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By Snieder R.

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Problem b: Show that for this velocity eld: v = 2 ^z. r This means that the vorticity is twice the rotation vector ^z. This result is derived here for the special case that the z -axis is the axis of rotation. 11) of Boas 11] it is shown with a very di erent derivation that the vorticity for rigid rotation is given by ! = v = 2 , where is the rotation vector. ) We see that rigid rotation leads to a vorticity that is twice the rotation rate. 3). This paddle-wheel moves with the ow and makes one revolution along its axis in a time 2 = .

However, the analogy of uid ow often is very useful to understand the meaning of the ux and divergence. 2) in this expression ^r is the unit vector in the radial direction and "0 is the permittivity. Compute the ux of the electric eld through a spherical surface with radius R with the point charge in its center. Show explicitly that this ux is independent of the radius R and nd its relation to the charge q and the permittivity "0 . Choose the coordinate system you use for the integration carefully.

Pay in particular attention to the sign of the di erent terms. 4) to a volume integral and show that the integral is indeed identical to zero. 3) is indeed satis ed and that in the application of Stokes' law you can choose any surface as long as it is CHAPTER 7. THE THEOREM OF STOKES 62 bounded by the contour over which the line integration is carried out. This is a very useful result because often the surface integration can be simpli ed by choosing the surface carefully. 2 Stokes' theorem from the theorem of Gauss Stokes' law is concerned with surface integrations.

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A guided tour of mathematical physics by Snieder R.


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