  The multiplicative (or geometric) calculus is a non-Newtonian calculus
derived from an arithmetic in which the operations of
addition/subtraction/multiplication are replaced by
multiplication/division/exponentiation. A major difference between the
multiplicative calculus and the classical additive calculus, and one that has
important consequences in the simulation of wave propagation problems, is that
in geometric calculus the role of polynomials is played by exponentials of a
polynomial argument. For example, whereas a polynomial of degree one has
constant (classical) derivative, it is the exponential function that has
constant derivative in the multiplicative calculus. As we will show, this
implies that even low-order finite quotient approximations|the analogues of
finite differences in the multiplicative calculus|produce exact multiplicative
derivatives of exponential functions. We exploit this fact to show that some
partial differential equations (PDE) can be solved far more efficiently using
techniques based on the multiplicative calculus. For wave propagation models in
particular, we will show that it is possible to circumvent the
minimum-points-per-wavelength sampling constraints of classical methods. In
this first part we develop the theoretical framework for studying
multiplicative partial differential equations and their connections with
classical models.
