Results 1 to 2 of 2

Math Help - [SOLVED] Checking answer from Green's Theorem

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    2

    [SOLVED] Checking answer from Green's Theorem

    Hey all,

    I'm working on some simple problems that use Green's Theorem, but my answers don't match up. I'm wondering why.

    "Use Green's Theorem to find the counter clockwise circulation for the field and the curve C the triangle bounded by y = 0, x = 1, and y = 3x. Also check your answer using line integral involving 3 separate lines."

    Sounds easy enough, so I set out my equations in Green's Theorem:

    \overrightarrow{V} = V_{x}\overrightarrow{i} + V_{y}\overrightarrow{j} = 2y\overrightarrow{i} + (x + 3xy^{2})\overrightarrow{j}

    \oint_{c} \overrightarrow{V}\cdot d\overrightarrow{r} =  \int\int \frac{\delta V_{y}}{\delta x} - \frac{\delta V_{x}}{\delta y} \; dx \; dy
    = \int_{x=0}^{x=1} \int_{y=0}^{y=3x} 3y^{2} - 1 \; dy \; dx
    = \frac{21}{4}

    And I believe that's right. Then, I use three line integrals to check my answer:

    \overrightarrow{V} = V_{x}\overrightarrow{i} + V_{y}\overrightarrow{j} = 2y\overrightarrow{i} + (x + 3xy^{2})\overrightarrow{j}
    <br />
\oint_{c} \overrightarrow{V}\cdot d\overrightarrow{r} =  \int \overrightarrow{V}\cdot d\overrightarrow{r_{1}} + \int \overrightarrow{V}\cdot d\overrightarrow{r_{2}} + \int \overrightarrow{V}\cdot d\overrightarrow{r_{3}}
    \overrightarrow{r_{1}} =  x\overrightarrow{i}
    \overrightarrow{r_{2}} = \overrightarrow{i} + y\overrightarrow{j}
    \overrightarrow{r_{3}} = x\overrightarrow{i} + 3x\overrightarrow{j}

    \Rightarrow \left (\int_{0}^{1} 2xy \; dx  \right ) + \left (\int_{0}^{3} 2y + xy + 3xy^{3} \; dy  \right ) + \left (\int_{1}^{0} 3x^{2} + 9x^{2}y^{2} \; dx  \right )

    Applying y = 0, x = 1, y = 3x in those integrals respectively:

    = \left (\int_{0}^{1} 0 \; dx  \right ) + \left (\int_{0}^{3} 2y + y + 3y^{3} \; dy  \right ) + \left (\int_{1}^{0} 3x^{2} + 81x^{4} \; dx  \right )
    = \frac{1141}{20}

    I must have done something incorrectly, but I just can't figure out what.

    -Xav
    Last edited by SpaXe; April 2nd 2010 at 12:31 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Apr 2010
    Posts
    2
    Never mind. I forgot to take the derivative of the r vectors. Silly me!

    Problem solved.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Green's theorem question needs checking
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 19th 2010, 08:42 AM
  2. [SOLVED] Help in checking my answer
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: September 24th 2009, 12:29 AM
  3. [SOLVED] Applying Green's theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 15th 2009, 05:14 PM
  4. Replies: 2
    Last Post: December 9th 2008, 02:55 AM
  5. [SOLVED] Checking an answer about dices :D
    Posted in the Advanced Statistics Forum
    Replies: 9
    Last Post: June 7th 2008, 07:16 AM

Search Tags


/mathhelpforum @mathhelpforum