1. Limited time only! Sign up for a free 30min personal 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!

Calculate the circulation of vector field

  1. Feb 26, 2013 #1
    Hello there,

    I've got a vector field which you can see here: Sketch of the vector field . It is: [itex]\vec{v} = \cos(x)\,\sin(y)\vec{i}-\sin(x)\,\cos(y)\vec{j}[/itex]

    Say I want to find the circulation around the square formed by [itex]-\frac{\pi}{2} \, \leq x \leq \, \frac{\pi}{2}[/itex] and [itex]-\frac{\pi}{2} \, \leq y \leq \, \frac{\pi}{2}[/itex]. I think that I should do this by finding the circulation along one of the sides, and multiply by 4 (I can tell from the vector field that the circulation along one of the sides is going to be equal to every other side.

    This is where I become unconfident. [itex]\int_{y = -\frac{\pi}{2}}^{y = \frac{\pi}{2}}\vec{v}\,\mathrm{d}\vec{r}[/itex] Please correct me if I am wrong. Any criticism is appreciated:

    My [itex]\mathrm{d}\vec{r} = \mathrm{d}y\vec{j}[/itex] along the y-axis, for the first side: [itex](\Delta y = -\frac{\pi}{2}, x = -\frac{\pi}{2})[/itex].

    [itex]\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(x)\,\sin(y) \, \mathrm{d}y -\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin(x)\,\cos(y) \, \mathrm{d}y[/itex]

    [itex]\cos(-\frac{\pi}{2}) = 0[/itex], so the first integral is equal to zero.

    [itex]-\sin(-\frac{\pi}{2})\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos(y) \, \mathrm{d}y = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos(y) \, \mathrm{d}y = [\sin(y)]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \sin(\frac{\pi}{2}) - \sin(-\frac{\pi}{2}) = 2[/itex]

    First question:
    Is the above correct?

    Second question:
    Can you think of any additional criticism?

    Thank you for your time.

    Kind regards,
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted