Part II
Summer Semester, 2007

(5 points)

Give a definition of the following function:

f(x)={({{cos(x) for x≤0}, {x f(x-1) otherwise}})

Plot it.

(10 points)

Consider an expression

f((Subscript[x, 1]),...,(Subscript[x, n])): (R^n)->(R^n),

(i.e. f((Subscript[x, 1]),...,(Subscript[x, n]))={(Subscript[f, 1])((Subscript[x, 1]),...,(Subscript[x, n])),(Subscript[f, 2])((Subscript[x, 1]),...,(Subscript[x, n])), ... ,(Subscript[f, n])((Subscript[x, 1]),...,(Subscript[x, n]))}).

Calculate the divergence of this function, i.e. the trace of its derivative matrix:

(∂_(Subscript[x, 1])) (Subscript[f, 1])+(∂_(Subscript[x, 2])) (Subscript[f, 2])+ ...+ (∂_(Subscript[x, n])) (Subscript[f, n]).

Give an example.

(15 points)

Write a substitution rule that replaces any polygon with its diagonals and colors them in some way. Apply this transformation to a 2D and a 3D graphics object (obtained from some plot statement).

(20 points)

Consider the following random motion. Let a particle move randomly from any point P such that length of the step is equal to (1/(1+||P((||)^2))), where ||P|| is the distance of the point from the origin. It means that the maximal step size is 1, but it is smaller and smaller as the particle is getting far from the origin.
Write a function for the step of the motion for any space! Iterate the process. Animate the motion {(Subscript[P, 0]),(Subscript[P, 1]),...,(Subscript[P, n])} of the particle using different ways (points, path, coloring).