Dimensional Analysis and the Buckingham Pi Theorem

Working from a simple example of a dimensional transformation, the essential elements are identified, and an abstract prototype transformation is defined. With the aid of careful definitions and a geometric interpretation of what happens in a dimensional transformation the Buckingham pi theorem is written down but not proved. The application of this theorem to the heat equation in spherical coordinates and one dimensional gas motion without boundary conditions serve to illustrate the usefulness of the theorem in physics.