Exercise 1 Consider the following 1-form: w (1-2)da, 2x1xgdzy. 1) Is there any function h such that dh = w? 2) Consider the following vector field: a Compute w(fi), then give a vector field that spans the Kernel of w. 3) Let الف =-(+)w. Find a function h(1,2) such that dhwy. What is the relationship between the level curves of h and the integral curves of fi? 4) Let the following vector field: 12= =(++) a -2) 231232) Show that f₂--Vh where Vh is the gradient of h. 5) Let V and compute LV to show that the singularity (0) of fa is asymptotically stable. What about its second singularity (0)? 6) Are the singularities of fi stable, asymptotically stable or unstable? Exercise 21) Find the state representation of the following differential equation: ++au+Buy cos(wt) 2) Let the following vector field: 8 f(x)=17 θα (ax+6x+3x); ara what is the link between it and the previous differential equation? 3) Find the singularity of the following dynamics *= f(x). 4) Study the stability of those singularities. 5) Set 80 for it = f(x) and find the matrix A such that f(x) = Az. Give the Laplace transformation of this linear dynamical system. 6. 6) Discuss the solution of this linear dynamical system based on the values of a and 7) Consider the following function: V(21)=++ Compute the LVV. For 8 > 0, does this result show the asymptotic stability of the singularity (0,0).