c) Use Stokes' theorem to evaluate $ (xy – y) i – In zj+ (x+ x²) k) · dr, where C is the intersection between the plane z = 3 and the cylinder 2 + (y - 1)? = 1 with the anticlockwise orientation viewed from above. %3D

Q: Sketch and describe, in your own words, the surface S parametrized as follows: r(u, v) = (3 cos v,…

A:

Q: Is the set c~(R* U {0}) “the entire plane without the non-negative reals" conformally equivalent to…

A: We need to evaluate the statement "the entire plane without non negative reals, ℂ-{ℝ∪{0}}, is…

Q: Let D be the unit disk: x + y² < 1. Evaluate I| 7exp(x² + y²) dx dy by making a change of variables…

A: Given:The integral

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

A:

Q: parametrizes the portion of the sphere x² + y² + z² = 9 that is above the xy-plane and below the…

A:

Q: Evaluate fff 1- x2 - y2-z2 dx dy dz where V is the volume of the sphere x? + y? + z2 = a? by…

A:

Q: 2.a. Write the equation of the surface r? + y? = 4y in cylindrical coordinates. b. Classify the…

A: To convert to the cylindrical coordinates, we need to make the following substitutions: x=rcos(θ);…

Q: Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y?…

A:

Q: Calculate the volume of solid S limited by the paraboloid z = 1 - (x² + y²) and the plane z = 0, as…

A: Volume of any solid is given by ∫∫∫dzdxdy=∫∫∫dzrdrdθ which is bounded by the region.  The the…

Q: FIs () Use spherical coordinates to evaluate 1- av where the solid E lies above the cone z=V+y*,…

A: We have to evaluate.I=∭Exzdv, using spherical coordinate  Consider, where the solid E lies above…

Q: Which of the following surfaces cannot be described by setting a spherical variable equal to a…

A: Given,ρ=k , ϕ= k , θ=k To check which one of the given surfaces cannot be described in the above…

Q: Apply Stoke's theorem to calculate S. (x + y – 4)dx + (2x – z)dy + (y + z)dz, where C is the…

A:

Q: Evaluate /| (r + y) dS, where S is the cone whose parametric equations are x = u cos v, y = u sin v,…

A:

Q: 2) Using Stokes theorem, evaluate pvdr where v= 2y'i– 2x° j+÷k_and the C is the curve of C…

A:

Q: In which part of the (x, y) plane the following equation is elliptic? Uzr + 4ury + (x² + 4y*) uyy =…

A:

Q: (b) Find a parameterization, in spherical coordiantes, for the ellipsoid 4x2 + 9y2 + 2²/4 = 1 that…

A:

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

A:

Q: 9. Use Stokes' Theorem to evaluate F. dr where F(z, y, 2) = (2 – y) i + (x + 2)j – (x + y) k and C…

A: The given problem is to evalute the contour integral using stokes theorem, where surface C is the…

Q: c) Use Stokes' theorem to evaluate (zy - u)i– In z j+ (x + z²) k) dr, | where C is the intersection…

A: According To Stokes Theorem∮CF⋅dr=∬S(∇×F)⋅ndS here n is unit normal to the surface.

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

A:

Q: Determine the parametric equations of the curve of intersection of the paraboloid x? – y? + z² = 0…

A: Detailed explanation mentioned below

Q: Prove that the locus of the foot of the perpendicular drawn x2 +² a? = 1 to any of its tangent om…

A:

Q: Use Stokes' Theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above.…

A:

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: 1. The part of the sphere x? + y² + z² = a² that lies within the cylinder x² + y² : = ax and above…

A: To Determine :-  The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax…

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

A:

Q: Exercise 2 Use spherical coordinates to evaluate f[[(9-x² - y³ ) dV , where E is the solid…

A: Introduction: When we perform a triple integral over a spherical region, it is convenient to follow…

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: Q2. Use the Stokes' Theorem to find fF.dr where F(x,y,z) = and C z=9-x²-y² on the xy plane oriented…

A:

Q: V9/ Compute I ,cxyz), uhere s is the part f the plane +x that lies inside the half - cyllnder…

A: Given: The Integral is ∬SxyzdS. The surface is the part of the plane S  z=y+x that is inside the…

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

A:

Q: Use the Stokes' theorem to evaluate p Fdr where F = 3yi + yj+ :k and C is the circle r2 + y? = 3 at…

A:

Q: 10. Using Stoke's theorem, evaluate y dx+ xz dy+x² dz where C is the boundary of the triangle cut…

A:

Q: 1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that…

A: According to the given information, It is required to calculate the parametric representation for…

Q: Let P(r, y, z) be the point at which the line with parametric equations x = -t, y=3+2t, z=1-t te R,…

A:

Q: Use Stokes' Theorem to find m # 0 satisfying F · dr = -5, whereF(r,y, z) = (3yx² + z³, y², 4yx²) e C…

A: Given:

Q: Identify the surface with the given vector equation. r(0,0) = 2sinocosei +2sinosinej +2/1+sin?ok a.…

A: The minimum number of parameters needed to represent a surface is two. Therefore, in general, the…

Q: Use Stokes' Theorem to evaluate  F · dr  where C is oriented counterclockwise as viewed from above.…

A:

Q: The surface area of the surface generated by revolving the parametric curve x-2t -2sin t, y-2-2cos…

A:

Q: . Let F(x, y, z) = (x³, y³, z³). Use Gauss's theorem to find ffsF dS if S is the sphere x? + y? + 22…

A:

Q: 17. F = |r|r, where r = xi + yj + z k, S consists of the hemisphere z = /1 – x² – y2 and the disk x?…

A: The solution are next step

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

A:

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: C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21,…

A: Strokes theorem:

Q: c) Use Stokes' theorem to evaluate + (zy – y) i – In z j+ (r + x*) k) · dr, where C is the…

A:

Q: Let F = · Use Stokes' Theorem to evaluate curi curlF · d§, where S is the part of the paraboloid z…

A: here i had written stokes theorem and done some basic calculation to find the curve of intersection…

Q: Use the polar coordinates to evaluate volumes : 1 2) Below z = 1– x² – y² inside x? + y2 above the…

A:

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: #1. Find a parametric representation of the part of the cylinder x² + z? = 9 that lies above the…

A: parametric equations for a cylinder of radius 3 x^2+y^2+z^2=r^2 Where r=3