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

1. 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. a. The domain tells you that u is positive in this function. Also, e^-v and e^v are >0 for all v (let me know if you don't understand this part).

So what do these facts tell you about ue^-v and ue^v ?

If you need help with the other parts, you need to tell me what D is.

Updated: April 24, 2012
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