3. Consider the parametric vector equation of a cone r(u, v) = u cos vi + u sin vj + uk with 1

Q: Describe the surface r(u, v) = ⟨2 cos u, 2 sin u, v⟩, for 0 ≤ u ≤ π and 0 ≤ v ≤ 1.

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

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

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Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: 11. Use Stokes' Theorem to evaluate OF dr where F=(xz,e",y+z) on the curve that is the intersection…

A: The equation of curl is F=xz,ey,y+z. The equation of plane is 6x+3y+z=12

Q: 5. Evaluate the integral (z²² + y² + z²)³/²dzdydx by changing rectangular coordinates to spherical…

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Q: Sketch the region in the plane consisting of points whose polar coordinates satsfy the given condi-…

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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: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

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Q: Find the area of the surface of revolution r(u, v)= 6sin(u)cos(v)i+ 6uj+6sin(u)sin(v)k, where 0SusT…

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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: This is a three part problem. Given: Find the line integral with respect to arc length (6x + 8y)ds,…

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Q: Let S be the disk enclosed by the curve C: r(t) = (cos q cos t, sint, sin p cos t), for 0sts2a,…

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Q: 21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward…

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Q: A parametric representation of the curve of intersection of the two surfaces x2 + 5y2 - z = 0 and z…

A: The given curves are: x2+5y2-z=0 &amp; z=4y2+36 ⇒x2+5y2-(36+4y2)=0⇒x2+y2-36=0⇒x2+y2=36

Q: Convert the integral 9-x²-y? Vr2 + y² + z² dzdydx to spherical coordinates then evaluate.

A: Consider the triple integral         I=∫03∫09-x2∫09-x2-y2x2+y2+z2 dz dy dx in cartesian coordinates.…

Q: Compute the surface area of the surface segment parametrised by x(u, v) = u² cos v, y(u, v) = u² sin…

A: Given, xu,v=u2cosv,yu,v=u2sinv,zu,v=u2where 1≤u≤3 and 0≤v≤2π To find the surface area.

Q: Show that the surface area of a sphere with radius a equals 4ra². r(u, v)= (a sin u cos v, a sin u…

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Q: Give a parametric description of the form r(u,v) = (x(u,v),y(u,v),z(u,v)) for the frustum of the…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Sketch the region in the plane consisting of points whose polar coordinates satsfy the given condi-…

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

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Q: Exercise 11. Compute the area of the pseudo-sphere, i.e. the surface of revolution obtained by…

A: Area= ∫2πrds=2π∫r.dr/sin(ϕ)∫2πrds=2π∫r.dr/sin(ϕ) Due to tangent property: r/sin(ϕ)=a,r/sin(ϕ)=a, and…

Q: Let S be the disk enclosed by the curve C: r(t) = (cos cos t, sint, sin p cos t), for 0sts2r, where…

A: We need to find circulation.

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

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Q: 2.8 Using the divergence theorem, show that the volume, V, enclosed by the surface I can be obtained…

A: As per the Gauss Divergence Theorem, the surface integral of the normal component of G vector…

Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

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Q: Consider the regular parametrised surface given by the following parametrisation V : [0, 1] × [–1,1]…

A: The surface is expressed by the following parametrization   ψ:[0,1]×[-1,1]→ℝ3, ψ(x,y)=xy4sin2xy (a)…

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: Evaluate Curlv•ñ , where v = 2xyi + (x² – 2 y)] + xzk and ñ is a unit vector normal to the surface…

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Q: Describe the surface r(u, v) = ⟨4 sin u cos v, 4 sin u sin v, 4 cos u⟩, for 0 ≤ u ≤ π/2 and 0 ≤ v ≤…

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Q: Evaluate Curlv.ñ , where v = 2xyi +(x² – 2y)j + xzk and ñ is a unit vector normal to the surface…

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Q: Use the theorem on saddle-node bifurcations from the lecture' to show that i = sin(r) sin(r) +…

A: Given dxdt=sinrsinx+cosx-er. Let us prove that 0,0 is the saddle-node bifurcation.   Theorem:…

Q: Let S be the surface parametrized by R (u, v) =< u, uv, uv

A: It is given that, the parametrized surface is R(u,v)=u, uv, uv2, 0≤u≤2, 0≤v≤3.  We have to calculate…

Q: - Prove that the contour integral – z)dz = -12i where C is the triangle with points 0, 3i and -4…

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Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

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Q: Let C be the positively oriented circle centered at the point zo with radius r>0. Use a…

A: We will first parametize the circle C

Q: Find the surface area of the helicoid (spiral ramp) with vector equation r(u, v) = (u coS v, u sin…

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Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

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Q: Consider the following surface parametrization. x = 3 cos(0) sin(4), y = 2 sin(0) sin(4), cos(4)…

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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: Compute the orthogonal projection of point A coordinates (2,0) onto the Set M (the closed unit disk…

A: I have given the answer in the next step. Hope you understand that

Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

A: Given ,x2+y2+z2 = 22       _________(I)and z = x2+y2     __________(II)Putting the value of z from…

Q: 2. Let W be the portion below the plane z = centered at the origin in R3. Give a domain for…

A: We will use relation between cartesian and spherical co ordinate to find domain of W

Q: Let F(r, y) = 4xî + 3yî and let i be the outward unit normal vector to the positively oriented…

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

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Q: Use Stokes' theorem to calculate , F.ds, where F = (z, x, y) and C is the boundary %3D of a triangle…

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Q: Use the polar coordinates to evaluate volumes : 2) Below z = V1- x2 – y2 inside x² + y2 = above the…

A: Using the polar coordinates we have to evaluate the volumebelow z=1-x2-y2 inside x2+y2=14 above xy…

Q: Bounded by the paraboloid z = 10 – 3x? – 3y2 and the plane z = 4 (using polar coordinates)

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

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Q: Evaluate the contour integrals cosh z dz 4z2 + 72 z +1 dz 223 – 322 – 2z and around the unit circle…

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