Calculate flux with normal form of Green's theorem

[dustbin]

Homework Statement

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 $R$.

Homework Equations

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

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 $\frac{5}{6}$.

 

[mighty2000]

I would like to add that Green's theorem is a powerful tool for calculating flux in two-dimensional regions. It relates the flux over a closed boundary to the double integral of the curl of the vector field over the region. This allows us to avoid tedious line integrals and instead focus on calculating simpler double integrals. In this case, we were able to use Green's theorem to find the flux of the given vector field over the boundary of the region bounded by the given lines.