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bio website math.poly.edu/~yang
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visits member for 5 years
seen 9 hours ago

Oct
28
comment Riemann's formula for the metric in a normal neighborhood
Of the modern proofs, I like the Jacobi field approach the best. You start by using the distance from a given point as one co-ordinate function and showing that you can extend the angular variables from the tangent space at the origin. This is most easily done using Jacobi fields $J_1, \dots, J_n$,. The metric in these co-ordinates is given by $g_{ij} = J_i\cdot J_j$, so its Taylor expansion is easily calculated using the formal solution to the Jacobi equation.
Oct
27
comment Riemann's formula for the metric in a normal neighborhood
Martin, I find Christoffel symbols highly unenlightening, so that's not a simple definition to me. It seems to me that both normal co-ordinates (up to second order only) and the Riemann curvature tensor arise pretty naturally if you search for co-ordinates that simplify the 2nd order Taylor expansion of the metric as possible. But since the Hessian of the metric is a 4th order tensor, that's still a bit tricky.
Oct
27
comment Riemann's formula for the metric in a normal neighborhood
It seems to me that it's first necessary to find a "simple" definition of the Riemann curvature tensor. I'm not sure how Levi-Civita defined it.
Oct
27
comment Uniqueness of scalar curvature
Robert, oops. You're right. I once knew stuff like that.
Oct
27
comment Uniqueness of scalar curvature
I don't have a proof, but it seems to me that more is true. Namely, any scalar function defined using a Riemannian metric and its derivatives up to second order that is equivariant under diffeomorphisms has to be a function of the scalar curvature.
Oct
23
comment Why do we teach calculus students the derivative as a limit?
Depends on what you mean. If you need it for your work, you'll learn it properly through what you do. That's always the best way, because you understand why it's needed. If you just want to learn it, you succeed either by having a really good teacher or studying it on your own (and not being satisfied until you understand it inside out).
Oct
23
comment Existence of minimizer in Sobolev space $H^{1,2}(\Omega)$
Sure it does. The functional is nonnegative and equal to zero if $u$ vanishes on the boundary but not in the interior.
Oct
18
awarded  Yearling
Oct
16
comment Regularity of the Minkowski functionnal of a convex
Use polar coordinates.
Oct
16
comment Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)
Yes, but without discussing the topological aspect. Uses quaternions.
Oct
16
answered Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)
Oct
15
comment How is this transformation related to the Legendre transform?
If you apply the Legendre transform to $-xf(x,y)$ in the $x$ variable and for fixed $y$, then you should get $s$ back. And, yes, it appears that $f$ is concave with respect to $y$ since it's obtained as an infimum over a family of concave functions.
Oct
15
comment How is this transformation related to the Legendre transform?
It looks to me like that, for each $y$, $-xf(x,y)$ is the Legendre transform of $-s(t,y)$.
Oct
11
awarded  Custodian
Oct
11
reviewed Leave Closed The periodic architecture underlying the natural numbers
Oct
11
comment Monge-Ampere type PDE
I agree with Robert that the question (and not just a comment) should contain an explicit description of how a Riemannian metric is constructed from the function $f$. Also, I don't recognize the PDE as written as an elliptic Monge-Ampere equation as it should be.
Oct
7
revised Variation of curvature with respect to immersion?
deleted 98 characters in body
Oct
7
answered Variation of curvature with respect to immersion?
Oct
7
revised Variation of curvature with respect to immersion?
deleted 16 characters in body
Oct
7
answered Variation of curvature with respect to immersion?