Definition Of A Double Integral

Hello,

I am currently reading about the preliminary work used to define the double integral, and there is one thing my author states, whose meaning I am not certain I understand. Here is an excerpt:

"The rectangles lying entirely within $\displaystyle R$ form an inner partition whose norm $\displaystyle || \Delta ||$ is defined as the length of the longest diagonal of the $\displaystyle n$ rectangles."

Could someone possibly explain the meaning of this to me, especially the part about the longest diagonal?

Thank you in advanced!

Re: Definition Of A Double Integral

It means that the rectangles in the partition doesn't have to be all of the same size, so their diagonals will be different. When the limit $\displaystyle ||\Delta|| \to 0$ will be taken, to smaller rectangles will also tend to "zero"