1. ## calculating flux

A vector field $\vec{G}$ in 3-space is defined outside the cylinder $x^2 + y^2 = 4$

$\vec{G} = \frac{6y\vec{i}-6x\vec{j}}{x^2+y^2}$

Find $\int\limits_C \vec{G} \cdot · d\vec{r}$ where C is the circle $x^2 + y^2 = 16$ in the xy-plane and oriented counterclockwise when viewed from above.

I am planning to use green's theorem to solve this problem and have found the curl to be:

\frac{12x^2}{x^2+y^2) is this true?? Then I need to parametrize the circle.. and integrate it..

2. Originally Posted by TheRekz
A vector field $\vec{G}$ in 3-space is defined outside the cylinder $x^2 + y^2 = 4$

$\vec{G} = \frac{6y\vec{i}-6x\vec{j}}{x^2+y^2}$

Find $\int\limits_C \vec{G} \cdot · d\vec{r}$ where C is the circle $x^2 + y^2 = 16$ in the xy-plane and oriented counterclockwise when viewed from above.

I am planning to use green's theorem to solve this problem and have found the curl to be:

\frac{12x^2}{x^2+y^2) is this true?? Then I need to parametrize the circle.. and integrate it..

the parameterization of the circle is $\vec r(t) =4\cos(t) \vec i +4\sin(t) \vec j$

The direct computation is quick

$\vec r'(t) =-4\sin(t) \vec i +4\cos(t) \vec j$

$\int\limits_C \vec{G} \cdot · d\vec{r} =\int_{0}^{2\pi}\frac{6(4\sin(t)(-4\sin(t))-6(4\cos(t)(4\cos(t)))}{16}dt$

$-6\int_{0}^{2\pi} \sin^2(t)+\cos^2(t)dt=-6\int_{0}^{2\pi}dt=-12\pi$