with γ(s)=t'(s) and by scalar multiplying i found a2(s)=-1/k(s) a2'(s)=k'(s)/k^2(s) and (k'(s)/k^2(s))-a3(s)τ(s)=0
where a3(s)=k'(s)/k^2(s)*τ(s) and i concluded to this that we wanted
i have this
γ'(s)=-a2(s)k(s)t(s)+a2(s)τ(s)b(s)+a2'(s)n(s)+a3'(s)b(s)-a3(s)τ(s)n(s)
we don't know γ'(s) i think that its difficult to find a2(s) and a3 .
if i differentiate i find this γ'(s)=a2(s)n'(s)+a2'(s)n(s)+a3'(s)b(s)+a3(s)b'(s) beacuse γ'(s)=t(s) i will replace this in this equation? and after that i am trying to replace the followings:
b(s)=t(s)xn(s) , n(s)=t'(s)/k(s), t'(s)=k(s)n(s), n'(s)=-k(s)t(s)+τ(s)b(s), b'(s)=-τ(s)n(s) ?
i tried to the derivative and i found this :
γ'(s)=a2(s)n'(s)+a2'(s)n(s)+a3'(s)b(s)+a3(s)b'(s) , we know that γ'(s)=0 beacuse we are on the surface of sphere , after that i tried to replace n'(s)=-k(s)t(s)+τ(s)b(s) and the other types?
my problem is:even γ: I-> R ^3 curve parameterized as to arc length (single speed) with curvature k (s)> 0 and torsion τ(s)>0. we assume that the γ is at the surface sphere with center the origin. Show that for any s we have:
γ(s)=-(1/k(s))*n(s) + (k'(s)/(k^2(s)*τ(s)))*b(s)
and he gives us a...
even γ: I-> R ^ 2 curve parameterized as to arc length (single speed) with curvature k (s)> 0 and torsion τ(s)> 0. I want to write the γ(s) as a combination of n(s), t(s), b(s). these are the types of Frenet.
the only thing i know is that the types of Frenet are t(s)=γ'(s) ...
Consider the surface S defined as the graph of a function z = 2x ^ 2 - y ^ 2
i) find a basis of the tangent plane Tp surface S at the point p = (-1,2, -2)
ii) find a non-zero vector w in Tp with the property that the vertical curvature at point p in the direction of vector w is zero
for...