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Line integral and greens theorem

  1. Mar 10, 2014 #1

    joshmccraney

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    1. The problem statement, all variables and given/known data
    [tex]\int \vec{F} \cdot d\vec{r}[/tex] where [itex]F=<y,0>[/itex] and [itex]\vec{r}=[/itex]unit circle.


    2. Relevant equations
    i'd prefer to do this one without greens theorem (using it is very easy).


    3. The attempt at a solution
    [itex]y=r\sin\theta[/itex] and [itex]x=r\cos\theta[/itex]. now [itex]\int \vec{F} \cdot d\vec{r}=\int r\cos\theta (\cos\theta dr -r\sin\theta d\theta)[/itex] where [itex]\theta [0,\pi][/itex] and [itex]r [0,1][/itex]
    but what do i do with my bounds of this single integral? please help! (i used the chain rule with the above substitutions to evaluate [itex]\int \vec{F_x}dx[/itex] since the [itex]dy[/itex] component seems to be zero)

    thanks!
     
  2. jcsd
  3. Mar 10, 2014 #2

    LCKurtz

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    ##r=1## on the circumference of the unit circle. You parameterization should be$$
    \vec r(t)=\langle \cos t, \sin t \rangle,~~\vec F = \langle \sin t , 0\rangle$$
     
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