Hi, I have a geometry question.

I need your help.

Thank you so much.

Problem 1: For the function of n x n matrices Fk : Mn,n + Mn,n given by Fk(A) = Ak with k > 1. Compute T2Fk explicitly. Also give a formula for the Hessian H(Fk) A(V, W). You can assume k 2 1. Hint: Perhaps solve the problem for k = 3 first, then provide the general answer. Problem 2: Consider the real numbers to be a 1-dimensional manifold. If there exists two vector fields on R with a non-zero Lie bracket, find an example of such. If no such vector fields exist, provide a proof. Hint: One systematic way to approach this problem would be to fix a vector field v and consider the equation [v, w] = 0 to be a differential equation in w. Problem 3: Compute the Lie bracket of the vector fields in R2. " (") -((2) ()) W. ( ) = ((2). (9) ) Do the flows commute