Recent content by sin123

I figured that much, if just for purely pedagogical reasons. For a while I didn't know how to use the positive entries of the matrix, until I realized that that means that the first octant is mapped to itself by the linear transformation. Follow the linear transformation by a projection and I...

Homework Statement Let A denote a 3x3 matrix with positive real entries. Show that A has a positive real Eigenvalue. Homework Equations This is a problem from a topology course, assigned in the chapter on fundamental groups and the Brouwer fixed point theorem.The Attempt at a Solution I...

I am an undergraduate student and I started looking for summer research opportunities. There are no opportunities at my own university and only a single REU (one of the most selective ones in the country, no hope of getting in) on the topic I am interested in. That's why I thought I might ask...

It makes sense intuitively. I am still struggling to come up with a rigorous argument why higher-order terms don't matter - I guess I am not sure how tangent angles are defined in complex-analysis-land to begin with. But at least now I get the idea for the proof. I can probably figure the rest...

Hahaha, yes, it should have occurred to me that I am in complex analysis and not multivariable calculus :) Let's see. In this case g(r) = r exp(i theta), and f(g(r)) = r^n exp(i n theta). Now my angle is n*theta. Makes sense. By pre- and post-composing with translations, I can WLOG assume that...

I am not sure. Suppose f(z) = z^n, and g is a line of slope m, say g(t) = t + m t i. Then f(g(t)) = (t + m t i)^n = \Sigma \left( \begin{array}{c} n \\ 2k \end{array} \right) (-1)^k m^{2k} t^n \, + \, i \, \, \Sigma \left( \begin{array}{c} n \\ 2k+1 \end{array} \right) (-1)^{k} m^{2k+1}...

It tells me that the first couple of coefficients are zero. But what's the connection between the Taylor series of a function and the angles in its image?

Homework Statement If z0 is a critical point of f(z), and m is the smallest integer such that f(m)(z0)=!= 0, then the mapping given by f multiples angles at z0 by a factor of m. Hint: Taylor's Theorem Homework Equations The Attempt at a Solution I don't even know where to...

P.S. This was supposed to go into the homework forum. I had windows open for both forums and I ended up using the wrong one. Let F be a map from S2 in R3 into R4, given by F(x,y,z) = (x^2 - y^2, xy, xz, yz) [ = (a,b,c,d)] Eventually I am supposed to show that this is an embedding...

Hi there, Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix...

Got it. Thanks!

I cannot tell you whether your limit exists or not, but I can tell you why you get different answers. In polar coordinates, when you take the limit as r -> 0 and t is fixed, you are "only" checking the limit along straight lines. It is equivalent to checking x = m y only. I don't know whether...

Suppose you have a continuously differentiable function f: R -> R with |f'(x)| <= c < 1 for all x in R. Define a second function F: R^2 \rightarrow R^2 by F(x,y) = (x + f(y), y + f(x)). F is supposed to be onto. Why is that so? Intuitively, I would say that F is locally onto by the...