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. I carelessly posted this question in the Topology&Geometry forum first:
https://www.physicsforums.com/showthread.php?t=341773
Homework Statement
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...
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...
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...