Thread: Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

1. Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

Let U denote the right half plane {(u,v)∈R^2 ∶u>0}. Define a function H: U -> R^2 by

H(u,v) = (ue^-v , ue^v)

a.Let Q denote the range of H. Show that Q = {(x,y) ∈R^2 ∶x>0,y>0}, the first quadrant
b.Find H^ -1 on Q
c.Compute DH and D(H^ -1)
d.Show that DH and D(H^ -1) are inverse to one another

2. Re: Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

What is D?

What have you tried?

3. Re: Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

Hi All
I don't need you guys to explain step by step and provide answer for my question. I just need some help to begin this one. like some kind of hint or path that I can walk through to solve this one. Because right now I feel like a total dummy for this question.
Thanks

4. Re: Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

First read the following quote from Tim Gower's blog.
Supervisee: I found this question rather difficult.

Supervisor: Well, what were your thoughts?

Supervisee: Erm … I don’t know really, I just looked at the question and didn’t know where to start. [By the way, never say that. Ever.]

Supervisor: OK, well the question asks us to prove that the action of G on X is faithful. So what does it mean for an action to be faithful?

Supervisee: Oh … er … no, I can’t remember. Sorry.

Supervisor: Have faithful actions been defined in lectures?

Supervisee: I’m not sure. Yes, I think so.

Supervisor: But hang on, if you weren’t sure what a faithful action was, did you not think to look up the definition in your notes?

Etc. etc. This is a fake difficulty because it is not a legitimate reason to get stuck on a question. If you don’t know a definition, you can look it up. (If you can’t find it in your notes, then type it into Google and the answer will be there for you in a Wikipedia article.) “I didn’t know where to start” is a well-known euphemism for “I was too lazy even to work out what the question was asking.” If you come to a supervision with fake difficulties, then you will waste time (not just yours, but that of your supervision partner) dealing with problems that do not require external help, and you will not pick up the mathematical tips that come from engaging with real difficulties.
Now, make sure you know the definitions of the following concepts:
(1) range of a function
(2) inverse of a function
(3) composition of functions

To test whether you understand these concepts, prove that Q ⊆ {(x,y) ∈ R^2 ∶ x>0,y>0} where Q is range of H. What is left in order to prove that Q = {(x,y) ∈ R^2 ∶ x>0,y>0}? Can you prove it? If not, what exactly is your difficulty?

5. Re: Let U denote the right half plane {(u,v)∈R^2 ∶u>0}

Originally Posted by mattturner83
Hi All
I don't need you guys to explain step by step and provide answer for my question. I just need some help to begin this one. like some kind of hint or path that I can walk through to solve this one. Because right now I feel like a total dummy for this question.
Thanks
It would help if you answered emakarov's questions! Until you do, no one has any idea what you are asking.