Math Is Fun Forum

Hello all,

Question is:

Show that (x,y) --> (x^2 - y^2, 2xy) maps the first quadrant {x>0, y>0}into the upper half plane {y>0}.

Now if i were to "brute force" this then I can easily do it. By "brute force" i mean I consider any arbitrary (x', y') in the upper half plane and show this point can be expressed as (x^2 - y^2, 2xy) where x>0, y>0 (first quadrant). This shows that every point in the upper half plane can be obtained. Then show that if (x,y) is chosen in the first quadrant, then the transformation cannot give a point in the lower half half plane.

So done. However, its not beautiful.

In the book they hint to use polar coordinates and then in the answers in the back they say:

(x,y) --> (x^2 - y^2, 2xy) becomes:
(r, 0) --> (r^2, 2*theta)

and I just dont see how?

Thanks for any help.

Let A be a set. Denote by dA the set of boundary points of A. Denote by intA the interior of A. Denote A' the closure of A.

It was asked of me to prove the following:

(i) dA' is a subset of dA
(ii) d(intA) is a subset of dA

and I have already done so. However, now I'm asked to find a example of a set A such that dA, dA', and d(intA) are all different. This set can be as weird as you like and you can consider any topology on any space.

I been trying to find such a set for some time but haven't been able. Such a set A cannot be open because then A = intA and obviously their boundary points will be the same. Similarly if A is closed A = A'

Definition: an Abelian group G is p-primary for any prime p, if for every g in G there is n>0 such that g*p^n = 0

So then given any arbitrary Abelian group, not necessary p-primary, we can define it's p-primary components:

Gp = {g in G : there exist n>0 such that g*p^n = 0} for any prime p.

So that means that for any prime p, an Abelian group has a p-primary component Gp.

So then there is a theorem that says that G is the direct sum (finite!) of it's p-primary components. My confusion is, since for any prime p there is a p-primary component, that means that there are infinitely many p-primary components. In the decomposition of G into the direct sum of it's p-primary components, which of the infinitely many components do you choose for the decomposition into a finite direct sum of components?

I don't know if I'm explaining myself. A group G has infinite components, but there is a finite decomposition into direct sum of those components, how do you choose those finite components from the infinite list?

Thanks.

Any idea how to prove this corollary?

Thanks.

The derivative of a matrix is the matrix containing the derivative of each entry. I thought this was by definition but it is not.

From the above definition of the derivative of a matrix, how do I derive that:

How do you prove this? Any ideas?

Any hints? Thanks.