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A theory for studying the dynamic scaling properties of branes andrelativistic topological defect networks is presented. The theory, based on arelativistic version of the level set method, well-known in other contexts,possesses self-similar ``scaling'' solutions, for which one can calculate manyquantities of interest. Here, the length and area densities of cosmic stringsand domain walls are calculated in Minkowski space, and radiation, matter, andcurvature-dominated FRW cosmologies with 2 and 3 space dimensions. The scalingexponents agree the naive ones based on dimensional analysis, except for cosmicstrings in 3-dimensional Minkowski space, which are predicted to have alogarithmic correction to the naive scaling form. The scaling amplitudes of thelength and area densities are a factor of approximately 2 lower than resultsfrom numerical simulations of classical field theories. An expression for thelength density of strings in the condensed matter literature is corrected.Comment: 46pp LaTeX, revtex4(preprint), 1 eps figure, revised for publication.  Note title chang

Level set method for the evolution of defect and brane networks