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How you can say if a line integral will be independant ot a given path

  1. May 9, 2014 #1
    1. The problem statement, all variables and given/known data

    Here is my problem :
    Screenshotfrom2014-05-09095713_zpsa598e6e1.png
    so far I've solved the line integral but I don't know what is the condition that must be met in order to be independant of the path given.
    I found the line integral to be: 27/28
     
    Last edited: May 9, 2014
  2. jcsd
  3. May 9, 2014 #2
  4. May 9, 2014 #3

    ehild

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    Try an other path between points (0,0,0) and (1,1,1) . What about a straight line, connecting them?

    ehild
     
  5. May 9, 2014 #4
    If the Curl of the vector field = 0 it is conservative and hence path independent.

    How would you find the curl of the vector field?
     
  6. May 9, 2014 #5

    LCKurtz

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    Open your text and look in the index for "curl"? Or Google it? Or look at the links given in post #2?
     
  7. May 9, 2014 #6
    I'm not the one looking for help, I was trying to give it :).
     
  8. May 9, 2014 #7

    LCKurtz

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    Woops! My bad. Well with all the hints, maybe the OP will sometime return to the thread.
     
  9. May 9, 2014 #8

    HallsofIvy

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    The integral of a vector function, [itex]\vec{F}[/itex], is independent of the path if and only if it is "a derivative". That is, if there exist a real-valued a function, f, such that [itex]\nabla\cdot f= \vec{F}[/itex]. That will be true for this vector function if [itex]f_x= xy[/itex], [itex]f_y= yz[/itex], and [itex]f_z= xz[/itex].

    We can check if that is true by looking at the mixed second derivatives: [itex]f_{xy}= x[/itex] and [itex]f_{yx}= z[/itex]. Those are NOT the same so this function is NOT independent of the path.
     
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