Hello. I study differential geomety but I think this is difficult. Please help me.
this is the problem.
Let surface X(u,v)=(u,v,u^2+v^2). Then a point p=(1,1,2) in on X.
principal vector at (1,1,2) is (1,1)&(1,-1). And principal curvature at (1,1) is 2/27. And pricipal curvature at (1,-1) is 2/3.
Let a curve a(t)=(t,1,t^2+1) on X. Then find normal curvature kn at (1,1,2) on a(t)⊆X .
I have a question.
1.Why the principal vector is not 3-dimension but 2- dimension?
If the principal vector above are not correct, how can I find the pricipal vector?
In fact, the suface is very smooth like surface of revolution but not suface of revolution. So I think that kn can't be applied latitude&longitude.( If the surface is revolution, I wanted Euler's thm using latitude&longitude as principal vector for finding kn.)
How can I find the principal vector? If you can explain with some pictiure, I really appreciate..
2.And how can I find the kn?
3.In fact, I have a solution. But I think the solution is a litte strange.
Because, it says, the direction of tangent vector of a'(t) is (1,0)& the angle between a'(t)& principal vector is π/4 (45').
But I can't understand why the direction of tangent vector of a'(t) is (1,0).
If you can explain with some pictiure, I really appreciate..