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!

Homework Help: Integrals with curl dot products

  1. May 1, 2010 #1
    1. The problem statement, all variables and given/known data

    1. Evaluate [tex]\int_{S}\int curl F \cdot N dS[/tex] where S is the closed surface of the solid bounded by the graphs of x = 4, z = 9 - y^2, and the coordinate planes.

    F(x,y,z) = (4xy + z^2)i + (2x^2 + 6y)j + 2xzk

    2. Use Stokes's Theorem to evaluate [tex]\int_{C}F\cdot T dS[/tex]

    F(x,y,z) = xyzi + yj +zk
    S: 3x+4y+2z=12, first octant


    2. Relevant equations



    3. The attempt at a solution

    1. For this one, I found the curl to be -6yi. However, I am at a loss as to how to get the N dS part without some sort of given equation for S? The book answer is 0.

    2.
    First I found the curl to be:
    [tex]xyj - xzk[/tex]

    I then used a theorem in my book to find N ds:
    3/2i + 2j + k

    Then I took the dot product:
    [tex]<0, xy, -xz> \cdot <\frac{3}{2}, 2, 1> = 2xy - xz[/tex]

    Integrating:
    [tex]\int^{4}_{0}\int^{4-\frac{4y}{3}}_{0}(2xy-x(-\frac{3x}{2} - 2y + 6)*dx*dy [/tex]
    which comes out to 64/27.

    The book answer is 0.

    Any pointers as to what I'm doing wrong would be appreciated.
     
  2. jcsd
  3. May 1, 2010 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    You can either use Stokes' theorem for this, and the answer will be immediate; or you can find the equation of the surface that bounds the volume described in the question and integrate directly. (If you are going to use the second method, I recommend you break the surface into 4 separate surfaces to make it easier, and start by sketching the volume so you can see what I mean)

    Shouldn't [itex]\textbf{n}dS[/itex] be a differential vector?:wink:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook