We describe a knowledge representation scheme in which expertise is encoded via expert-supplied mappings, or knowledge functions. This functional representation technique was originally developed for the Selection Advisor for Initial Value Software (SAIVS), a prototype system for recommending ordinary differential equation software from numerical subroutine libraries. We discuss the deficiencies in previous knowledge representation schemes that motivated the development of functional scheme, and then present the method. We propose other classes of mathematical software to which the existing SAIVS shell may be applied. Recently, the representation has been adapted for use in the Parallel Object Matching System (POMS), operational system for scheduling parallel scientific observations on the Hubble Space Telescope. Our experience with the scheme in these two very different areas suggests that it is a generally useful method with potentially widespread scientific applications. We discuss the technique’s advantages and limitations, as observed in SAIVS and POMS.