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Gauss' Theorem/Stokes' theorem

  1. Oct 3, 2006 #1
    Hi guys,

    I am having trouble with this "simple" problem involving these two theorems:

    Find the value of the integral (A dot da) over the surface s, where A = xi - yj + zk and S is the closed surface defined by the cylinder c^2 = x^2 + y^2. The top and bottom of the cylinder are z= 0 and z=d.

    From common sense, integrating circular layers from z=0 to z=d should give the volume of a cylinder. The book doesn't have any sample problem so I don't know which theorem to apply, and how.

    Here's a more complicated question:

    Find the value of the integral (curl A da) over the surface s, where A = yi + zj + xk and S is the closed surface defined by the paraboloid z=1-x^2-y^2 where z >=0

    I appreciate any help.
     
  2. jcsd
  3. Oct 3, 2006 #2

    Meir Achuz

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    "Find the value of the integral (A dot da) over the surface s, where A = xi - yj + zk and S is the closed surface defined by the cylinder c^2 = x^2 + y^2. The top and bottom of the cylinder are z= 0 and z=d."
    Find div A. (It is a constant.) Then just multiply by the volume of a cylinder.
     
  4. Oct 3, 2006 #3

    Meir Achuz

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    "Find the value of the integral (curl A da) over the surface s, where A = yi + zj + xk and S is the closed surface defined by the paraboloid z=1-x^2-y^2 where z >=0"
    By either the div theorem or Stokes' theorem, the integral of curl over a closed surface=0. Prove it.
     
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