Math 302, Assignment 9, Fall 2000

Due Wed Nov 1 (at the beginning of class)

1. Consider the map f from E^2 to E^2 given by f(x,y) = (ax + by, cx + dy), where a,b,c and d are constants.
   1. Compute the partial deriviates of f.
   2. Compute the distortion in the horizontal direction.
   3. Compute the distortion in the vertical direction.
   4. Compute the distortion in the direction (1,1).
   5. When is the distortion the same in every direction at a given point?
   6. When are the partial derivates of f perpendicular?
   7. When is f conformal?
   8. When is f a local isometry?
   9. When is f one-to-one?
   10. When is f onto?
   11. Does f take lines to lines?

2. Recall that we defined chi from E^2 to H^2 as follows: Pick a point O. To find chi of (u,v), start at O; travel first distance u to the right along an annular curve (left if u is negative); and then distance v up a vertical geodesic (down if v is negative). Let P denote chi of (u,v).
   Do you get the same point if you start at O; travel first a distance v up a vertical geodesic (down if v is negative); and then travel distance u to the right along an annular curve (left if u is negative)? Let P' denote this new point.

   Do the points P and P' lie on the same annular strip? If not, which lies farther up?

   Do the points lie on the same radial geodesic? If not, which lies farther right?

   
   



3. Let C be the intersection of the unit circle with center (0,0) with the upper half plane U. Let z be the upper half plane map from U to H^2.
   1. Find a paramterization of the semi-circle C, that is a map f from an interval (a,b) in R to this semi-circle.
   2. Use this to find the length of this semi-circle C.
   3. Let D = z(C) be the image of the semi-circle C under the upper half plane map; this is a curve in H^2. The map g from (a,b) to H^2 given by g(t) = z(f(t)) gives a paramaterization of D. Use this to find the length of the image D.
   Hint #1: Use the formula for the distortion of z. Hint #2: you may need to look up this integral, if you have forgotten it. It will be in any calculus book.

Journal for Friday is under ``handouts''. No journal for Monday.