In mathematical terms, a partial differential equation (PDE) is any equation involving a function of more than one independent variable and at least one partial derivative of that function. The order of a PDE is the order of the highest order derivative that appears in the PDE. The principal part of a PDE is the collection of terms in the PDE containing derivatives of order equal to the order of the PDE. The following example illustrates these definitions and introduces the two most common notations for expressing partial derivatives.
If is a function of the two independent variables x and y, then
is a PDE of first order whose principal part is
Using subscript notation a more compact way to express this PDE is
in which case we would say that the principal part is .