Recent content by Demon117

nevermind.

Well, here is the issue. Suppose I have a helicoid parameterized by Y(u,\theta) = (sinh(u)cos(\theta), -sinh(u)sin(\theta), \theta). For some point on this surface with the coordinate (u,\theta), how can one easily compute the geodesic passing through that point using the first fundamental...

I've been thinking about this quite a bit. So it is clear that one can determine the Christoffel symbols from the first fundamental form. Is it possible to derive the geodesics of a surface from the Christoffel symbols?

I am scratching my head on all of this. I had heard, and I don't remember if it was from a textbook or from a professor that continuity of the wave function and it's derivative imply conservation of momentum across a boundary. Anyone care to elaborate or expel this idea?

Yes, now that makes sense. Our notation uses m rather than \lambda. I appreciate your help.

What is the significance of \lambda(E) in this case? Could you define that for me? It seems rather arbitrarily chosen to me.

Ok, so I have taken care of that. Thanks for the suggestions, they were quite helpful. I have also shown the second part quite well enough. The only thing that remains is the following: (c) If h is a bounded continuous function on R^{2}, using the previous part, give a reasonable argument as...

Actually, I hadn't caught that until you pointed it out. I am not familiar with that theorem.

Homework Statement (a) Let \alpha:I=[a,b]→R^2 be a differentiable curve. Assume the parametrization is arc length. Show that for s_{1},s_{2}\in I, |\alpha(s_{1})-\alpha(s_{2})|≤|s_{1}-s_{2}| holds. (b) Use the previous part to show that given \epsilon >0 there are finitely many two...

Homework Statement What is the motion of the guiding center of a particle in the field of a straight current carrying wire? What happens to the particle energy? Homework Equations The field is tangential to the Amperian loop, so the magnetic field is simply: \oint B\cdot dl =...

I agree with the sign issue but I think it does make a huge difference because N'\cdot n \ne N'\cdot N by definition of the Darboux frame, and by that same definition N'\cdot n=\tau_{g}. So II(T,N)\ne \tau_{g}.

That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by II(T,N)=\tau_{g}. But by inspection this wouldn't make sense. I'm just confused a little I guess.

Yes that is correct. I made a mistake in my initial statement. N(s) is the unit normal field to the surface at \alpha(s). Thanks for allowing me to clarify this. That means n(s) is the normal to the curve \alpha. I will look into this. The second fundamental form defined by...

I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field...

Readjustment So I have new information now. Apparently showing that the gradient of a level set does not vanish somehow also shows that a set defined as above is invariant under three translations. How is that the case? With that in mind, the gradient of the above is...