# Differential Geometry | Topology Answers

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let r(t)= (e^kt cos t, e^kt sint) find the arc length of r starting at the point (1,0)
find the arc length of r(t)=(e t sin t , e t cos t ,e t)
Bob claims that he can map rectangle 1 to rectangle 2
Find the radius of curvature of follium decart x^3+y^3=3axy at the point (3a,3a)
find the radius of curvature at the point (1,1) of the folium x^3+y^3=2yx
Trace the curve (x/a)^2/3 + (x/a) ^2/3=1
Find the asymtotes of the curve xu(x^2-y^2) +x^2+y^2-a^2.show that the eight points of intersection of the curve with its asymtotes lie on a circle whose centre is at the origin
For what n TS^n=R^n?
Q no. 3) let f(t)= { e^-1/t^2 if t not equal to 0 {0. If t=0 You may assume f is C ^infinity with f^(n) =0 for all n. Let a(t) be given by a(t) = {(t,f(t),0) if t&lt;0 {(0,0,0) if t=0 {(t,0,f(t) if t&gt;0 a) prove that a is regular and c^infinity . b) show that k=0 at t=0 a consist of curves in two different planes joined together at a point where k=0. Q no. 4) let a(s) be a unit speed curve A) prove that the tangent spherical image of alpha (a) is a constant curve iff. a is straight line. B) prove that the binormal spherical image of a is a constant curve iff. a is a plane curve. C) prove that the normal spherical image of a is never constant. Qno. 6 ) let a(s) be a unit speed curve k&gt;0 . Let s star be arc length on the normal spherical image. Prove k= | ds star /ds| iff. a (alpha) is a plane curve.
if A = 5t2 + tj-t3k and B = sin t1-costj evaluate d/dt (AXB)
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