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Integral through a path in 2D (or ND) What's the usual definition ?

  1. Dec 26, 2012 #1
    Integral through a path in 2D (or ND) What's the usual "definition"?

    [Bold letters are vectors. eg: r]
    We have a scalar function f(r) and a path g(x)=y.
    I see two ways to reason:
    (1) The little infinitesimals are summed with the change of x and on the change of y separately.
    (2) The little infinitesimals are summed with the change of r.

    For example:
    The scalar function is f(r)=1
    The path is the straight line x=y, from x=0 to x=1.
    (1) ∫dx+∫dy=1+1=2 ∫dx from 0 to 1, and since x=y, ∫dy from 0 to 1.
    (2) ∫dr=√2 It's a straight path so ∫dr from 0 to √2.

    What is the regular way to take an integral through a path?
    (1) treats x and y totally independently, (2) seems more "physical/relative" but harder to calculate
     
  2. jcsd
  3. Dec 26, 2012 #2

    mfb

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    Re: Integral through a path in 2D (or ND) What's the usual "definition"?

    A path integral is the integral along the length of the path, so (2).
    I think (1) is not possible (in a meaningful way) for general paths.
     
  4. Dec 26, 2012 #3
    Re: Integral through a path in 2D (or ND) What's the usual "definition"?

    Like mfb said, 99% of the time when you're asked for a line integral of a scalar field you'll want it with respect to arc length, and then you'll want the integral with ds in it. As you showed in your post, ##\int_C{F(x,y) dx}## + ##\int_C{F(x,y) dy}## ≠ ##\int_C{F(x,y) ds}##, so the line integral is defined like (2) in your post.

    When you are doing line integrals in a vector field ##\vec{F}(x,y) = <P,Q>##however, you'll find out that ##\int_C{\vec{F}(x,y) \cdot d\vec{r}} = \int_C{P dx} + \int_C{Q dy}##, so then you'll use line integrals with regards to dx and dy.
     
    Last edited: Dec 26, 2012
  5. Dec 26, 2012 #4
    Re: Integral through a path in 2D (or ND) What's the usual "definition"?

    Thank you for your answers.
    I think it completely clears it up. (feel free to add anything if you want of course)
     
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