ABSTRACT
The fundamental purpose of the integration approach employed here is to obtain theorems that describe the transformation of derivatives from one spatial scale to another. This transformation of scale is accomplished by integrating smaller scale equations to obtain forms relevant at larger scales. Integrations of these small scale equations assume a variety of forms depending upon the application. For example, a single averaged value may be desired for an entire region. Alternatively, averaged values for smaller sub-regions within a large region may be needed. The primary difference between these two very different types of averages lies in selection of the length scale for integration. Three spatial scales of integration are considered here. In order of increasing size they are referred to as microscopic, macroscopic, and megascopic scales.
