Prove that at the point of intersection of the surfaces x² + y² = 2², z = a tan` X where y = x tan 0, the radius of curvature and torsion are given by a(2+0²) ³/2 (8+50+0¹) ¹² P = and o = a (8+50² +0¹) 6+0²

Q: Prove that at the point of intersection of the surfaces x² + y² = z², z = a_tan` (*) vhere y = x tan…

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Q: . A donut with major radius R and minor radius r can be represented parametrically by r(u, v) =…

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Q: If a parametric surface given by r₁(u, v) = f(u, v)i + g(u, v)j + h(u, v)k and −4 <u≤ 4,-2 ≤v < 2,…

A: The surface is  and The surface area of the parametric surface given by r1(u,v) =

Q: Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive…

A: We will use the Gauss–Bonnet theorem to prove the given statement.

Q: What is the tangent plane equation of sphere f(x, y. :)=x² + y² += =6 at point (1,-1,2)? O a.…

A: I hope that this solution will help you .Thankyou. Best regards.

Q: ind the parametrization of the curve defined by the intersection of the two surfaces y = x^2 − 3z^2…

A: To find the parametrization of the curve defined by the intersection of the two surfaces  and ,…

Q: Calculate the curvature of r (t) = 〈t,t^2,t^3 〉 at the point (1,1,1)

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Q: where y . Prove that at the point of intersection of the surfaces (4) y __x² + y² = z², z = a tan` =…

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Q: let AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three…

A: Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

Q: PCt) =<5-t, 4t -3,3t7 uking are length Reparami tize parameter from the point (4,113).

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Q: 1) Prove Green's Theorem for F = 3xy'i - 4xj around the curve C bounded %3D 160 by y = 2x and y =…

A: These questions are about green's theorem around any given curve c.

Q: Integrate h(x, y) = (x+y)i + (y²-x)j over the closed curve that begins at (-7,0), goes along the…

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Q: An equation for the tangent plane to the surface (xty at the point P(1, 1, 4) is a) 8r + 8y-:= 0 b)…

A: The given surface is  f(x,y)=z=(x2+y2)2  ...........(1); we have to find the tangent plane at point…

Q: If a parametrized curve is given by: r⃗ (t)=⟨2t^3,10ln(t),5t^2+9t⟩ At t==2, the position vector is…

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Q: Find the lines of curvature of a surface x=cos⁡u y=sin⁡v z=v

A: Given: Surface: x=cos u, y=sin v, z=vTo find: The lines of curvature of the given surface.

Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: Consider the surface S of equation 2 = x² + y*" An equation for the tangent line tangent to S at…

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Q: Determine the area of the parametric surface S with equation R(u, v) = (u²+v², 2uv, u² −v²), where 0…

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Q: If the circles x\power{2}+y\power{2}+2a'x+2b'y+c'=0 and 2x\power{2}+2y\power{2}+2ax+2ay+c=0…

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Q: Consider a plane curve defined by the parametric equations :  x(t)=acos^3 (t) and  y(t)=asin^3 (t)…

A: The curvature κ(t) of a plane curve defined by parametric equations x(t) and y(t) is given…

Q: The tangent plane to the surface 2x² + 3y² = 5z² at (1, 1, 1) has an equation ax +by+cz = d.What is…

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Q: A particle is moving in space with position function RIt) = (cos(4t), sin(4t), 5- 3t. a) Determine…

A: We will find arc length and use it to parametrize the curve

Q: Find the angle at which the circles x² + y² + x + y = 0 and x² + y² + x − y = 0 ir If the circles x²…

A: The angle of intersection of the two circles is given by cosθ=r12+r22-C1C22r1r2where r1 and r2 are…

Q: Let C be the curve of intersection of the surfaces y = V2 x2 and z 2/2 ху. (a) Compute the curvature…

A: Given C be the curve of intersection of the surfaces y=2x2 and z=223xy

Q: erustion for the tangena plare to the Surface = 12 at the point L-2,1,3) Find an Siven by x2y2z

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Q: 3. Consider the curve 7() = i+(Vī lm(1) +-R. b) Find the curvature at the point where t =1. Hint:…

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Q: (b) Find the two points on the surface x² + y² + 24 = 2 where the tangent plane is parallel to the…

A: The two points on the surface x4+y4+z4=2 where the tangent plane is parallel to the plane x-z=5.

Q: A secant to two congruent circles, which is parallel to the line of centers OO', meets the first…

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Q: drawn from the point A(2,10) on the parabola y = x^2 + bx + c Since the tangent is parallel to y =…

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