Recent content by Geometrick

Homework Statement You are at P(-1,1) on the surface z = (y-x^2)^3. What direction should you move from P so that your height remains the same? Homework Equations The Attempt at a Solution So I basically do not want my height z to change. In this case, I will take a vector...

Thank you so much!

Just a quick question, is the 2nd diagram in Fig 14.1 of order g? I don't see why there are two powers of g in this diagram. We have 2 propagators and one counterterm, giving us one power of g is Y = O(g). Can anyone clear up this quick question? Basically, I'm asking, is it ~ g and hence...

How come there is no h^2 term after we do it this way? Using the definition, I don't see why I only get an h and not h^2.

Apparently there is a technique to get the equivalence without using the Riemannian metric, but just using a Partition of Unity. Having said that, I know that having a Partition of Unity implies the existence of a Riemannian metric, but the book I am using, a Riemannian metric is not even...

One form <-----> vector field How exactly does having a one form yield a vector field in a smooth way? I understand it's a duality relationship, but can anyone give me some more insight into this?

That didn't help me too much, I just looked up the definition of submanifold (explicitly) and just used that. It works quite nicely.

Homework Statement Show that the set of all 2x2 matrices of rank 1 is a submanifold of R^4 Homework Equations The Attempt at a Solution The hint in the book was to show that the determinant function is a submersion on the manifold of nonzero 2x2 matrix M(2) - 0. This is easy to...

I'm currently a grad student and I was hoping some of the more experienced members of this board could give me some guidance. Due to some unfortunate circumstances, I'm really wondering if being in a PhD program is the best possible option for me. I do not want to get into specifics, seeing...

Here is a better question: Is R^3 - {n non-intersecting lines} homeomorphic to R^3 - {n parallel non intersecting lines}?

Homework Statement In R^3, if we have p non-intersecting lines, is it true that there exists a plane Y such that all the lines intersect Y? Up to rotations.Homework Equations The Attempt at a Solution No real attempt, this is just something I found to be interesting. Any thoughts?

Why is there no satisfactory way to define the energy density of a gravitational field? Why must we use global energy concepts? I'm currently reading proofs of the Positive Energy Theorem, but none of the references explains why this is so. Can anyone explain it or point me to a good reference?

Hmm I see. I think once I pass my prelims I will take some programming courses, who cares if they are undergrad courses, I need to start somewhere. I thought the math-finance world was really on the decline? How easy would it be for a math PhD from a top 15 program with good programming...

I have some questions about what direction to take in grad school. First of all, I don't really think I'm going to pursue the academia route. With this being the case, I'm wondering what a math PhD's prospects in industry are. I plan on pursuing PDEs and mathematical physics. Some of the...

I don't know if that's true; a point removed on the projective plane corresponds to a point, antipodal point pair being removed on the sphere, i.e. the sphere is a 2-1 covering of the projective plane. So removing a point on the projective plane corresponds to removing 2 points on the sphere.