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(God, this has been a boring and tedious chapter!)

Especially because we have introduced several different tools and concepts, but not used them yet to understand particularly breathtaking physical occurrences, it is probably good to note what we have learned.

We have found that fields are the complements of objects, being smooth and featureless on small scales, whereas objects are precisely ``features''. For approximate descriptions, sometimes the language of objects is convenient, and sometimes the language of fields is, as is the case for fluids. Both, however, can be real, physical quantities, for which we can describe the configurations possible and predict the evolution. For fundamental descriptions, however, fields possess the important advantage of explicit locality, which objects requiring action-at-a-distance descriptions do not have. This makes field descriptions compatible with Special Relativity, and seemingly requires field language as the basis for any fundamental theory.

For either fundamental or large-scale approximate fields, the configurations are often conveniently described in terms of how much of various modes are present, and the dynamics are usually expressed so as to relate the evolution of the field someplace to the averaged values of the same or other fields around that place. Properly defined and on small scales, how the averages are computed must not have any effect on the answer.

Such dynamics may or may not produce waves as possible excitations, and it may or may not obey a property of superposition, whereby disturbances `sum', as they do for pond ripples or sound. Wavelike behavior arises in cases where some part of the field law behaves like a spring, and some other part behaves like an inertia, because waves are the field counterpart to the oscillations of objects, in a sense.

An important class of fields are the flow fields, which are charactirized in terms of currents. A useful way to refer to currents is in terms of how much of something they transport from one place to another, because that description relates currents to conservation laws. Such currents also naturally lend themselves to being mapped by means of arrows, which provides a way to represent the flow field mathematically. In general, flow fields can be very complex, but have two simple characteristics, those of divergence and circulation. The observation that the divergence out through any surface is constant (indeed zero) for water is another way to express the conservation law that applies to water. The curious observation that any circulation field automatically has zero divergence provides a powerful and very general property of circulation fields, wherever they may arise.

Next: References Up: Classical Fields and Previous: THE DIVERGENCE OF

Thu Aug 31 12:01:42 CDT 1995