1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Line Integral

  1. Sep 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Hi there! So I'm working on an old homework problem for review so that I actually have the solution, but I'm not sure how to compute a certain part. Here it is:

    $$\int \int rcos^2(\theta)dr - r^2cos(\theta)\sin(\theta)d\theta$$

    2. Relevant equations
    The solution involves (what I think is) a rewrite of $$dr$$ in terms of $$d\theta$$, but I don't follow the substitutions the author makes.

    3. The attempt at a solution
    Any thoughts on how to approach a problem like this? Thanks!
     
  2. jcsd
  3. Sep 28, 2014 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You have titled this thread "line integral". But it doesn't look like a line integral since it is a double integral. You haven't given us a statement of the problem or any equations to work with. We aren't mind readers.
     
  4. Sep 28, 2014 #3
    Well, the original problem is to compute the line integral of $$v= r\cos^2(\theta)\hat{r} -r\cos(\theta)\sin(\theta)\hat{\theta} +3r \hat{\phi}$$ around a path depicted in the text. The path, in terms of spherical coordinates, runs as follows, broken into 3 segments:

    1) $$r:0 \rightarrow 1, \theta = \frac{\pi}{2}, \phi = 0$$

    2) $$r=1, \theta= \frac{\pi}{2}, \phi: 0 \rightarrow \frac{\pi}{2}$$

    3) $$r: 1 \rightarrow \sqrt{5}, \phi = \frac{\pi}{2}, \theta: \frac{\pi}{2} \rightarrow \arctan(1/2)$$

    The integral comes from computing the dot product of v with dl along the third path. I didn't inclue the rest of the problem since my question is one of how to reduce th double integral to a single integral in theta, as the author did in the solution.
     
  5. Sep 28, 2014 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Assuming that (3) is a straight line, you can parametrize it using the angle ##\theta## as the curve parameter. You can then use the relation ##dr = \frac{dr}{d\theta} d\theta## (essentially the chain rule) to rewrite the differential. Note that what you want for ##r(\theta)## is a linear function such that ##r(\pi/2) = 1## and ##r(\arctan(1/2)) = \sqrt 5##.
     
  6. Sep 28, 2014 #5
    Ah thank you! That's the step I was missing, 3 is a straight line. This is exactly what I needed.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Line Integral
  1. Line Integral (Replies: 1)

  2. Line integral (Replies: 2)

  3. Line Integrals (Replies: 4)

  4. Line Integrals (Replies: 4)

Loading...