Differential Geometry | Topology Answers

Q.Find equation of the osculating plane and osculating circle of the curve at the given point.
γ(t)=(2 sin3t, t, 2 cos3t), (0, π, -2)

Q. Compute the torsion of the following curves
(i) γ(t)=4/5cos t, 1-sin⁡t,(-3)/5cos t
(ii) γ(t)=(t, cosht)
(iii) γ(t)=4/5 (〖cos〗^3t,〖sin〗^3t)

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Q.Torsion is defined only when K(S)≠0 (why?)

Q. Compute the curvature of the following curves
(i) γ(t)=4/5cos t, 1-sin⁡t,(-3)/5cos t
(ii) γ(t)=(t, cosht)
(iii) γ(t)=4/5 (〖cos〗^3t,〖sin〗^3t)
For the astroid in (iii), show that the curvature tends to ∞ as we approach one of the points (±1,0), (0,±1)

Let X={a,b,c,d} with the topology T={ ϕ,{a},{a,b},{a,b,c,d}}.Find close set.

true/false? prove.
1/2 is a limit of the interval ]− 2.5,1.5[

A space
d(X,T)
is called a discrete topological space if every subset of X is _________

Find a Peano Curve which fills out the unit square in E^2

What is basis of usual topology?

Let U and V be open dense subsets of X. Prove that U intersection V is also dense in X.