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Verification of the divergence theorem on two given surfaces

Verify the divergence theorem (∫∫ (F.n) ds = ∫∫∫ (grad.F) dV) for the following two cases:

a. F = er r + ez z and r = i x + j y where s is the surface of the quarter cylinder of radius R and height h shown in the diagram below.

b. F = er r^2 and r = i x + j y + k z where s is the surface of the sphere of radius R centered at the origin as shown below.

Note: er, ez, i, j and k are unit vectors.


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In cylindrical coordinates,

dV =

F =


So, = 3 = 3

To find the surface integral, we need to consider the following surfaces ...

Solution Summary

Divergence theorem is verified for two given surfaces (quarter cylinder and a sphere) using given field vectors. Knowledge gained in learning cylindrical coordinates systems and spherical coordinates systems are heavily used in the solution. Solution is prepared mainly using equation editor and it is presented in a 3-page word document.