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# Differential Geometry | Topology Answers

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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)

Q.Torsion is defined only when K(S)≠0 (why?)
Q.Torsion is defined only when K(S)≠0 (why?)
Q. Show that if curvature K(t) of a regular curve γ(t) is >0 every where, then k(t) is a smooth function of t. Give an example to show that this may not be the case without the assumption that k>0.
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)
Q. Show that if curvature K(t) of a regular curve γ(t) is >0 every where, then k(t) is a smooth function of t. Give an example to show that this may not be the case without the assumption that k>0.
Q.Torsion is defined only when K(S)≠0 (why?)
Let X={a,b,c,d}, A={ } be a subset of X, T={ ϕ,{a},{a,b},{a,b},X} be a topology on X. Find limit points
Let X={a,b,c,d} with the topology T={ ϕ,{a},{a,b},{a,b,c,d}}.Find close set.
Let X={a,b,c,d} with the topology T={ ϕ,{a},{a,b},{a,b,c,d}}.Find closure.
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