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

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- Mar 18th 2008, 07:25 AMenterprise-solutionsFind curves C1 and C2 that are not closed
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 - Mar 18th 2008, 07:41 AMTheEmptySet

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

$\displaystyle (\frac{\pi}{2},0) \mbox{and} (0,0)$