How Do Gauss' and Stokes' Theorems Apply to These Integral Problems?

[superpig10000]

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.

 

[Meir Achuz]

"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.

 

[Meir Achuz]

"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.