Differential Geometry | Topology Answers

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<0
{(0,0,0) if t=0
{(t,0,f(t) if t>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>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)