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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. In this coordinate system r haspa di erent meaning than in cylinder coordinates because in spherical coordinates r = x2 + y2 + z 2 .

This is most easily accomplished by determining from the expressions above the ow eld at some selected lines such as the x- and y-axes. One may also be interested in computing the streamlines of the ow. These are the lines along which material particles ow. The streamlines can be found by using the fact that the time derivative of the position of a material particle is the velocity: dr=dt = v(r). 11) leads to two coupled di erential equations for x(t) and y(t) which are di cult to solve. Fortunately, there are more intelligent ways of retrieving the streamlines.

However, particles with a magnetic charge (usually called \magnetic monopoles") have not been found in nature despite extensive searches. 13) but we should remember that this divergence is zero because of the observational absence of magnetic monopoles rather than a vanishing coupling constant. 4 The divergence in cylinder coordinates In the previous analysis we have only used the expression of the divergence is Cartesian coordinates: v = @xvx + @y vy + @z vz . As you have (hopefully) discovered, the use of other coordinate systems such as cylinder coordinates or spherical coordinates can make life much simpler.

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A guided tour of mathematical physics by Snieder R.

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