Picards theorem

1. Oct 21, 2011

georg gill

I was wondering if anyone could explain the mathematic signs in the first line in detailed proof
in the link here. What do this mathemathical sentence mean sign by sign?

http://bildr.no/view/1002076

2. Oct 22, 2011

LCKurtz

You mean the line where he indicates that the closed unit balls in this case are just closed intervals?

3. Oct 23, 2011

georg gill

The mathematical signs in dotted area here between: Let ........... be the compact cylinder where f is defined this is

Which is just below the header detailed proof. It looks like the sign for cross product to me but how does that make a cylinder?

4. Oct 23, 2011

LCKurtz

That is the symbol for the Cartesian Product of the two sets. The Cartesian product of A and B is:

A x B = {(a,b): a ε A and b ε B}

5. Oct 28, 2011

georg gill

I have read some about the cartesian product with a deck of cards as example which has 13 different cardvalues and 4 different colors which make a deck of cards have cartesian product equal 52.

But how can a cartesian product descripe a cylinder?

6. Oct 28, 2011

LCKurtz

It is using cylinder in a more general sense than a common circular cylinder. If you take a circle in the xy plane and take its Cartesian product with the z axis you get a what anyone would call a cylinder. But you can take any region, such as a square in the xy plane and cross it with the z axis. You get an infinitely long square cross section block. Just as you would call the surface of that block a cylindrical surface, you would also call the block itself a cylinder. It just isn't round.

7. Nov 9, 2011

georg gill

This is the whole proof

http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem#Detailed_proof

I wonder is t who is the variable for I a parameter for two dimensions (thoose two dimensions one could call that x and y-axis?) where as y is a variable for B which makes the third dimensions (one could call that one z-axis)

and does the points of I make a circle and B make a line on the z-axis to make the points on the surface of a cylinder?

And what does M=sup||f|| mean?

8. Nov 9, 2011

LCKurtz

Think of a t-y plane instead of xy plane. You are looking for a solution of the DE with y(t0)= y0. Ia is just the closure of the open interval of length 2a about t0: $I_a=\overline{(t_0-a,t_0+a)}= [t_0-a,t_0+a]$.

Simarly, Bb is the closure of an interval of length 2b about y0 on the y axis: $B_b=\overline{(y_0-b,y_0+b)}= [y_0-b,y_0+b]$. Your picture looks like this:

Your Cartesian product in this case is just a rectangle in the ty plane. What is confusing you is that the author is writing it in a more general notation to use the general Banach Fixed Point Theorem.

For continuous functions and a closed region, the sup of a function is the same thing as its maximum.