1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that lies below the plane z = V2.

Q: (c) The cone r² = 4(y² + z?) for (0 <x < 4

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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: r2 0, Is es 2

A: We draw the graph of following function.

Q: 2 Sketch the parametric surface x = Vi u +v ,y = u, z = v.

A: We will sketch the parametric surface x=u2+v2, y=u, z=v. Notice that x2=u2+v2⇒x2=y2+z2.

Q: Write the sphere equationil odi lo anoitaupo ontemeeg bas Totoov bnif (d) x2 + y? + z2 + 2x - 4y –…

A:

Q: osen y

A: Is it sen y =x

Q: Find the area of the part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 +…

A: Given, x2+y2+z2=4z  and  z=x2+y2 Equation of the sphere and paraboloid in cylindrical coordinates is…

Q: Classify the quadric surface. 22 = x² + y? + 1 81 ellipsoid elliptic cone ollintio naraboloid

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Q: ? Find the surface area of the plane ( 15x-5z+20y%3D35) if it is in the first octant

A: Here we have to find the surface area of the plane 15x-5z+20y=35 ................... (1) for the…

Q: By transforming into spherical polar coordinates, evaluate JJ) 1-x². ff - (- dxdydz , taken…

A:

Q: 1. Find the area of the part of the surface z = a? + 2y that lies above the triangle with vertices…

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Q: Determine the projection of the following surface on the xz plane. (xz trace) 4z – + = 0 O Parabola…

A: Solution is given below:

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

A:

Q: Sketch the xy-trace of the sphere whose equation is (x−3)2 +(y−2)2 +(z+4)2 =52.

A: We can find the xy trace by setting z = 0 in the original equation.

Q: Find the area of the following surface using a parametric description of the surface.   The cap of…

A: Consider the surface that is the cap of sphere:

Q: Consider the three-dimensional Euclidean space with coordinates x, y and z. The intersection of the…

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Q: z+v" = 2 +3: = 2

A:

Q: state the type of the quadric surface and describe the trace obtained by intersecting with the given…

A: The given quadratic surface is  x2+y42+z2=1 , y=0

Q: 4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the…

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Q: 6. Consider the sphere a2 + y? + (z-3)2 = 1. Find the point P (a, b, c) on the sphere such that the…

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Q: Consider a sphere E with center at the origin and radius equal to 1, following the Beltrami-Klein…

A: Substitute the parametric coordinates of the line in the equation of sphere and solve to get the…

Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? = 32 and the coordinate…

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Q: Sketch the curve of intersection of the surfaces. The ellipsoid 2x + 2y° + = 288 and the paraboloid…

A:

Q: Determine the projection of the following surface on the plane of the xy-axes -2y? + + =1 3 3 O…

A: -2y2+x23+z23=1To find the projection

Q: Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the…

A: Given surface is x2a2+y2b2+z2c2=1 To find the equation of the plane, find the normal of the given…

Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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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: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

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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: 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: 9. The Whitney umbrella surface is given parametrically by I= uv, y = v, 2= u. A picture of this…

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Q: 7. Use a program to plot the surface generated by the following equation: 4x – 16y + z? = 16

A: we need to plot the following surface generated by, 4x2-16y2+z2=16

Q: Locate and classify the stationary points on the surface z=xy2-2xy-2x2-3x, remembering that  x, y ,…

A: Solving for stationary points of given function using partial derivatives

Q: From the stereographic projection of the Lobachevsky plane from the north pole (0, 0, -R) on disk D…

A:

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Let o be the surface of the sphere given by r² + y? + z² = a². Evaluate the surface integral [S,(r²…

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Q: Find a parametric representation for the surface. the part of the plane  z = x + 2  that lies inside…

A: We need to find parametric representation.

Q: Please help.

A: Note: The tangent plane to the surface given by f(x, y, z) = 0 at (x0, y0, z0) has the equation,…

Q: Determine the projection of the following surface on the xz plane -y+ 름 =1 O Ellipse O Parabola O…

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Q: Describe and sketch the surfaces in space (R') defined by the following equa- tions: (a) y = -z +1…

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Q: Evaluate the surface integral fF- dS where F = (52,-52, 5y) and S is the part of the sphere a² +2+22…

A: Solution

Q: If a parametric surface given by ri (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -3 <u < 3, –4 < v<…

A: We solve this

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: Find the area of the part of the sphere x+y +z = 16z that lies inside the paraboloid z =x²+y?.

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Q: The part of the sphere x^2+y^2+z^2=4z that lies inside the paraboloid z=x^2+y^2

A:

Q: Which of the following represents a plane surface whose x-, y-, and z-intercepts are a, b, and c…

A: Given questions are solved.

Q: (b) Use Stokes' Theorem to evaluate (f, curlF .n dS, where F(x,y, 2) = x²yzi + yz?j + z³e™Yk and S…

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Q: Express the area of the given surface as an iterated double inegral in polar coordiantes, and then…

A: We need to find the surface area in the portion of the sphere x^2+y^2+z^2=16 between the planes z=1…

Q: 1.Find the area of the surface. a. The part of the plane 2x + 5y + z = 10 that lies inside the…

A: Note : As per our guidelines we can supported to answer only first three subparts

Q: Determine the projection of the following surface on the plane of the xy-axes -2y? + 물 + 물 =1 O…

A: Given data: -2y2+x23+z23=1 ..........................(1) Solution: We have to find projection of…