1. ## [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:

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

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

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

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

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

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

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

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

-Xav

2. Never mind. I forgot to take the derivative of the r vectors. Silly me!

Problem solved.