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Line integrals with respect to x and y

  1. Jan 4, 2008 #1
    1. The problem statement, all variables and given/known data
    I am having a bit of trouble relating the line integral of a function with respect to arc length with the line integrals with respect to x and y.
    Last edited: Jan 4, 2008
  2. jcsd
  3. Jan 4, 2008 #2


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    It would be best to ask a specific question.
  4. Jan 5, 2008 #3
    Let's say we have a curve [tex]C[/tex] parameterized by a function [tex]\textbf{r}(t)=x(t)\textbf{i} + y(t)\textbf{j}[/tex]. Differentiating with respect to [tex]t[/tex] we get

    [tex]\frac{d\textbf{r}}{dt} = \frac{dx}{dt}\textbf{i} + \frac{dy}{dt}\textbf{j}.[/tex]

    Multiplying through by [tex]dt[/tex], we get

    [tex]d\textbf{r} = dx\textbf{i} + dy\textbf{j}.[/tex]

    Plugging into the line integral, we get

    [tex]{\int_C \textbf{F} \cdot d\textbf{r} } = {\int_C (M\textbf{i}+N\textbf{j}) \cdot (dx\textbf{i} + dy\textbf{j})}={\int_C Mdx + Ndy}[/tex]

    where [tex]\textbf{F}=M\textbf{i}+N\textbf{j}.[/tex]
  5. Jan 5, 2008 #4

    Gib Z

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    foxjwill is either a physicist or an engineer =]
  6. Jan 5, 2008 #5
    Neither, actually. ;) I'm a high school senior who's really into math. Is the way I formulated the answer the way a physicist or engineer would do it?
  7. Jan 5, 2008 #6

    Gib Z

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    "Multiplying through by dt" lol
  8. Jan 5, 2008 #7
    lol. Thought it was that. I got that terminology from my physics teacher.
  9. Jan 5, 2008 #8

    Gib Z

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    We'll, theres a lot of Physicists/Engineers on these forums, so I won't say anymore =]
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