A guided tour of mathematical physics by Snieder R.

By Snieder R.

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Problem b: Show that the ux through the surfaces perpendicular to '^ is to rst order @v in d' given by @'' drd'dz . Problem c: Show that the ux through the surfaces perpendicular to ^z is to rst order in dz given by @v@zz rdrd'dz . 2) is given by rdrd'dz . 8) without using Cartesian coordinates as an intermediary. 5. IS LIFE POSSIBLE IN A 5-DIMENSIONAL WORLD? 37 In spherical coordinates a vector v can be expended in the components vr , v and v' in the directions of increasing values of r, and ' respectively.

17) over the mass. Gauss's theorem can then be used to compute the integral of the divergence of the gravitational eld. For the curl the situation is similar. 5) we computed the magnetic eld generated by a current in a straight in nite wire. The eld equation r B = 0J (5:12) again was used to compute the eld away from the wire. 13) contained an unknown constant A. 12) was only used outside the wire, where J = 0. 5) therefore did not provide us with the relation between the eld B and its source J.

6) which is axially symmetric around the z -axis. 6) is still arbitrary. We know that both the rotation around the z -axis and the shear are a source of vorticity. 11) the vorticity vanishes, with A a constant that is not yet determined. Make a sketch of this ow eld. The vorticity of this ow vanishes despite the fact that the ow rotates around the z -axis (but not in rigid rotation) and that the ow has a nonzero shear. The reason that the vorticity vanishes is that the contribution of the rotation around the z -axis to the vorticity is equal but of opposite sign from the contribution of the shear, so that the total vorticity vanishes.

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