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Thread: Another line integral question

  1. #1
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    Another line integral question

    The question
    Consider the vector field F(x, y, z) = (yz + 1)i + xzj + (xy + $\displaystyle 2e^{2z}$)k.

    Evaluate $\displaystyle \int_C{F.dr}$ along the path C from (0, 0, 0) to (1, 0, 0), following the helix (x, y, z) = (cos(t), sin(t), t) from (1, 0, 0) to (1, 0, 4$\displaystyle \pi$) and then the straight line from (1, 0, 4$\displaystyle \pi$) to (0, 0, 0).

    My attempt
    I first found a scalar potential $\displaystyle \phi$ with the property $\displaystyle \nabla{\phi}$ = F, which is $\displaystyle xyz + x + e^2z$

    So the vector field is conservative. I noticed that the path is actually a closed loop, so if I'm not mistaken, I can apply the Fundamental Theorem of Line Integrals. Thus, the solution should be 0, right? However my book says -1. :/

    Where have I gone wrong? Thanks.
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  2. #2
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    Let $\displaystyle \phi = xyz + x + e^{2z}. $ Then the line integral is

    $\displaystyle \left.\phi\right|_{p_1}^{p_2}$ where $\displaystyle p_1 = (0,0,0)$ and $\displaystyle p_2 = (1,0,0). $ When you substitute you get

    $\displaystyle (0 + 0 + e^0) - (0 + 1 + e^0) = -1$.
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  3. #3
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    Why did you choose $\displaystyle P_2$ to be (1, 0, 0)? I thought we used the initial and end points of the path?
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  4. #4
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    Because that's what the question said - from (0,0,0,) to (1,0,0)

    ($\displaystyle p_1 \to p_2$)
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  5. #5
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    Oh, oops, for some reason I thought it was a line from (0, 0, 0) to (0, 0, 1) then the helix then the line. >_<

    That'd explain my confusion! Thanks!
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