Prove that at the point of intersection of the surfaces x² + y² = z², z = a_tan` (*) vhere y = x tan 0, the radius of curvature and torsion are given by a(2+0²) ³/2 (8+50+0¹) ¹/2 p = and o = a (8+50² +0¹) 6+0²

Q: 3. (a) Show that the two surfaces S1 : z = xy and S2 : z =3x^2/4 - y^2 perpendicularly at the point…

A: As per our guideline, we are supposed to solve only first subpart. Kindly repost other subpart as…

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: Q9)The curvature of the circle * x^2+ y^2=9 is equal to 1/2 F O

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

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Q: (i) find the curvature of the function f(x) = 4x^2 at the point (0,0).  (ii) find the equation for…

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Q: Determine the curvature of the curve x=y^3 at point (1, 1). a) 10.89 b)11.12 c) 8.98 d) 12.2

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

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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: a(0)=(r(0)cos0,r(0)sin@) Find the curvature of the curve. What are the coefficients a and b if its…

A: (a) Curvature of the curve αθ=rθcosθ, rθsinθ: Let α1θ=rθcosθα2θ=rθsinθ Then, the curve αθ is given…

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: Calculate the Foĥds? where S is Surface bo undade by Xャそ=6, x*+y=4, Z=0 ana F= (x*vcosz)& + (xy+…

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Q: True or false: the curvature of,k, for x^2+y^2=100 is larger than for x^2+y^2=1/4

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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: Find the equation of the tangent plane to the given surface at the specified point   2x=y^2+z^2-9 at…

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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: Find the curvature κ of r(t) = at t = 0. Simplify your answer

A: use curvature formula for vector valued function.

Q: Find the equation of the tangent plane to the sphere x² + y2 + z² = 1 at the point (1/ V3, 1/V3, 1/…

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Q: find the curvature of y=x^4 at (1,1).

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Q: Give an arclength parametrization of the line which is the intersection of the tangent planes of…

A: Let First, find the tangent planes at (1, -1, 0), (1, 2, 9).     Tangent plane at (1, -1, 0):…

Q: find the Cerrvature of the Curve x:(0,1|R? defined by x(t)=14Cost, 1sint,o), where %3D is coostant 2

A: α(t)=(1rcost ,1rsint ,0)r is constantTo find the curvature of the given curve

Q: At what point does the curve have maximum curvature? y = 2ex (x, y) = What happens to the…

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Q: If 7 (t) = then the curvature at the point (2,0,0) is (A) (B) 25 (c); (D)를 (E) -

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Q: What is the curvature of r(t) =e^t cos(t) i + e^t sin(t) j?   A) e^ (-t)/squareroot(2)   B) e^t/…

A: The curvature of a curve given by the vector function , we need to determine the magnitude of the…