…

A: From given data,

Q: Consider the surface S shown in the gure and whose parameterization is given by: r(u, v) = (sin v,…

A: Consider the surface S shown in the gure and whose parameterization isgiven by: r(u, v) = (sin v, u…

Q: Use cylindrical coordinates to evaluate V9-x² r9-x²-y² a² dz dy dx -3 J- 9-x² Jo Do not forget to…

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Q: Match the given parametrizations to the surfaces shown in the figures. Answer Bank u cos (v) ((u² +…

A: The complete solutions are given below

Q: find a parametrization of the surface. 1.The portion of the cylinder (x - 2)2 + z2 = 4 between the…

A: 1) Given information:

Q: A) N = (0,4 cos u, 2 sin u) %3D B) N = (0, –4 cos u, –2 sin u) C) N = (1,0,0) D) N = (0, –2 sin u, 4…

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Q: III Find the area of the portion of the cylindrical surface y = 4- a2 in the first octant that is…

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Q: 1. Find the closest point of the surface ry + xz + yz = 1 to the origin. %3D

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Q: Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,…

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Q: Are there any points on the hyperboloid x 2 − y 2 − z 2 = 1 where the tangent plane is parallel to…

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Q: 8. Find a Cartesian equation for the surface whose spherical equation is p= 2 sin() sin(0).

A: Solve the following

Q: Given the surface z = In(5x2 + 2xy - 3y2 – 12). (a) Find the equation of the tangent plane and the…

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Q: 29. Find parametric equations for the tangent line to the cur ve of intersection of the paraboloid z…

A: Given paraboloid is z=x2+y2 ellipsoid is x2+4y2+z2=9 and the point (1,-1,2) find the parametric…

Q: The planet is spherical with a radius of 1 unit in length. Suppose that in a coordinate system whose…

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Q: Consider the spherical surface E of center at C= (0,0.2) and radius 3. There are two planes parallel…

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Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…

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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: Find a parametrization for the curve y = Vx with terminal point (0, 0) using the angle 0 in the…

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Q: The parametric form of the solutions of the PDES -y žz + x zy = 0 , z (s, s²) is z(s, 1) = s cost…

A: Second option is correct

Q: A solid lies above the cone 2z = v x? + y² and outside the sphere x2 + y2+ z² = 6z. Write a…

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Q: Calculate the area of the parametric surface given by: r(u, v) = 2 sin u cos vi + 2 sin u sin vj + 2…

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Q: XII. $F•d$ where F = (x,y,z²) and S is sphere of radius 2 with center at 0.

A: F→=x,y,z2, surface S is sphere of radius 2 with centre O=0,0,0To find: ∯SF→.dS→

Q: Here's a picture of the parametric surface r(u, v) = (u cos v, u sin v, uv) for 0 ≤ u ≤ 12,0 ≤ v ≤…

A: As per the question we are given a parametric surface r(u, v) and we have to find the surface area…

Q: area of the portion of the cylindrical surface y = 4 − x2 in the first octant that is between the…

A: Given: The cylindrical surface is, y=4-x2 ........(1) And the xy-plane is, x2+2x+y-z =4

Q: u' cos u?, r? EXAMPLE 6.6.3 For a surface of revolution defined by x u sin u?, x3 = f (u'), where f…

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Q: 1. Identify the parametric surface S given by: r(u, v) = 3 cos ui+ vj + 3 sinu k where 0 <u < 2n and…

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Q: Find parametric equations for the line tangent to the curve of intersection of the surfaces tan-1(x?…

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Q: 5. Given the surface z = In(-2x² + 5y2 – xy). (a) Find the equation of the tangent plane and the…

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Q: 1. Let S be the part of the plane 2.x – 2y + z = 1 inside the “cylinder" whose walls are given by…

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Q: 30. The equation of the tangent plane to the surface z = x? – 2y² at x = 1, y = 0 is (а) 2 — : 2т —…

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Q: Find an equation in rectangular coordinates for the surface represented by the cylindrical equation.…

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Q: Find an equation of the form r = f(0) in cylindrical coordinates for the surface 6x2 + 6y² = 13. %3D

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Q: Find the parametric equations for the curve of intersection between the cylindrical surface S : x +…

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Q: The parametric surface R (r, 0) = A = ) = r cos (i + r sin 0j + 0k, for 0 ≤ 0 ≤ 6π, 0 ≤ r ≤ 1, is a…

A: Given parametric surface  R→r,θ=rcosθi+rsinθj+θkfor 0≤θ≤6π, 0≤r≤1

Q: Find the area of the surface obtained by rotating the curve of parametric equations T = 2 cos 0, y =…

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Q: A solid lies above the cone 2z = v x² + y² and outside the sphere x2 + y² + z² = 6z. Write a…

A: Given: Solid lies above the cone is 2z=x2+y2 and outside the sphere x2+y2+z2=6z

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

A: