Greens Theorem Handout #2

Red

Let $F=\langle xy, -xy\rangle$ and suppose that C bounds the rectangle $0\leq x\leq1 $, $ 0\leq y \leq 1$. Use Green's Theorem and the divergence theorem to compute

$$\oint_C F\cdot dr$$ and $$\oint_C F\cdot dn$$

lusta

I'm just gonna rewrite this so I can see it better,

$F = <xy,-xy>$ and suppose $C$ bounds the rectangle $0 \leq x \leq 1$ and $0\leq y \leq 1$ Use greens theorem and divergence theorem to find $$\int_C F \cdot dr \text{
and } \int_C F \cdot dn$$

lusta

The first integral we can translate using Greens Theorem to be the integral over the rectangle $$\int_C F\cdot dr = \int_0^1 \int_0^1 Q_x - P_y dA$$
After solving for the partial derivatives $Q_x$ and $P_y$ we find $$\int_0^1 \int_0^1 Q_x - P_y dA = \int_0^1 \int_0^1 -y-x dxdy = -1$$

Now we can solve the second integral using greens theorem $$\int_C F\cdot dn = \int_0^1 \int_0^1 P_x + Q_y dA = \int_0^1 \int_0^1 y-x dx dy = 0$$