The Unapologetic Mathematician

Mathematics for the interested outsider

Iterated Integrals II

Let’s get to proving the assertions we made last time, starting with

\displaystyle{\int\limits_-}Rf(x,y)\,d(x,y)\leq{\int\limits_-}_a^b{\int\limits^-}_c^df(x,y)\,dy\,dx\leq{\int\limits^-}_a^b{\int\limits^-}_c^df(x,y)\,dy\,dx\leq{\int\limits^-}Rf(x,y)\,d(x,y)

where f is a bounded function defined on the rectangle R=[a,b]\times[c,d].

We can start by defining

\displaystyle F(x)={\int\limits^-}_c^df(x,y)\,dy

And we easily see that \lvert F(x)\rvert\leq M(d-c), where M is the supremum of \lvert f\rvert on the rectangle R, so this is a bounded function as well. Thus the upper integral

\displaystyle\overline{I}={\int\limits^-}_a^bF(x)\,dx={\int\limits^-}_a^b{\int\limits^-}_c^df(x,y)\,dy\,dx

and the lower integral

\displaystyle\underline{I}={\int\limits_-}_a^bF(x)\,dx={\int\limits_-}_a^b{\int\limits^-}_c^df(x,y)\,dy\,dx

are both well-defined.

Now if P_x=\{x_0,\dots,x_m\} is a partition of [a,b], and P_y=\{y_0,\dots,y_n\} is a partition of [c,d], then P=P_x\times P_y is a partition of R into mn subrectangles R_{ij}. We will define

\displaystyle\begin{aligned}\overline{I}_{ij}&={\int\limits^-}_{x_{i-1}}^{x_i}{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\\\underline{I}_{ij}&={\int\limits_-}_{x_{i-1}}^{x_i}{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\end{aligned}

Clearly, we have

\displaystyle{\int\limits^-}_c^df(x,y)\,dy=\sum\limits_{j=1}^n{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy

and so we find

\displaystyle\begin{aligned}{\int\limits^-}_a^b{\int\limits^-}_c^df(x,y)\,dy\,dx&={\int\limits^-}_a^b\sum\limits_{j=1}^n{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\\&\leq\sum\limits_{j=1}^n{\int\limits^-}_a^b{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\\&=\sum\limits_{j=1}^n\sum\limits_{i=1}^m{\int\limits^-}_{x_{i-1}}^{x_i}{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\end{aligned}

That is

\displaystyle\overline{I}\leq\sum\limits_{j=1}^n\sum\limits_{i=1}^m\overline{I}_{ij}

and, similarly

\displaystyle\underline{I}\geq\sum\limits_{j=1}^n\sum\limits_{i=1}^m\underline{I}_{ij}

We also define m_{ij} and M_{ij} to be the infimum and supremum of f over the rectangle R_{ij}, which gives us the inequalities

\displaystyle m_{ij}(y_j-y_{j-1})\leq{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\leq M_{ij}(y_j-y_{j-1})

and from here we find

\displaystyle m_{ij}\mathrm{vol}(R_{ij})\leq{\int\limits_-}_{x_{i-1}}^{x_i}{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\leq{\int\limits^-}_{x_{i-1}}^{x_i}{\int\limits^-}_{y_{j-1}}^{y_j}f(x,y)\,dy\,dx\leq M_{ij}\mathrm{vol}(R_{ij})

Summing on both i and j, and sing the above inequalities, we get

\displaystyle L_P(f)\leq\underline{I}\leq\overline{I}\leq U_P(f)

and since this holds for all partitions P, the assertion that we’re trying to prove follows.

The second assertion from last time can be proven similarly, just replacing F(x) by the lower integral over [c,d]. And then the third and fourth assertions are just the same, but interchanging the roles of [a,b] and [c,d]. Finally, the last assertion is a consequence of the first four. Indeed, if the integral over R exists, then the upper and lower integrals are equal, which collapses all of the inequalities into equalities.

About these ads

December 17, 2009 - Posted by | Analysis, Calculus

1 Comment »

  1. [...] copy of these three for each index between and . The proofs of these are pretty much identical to the proofs in the two-dimensional case, and so I’ll just skip [...]

    Pingback by Iterated Integrals IV « The Unapologetic Mathematician | December 21, 2009 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 389 other followers

%d bloggers like this: