Results 1 to 2 of 2

Math Help - Help on multiple integrals?

  1. #1
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2

    Unhappy Help on multiple integrals?

    Dear friends,

    I have a problem on multiple integrals.
    Now, I explain it below:

    Let x=[x_{1},\ldots,x_{n}] be a vector, E:=[a_{1},b_{1}]\times\ldots\times[a_{n},b_{n}] be a rectangle, where a_{i},b_{i} are reals for i=1,\ldots,n, and u:E\to\mathbb{R} be a continuous function, which vanishes on the boundary \partial E.
    Then, the following is written
    \int\limits_{E}\bigg\{\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]\int\limits_{a_{i}}^{b_{i}}\bigg|\frac{\partial}{\  partial s_{i}}u(x;s_{i})\bigg|d s_{i}\bigg\}dx

    =

    <br />
\Bigg(\int\limits_{a_{i}}^{b_{i}}\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]d x_{i}\Bigg)\Bigg(\int\limits_{E}\bigg|\frac{\parti  al}{\partial x_{i}}u(x)\bigg|dx\Bigg)
    for i=1,\ldots,n, where u(x;s_{i}):=u(x_{1},\ldots,x_{i-1},s_{i},x_{i+1},\ldots,x_{n}).

    How can we get the second line from the first one?
    By partial integration (but how)?
    Last edited by bkarpuz; September 8th 2008 at 10:51 AM. Reason: Defined u(x,s_{i}).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2
    Dear friends,

    let me tell the answer, I just found it, and now I think that there is nothing interesting about it.
    Clearly, we have
    <br />
\int\limits_{E}dx=\int\limits_{a_{1}}^{b_{1}}\ldot  s\int\limits_{a_{n}}^{b_{n}}dx_{1}\ldots dx_{n}=\int\limits_{a_{1}}^{b_{1}}\ldots\int\limit  s_{a_{i-1}}^{b_{i-1}}\int\limits_{a_{i+1}}^{b_{i+1}}\ldots\int\limit  s_{a_{n}}^{b_{n}}\int\limits_{a_{i}}^{b_{i}}dx_{1}  \ldots dx_{i-1}dx_{i+1}\ldots dx_{n}dx_{i}<br />
    since the integral bounds are constants.
    Hence, it suffices to show that
    \int\limits_{a_{i}}^{b_{i}}\bigg\{\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]\int\limits_{a_{i}}^{b_{i}}\bigg|\frac{\partial}{\  partial s_{i}}u(x;s_{i})\bigg|ds_{i}\bigg\}dx_{i}\qquad(*)<br />

    <br />
=<br />

    <br />
\Bigg(\int\limits_{a_{i}}^{b_{i}}\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]d x_{i}\Bigg)\Bigg(\int\limits_{a_{i}}^{b_{i}}\bigg|  \frac{\partial}{\partial s_{i}}u(x;s_{i})\bigg|ds_{i}\Bigg)<br />

    <br />
=<br />

    <br />
\Bigg(\int\limits_{a_{i}}^{b_{i}}\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]d x_{i}\Bigg)\Bigg(\int\limits_{a_{i}}^{b_{i}}\bigg|  \frac{\partial}{\partial x_{i}}u(x;x_{i})\bigg|dx_{i}\Bigg).<br />

    From (*), we see that \int\limits_{a_{i}}^{b_{i}}\bigg|\frac{\partial}{\  partial s_{i}}u(x;s_{i})\bigg|ds_{i} is independed from the i^{th} component x_{i}, and thus the equality is obvious.
    Hence integrating both sides of (*) by
    <br />
\int\limits_{a_{1}}^{b_{1}}\ldots\int\limits_{a_{i-1}}^{b_{i-1}}\int\limits_{a_{i+1}}^{b_{i+1}}\ldots\int\limit  s_{a_{n}}^{b_{n}}dx_{1}\ldots dx_{i-1}dx_{i+1}\ldots dx_{n},<br />
    we get the desired result.

    Please, tell me if I am wrong, thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Multiple integrals help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 28th 2009, 01:21 PM
  2. Multiple Integrals
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 7th 2009, 05:37 PM
  3. multiple integrals
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 7th 2009, 02:45 PM
  4. Multiple Integrals
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 19th 2008, 01:17 PM
  5. multiple integrals
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 11th 2006, 08:22 AM

Search Tags


/mathhelpforum @mathhelpforum