Calc 4 Student, Please help me understand Existance and Uniqueness

[danerape]

Ok, after going thru the proof, the only think that still eludes me is the region of definition given in the theorem itself. The rectangle where f and the partial of f with respect to y are known to be continuous.

I am thinking of this three dimensionally, and I do not know if this is the correct way to think about it? In other words, I am imagining that f and its partial are continuous "ON" the rectangle, not "IN" the rectangle. I have seen the theorem written with both words.

My 3-d take on this is that f and its partial are known continuous at every point, (x,y), within the confines of the rectangle. I am imagining f and its partial to be graphed in the z direction in other words.

I think I am confused about this because of the nature of picard iteration, where y is a function of x. I am considering y to be independent of x in my thinking, this is why I am not sure it is right.

So, is it true that continuity of f and its partial exist at every interior point in the rectangle?

Is it correct to think of this three dimensionally, as if f and the partial are being graphed ON the rectangle in the z direction.

ALSO, I HAVE A ROUGH DRAFT OF A PAPER I AM WRITING FOR STUDENTS WORKING AHEAD LIKE MYSELF, I THINK YOU CAN GET A BETTER JIST OF MY UNDERSTANDING THERE, ON PAGE 2.

Thanks

Dane

 

[danerape]

Could this in some ways be analogous to thinking of a direction field? Even though we know y to be a function of x while graphing the direction field of y'=f(x,y), we still graph lineal elements, which seems analogous to graphing in the z direction. Does f being continuous in, or on the rectangle insinuate that the direction field exist in that rectangle?

 

[danerape]

Also, any critique of the paper is certainly welcome, before I submit it I will have it reviewed to make sure all is well.

Thanks

Dane

PS, pretty hard to understand for a mining engineering major, lol