  Using standard calculus, explicit formulas for one-, two- and
three-dimensional homotopy operators are presented. A derivation of the
one-dimensional homotopy operator is given. A similar methodology can be used
to derive the multi-dimensional versions. The calculus-based formulas for the
homotopy operators are easy to implement in computer algebra systems such as
Mathematica, Maple, and REDUCE. Several examples illustrate the use, scope, and
limitations of the homotopy operators. The homotopy operator can be applied to
the symbolic computation of conservation laws of nonlinear partial differential
equations (PDEs). Conservation laws provide insight into the physical and
mathematical properties of the PDE. For instance, the existence of infinitely
many conservation laws establishes the complete integrability of a nonlinear
PDE.
