Self-affine vector measures and vector calculus on fractals

In this paper, we construct an IFS framework for studying self-similar vector measures. These measures have several applications, including the tangent and normal vector measure "fields" to fractal curves. Using the tangent vector measure, we define a line integral of a smooth vector field over a fractal curve. This then leads to a formulation of Green's Theorem (and the Divergence Theorem) for planar regions bounded by fractal curves. The general IFS setting also leads to "probability measure" valued measures, which give one way to coloring the geometric attractor of an IFS in a self-similar way.