# Calculate flux with normal form of Green's theorem

1. Jun 11, 2013

### dustbin

1. The problem statement, all variables and given/known data

Let $R$ be the region bounded by the lines $y=1$, $y=0$, $xy=1$, and $x=2$. Let $\vec{F} = \begin{bmatrix} x^4 & y^2-4x^3y \end{bmatrix}^T$. Calculate the outward flux of $\vec{F}$ over the boundary of $R$.

2. Relevant equations

Green's theorem (normal form): $\int_{\partial R} F_1\,dy - F_2\,dx = \iint_R F_{1_x} + F_{2_y}\,dx\,dy$.

3. The attempt at a solution

We have $F_{1_x}+F_{2_y} = 4x^3+2y-4x^3 = 2y$. Then
\begin{align*} \iint_R 2y\,dx\,dy &= \int_0^1\int_0^x 2y\,dy\,dx + \int_1^2\int_0^{\frac{1}{x}} 2y\,dy\,dx \\ &= \int_0^1 x^2\,dx + \int_1^2 \frac{1}{x^2}\,dx \\ &= \frac{1}{3} + \frac{1}{2} = \frac{5}{6} \ . \end{align*}
By Green's theorem, the flux is $\frac{5}{6}$.