Hi, I have geometry questions.

I need your help.

Thanks a lot.

Problem 1: (a) Consider the function FR : C -C defined as Fr(z) = 2*, with Fo(z) = 1. Compute the derivate of Fk, provided k 2 0 is an integer. Hint: Try writing Fr as the composite of the functions Ak : C + Ck with Ak(z) = (2, z, . .., z) E Ck with mk : Ck > C given by mk(21, Z2, . . . , Zk) = 2122 . . . 2k . (b) Let S' = {v E R2 : (v) = 1} and consider R2 to be the complex numbers R2 = C. Let be the function fk (2) = zk where k E Z. Compute the derivative of fk, i.e. D(fk) = (h). (c) When does the implicit function theorem apply to fk? i.e. determine the regular values of fr for all k. (d) Describe fx (1) when k > 0 as a set. Is it a manifold? Of what dimension? Problem 2: Compute the derivative of the function f(A) = A* where A is an n x n matrix, and k > 0 is an integer, f : Mn,n + Mn,n