Questions

How do we see whether we a number c is in the Mandelbrot set M or not? 0 and apply the coordinate change map T again and again and look what happens. If this escapes to infinity, then c is not in M, otherwise it is in M. Take c=2+3i
```
     0 goes to 0^2+c = 2+3i
     2+3i goes to (2+3i)^2+ (2+3i) = -3+15i
    -3+15i goes to ((-3+15i)^2+ (2+3i) = -214-87 i
```
We see that the parameter c=3+3i is not in M. Lets do an other example: z=i;
```
     goes to 0^2+i = i
     i goes to i^2+i=-1+i
     1+i goes to (-1+i)^2+i =-i
     -i goes to =1+i which again goes to -1
```
as this says bounded, the parameter c=i is in M. In the homework, you show also that if |c|>2, then M is not in M.

Here is a 2 line code plotting the Mandelbrot set

F[c_]:=Module[{z=c},Do[z=N[z^2+c];If[Abs[z]>2,z=10],{200}];If[Abs[z]>2,0,1]];
ContourPlot[F[x+I*y],{x,-2,1},{y,-1.5,1.5},Axes->True,PlotPoints->100]

XAos, a mandelbrot viewer.

Q: I was just wondering if problem sets need to be in LaTeX or if we can handwrite our solutions (on a tablet)? A: While LaTeX is not forbidden, we recommend going with handwritten solutions, either on paper or on a tablet. The later is not very different than paper today. This is just by experience having seen many students after some time keeping up with the latex version. Also, for exams, we require handwritten solutions, so, it can be beneficial to write also by hand for the practice work. Of course, one can use a tablet to write. For exams, there will be paper.

Q: Why bother with questions?

A: It is one of the most important skills to be able to ask questions. There are no stupid questions. If you ask a question, you might feel dumb for a moment, if you don't ask questions, you remain dumb for the rest of your life. We hope in this part to document a bit some questions which have come up.

Math 22b Spring 2022

22b Vector Analysis and Linear Algebra

Questions