f = gradƒ, where ƒ(x,y)=sin(x-2y)
Find curves C1 and C2 that are not closed and satisfy the equations
i) ∫C1 F•dr = 0
ii) ∫C2 F•dr = 1
1) for the first one since we want the integral equal to zero, The sine function equals zero when its argument is zero so set
$\displaystyle x+2y=0$ and we get..
$\displaystyle y=-\frac{x}{2}$
so any curve on that path will give an integral of zero.
for #2
using the fundemental theorem for line integrals
$\displaystyle \int_c= \overrightarrow{F} \cdot d \overrightarrow{r}=\nabla{f(r_2)}- \nabla{f(r_1)}$
so we need to pick a path that the difference of it's end point is 1.
so try
$\displaystyle (\frac{\pi}{2},0) \mbox{and} (0,0)$