1. The problem statement, all variables and given/known data Consider the function f= sin(4pix)cos(6pix) on torus T^2=R^2/Z^2 a) prove this is a morse function and calculate min, max, saddle. b) describe the evolution of sublevel sets f^-1(-inf, c) as c goes from min to max 2. Relevant equations grad(f)= <partial x, partial y> show hessian matrix not equal to zero 3. The attempt at a solution From what I understand 1st need to find critical points. so take grad and set equal to zero 2nd use hessian matrix with those critical values that i found before and see if non zero BUT, i dont know what torus T^2=R^2/Z^2 looks like. What does the T^2 mean? I believe R^2/Z^2 is just the xy graph because z has been removed. so its like 3D but if remove z then 2D so is this a square flat torus? once I know the shape then I can do the part b part since all you have to do is fill the shape with "water" and see how the topology changes within the critical values. So is this correct? Since its cos and sin how do i know which critical values to pick and are within the domain.