  A theory for studying the dynamic scaling properties of branes and
relativistic topological defect networks is presented. The theory, based on a
relativistic version of the level set method, well-known in other contexts,
possesses self-similar ``scaling'' solutions, for which one can calculate many
quantities of interest. Here, the length and area densities of cosmic strings
and domain walls are calculated in Minkowski space, and radiation, matter, and
curvature-dominated FRW cosmologies with 2 and 3 space dimensions. The scaling
exponents agree the naive ones based on dimensional analysis, except for cosmic
strings in 3-dimensional Minkowski space, which are predicted to have a
logarithmic correction to the naive scaling form. The scaling amplitudes of the
length and area densities are a factor of approximately 2 lower than results
from numerical simulations of classical field theories. An expression for the
length density of strings in the condensed matter literature is corrected.
