Consider the following PDE "system" consisting of a single pde.
This system automatically satisfies the conditions for being a divergence mentioned in the Description:
Hence is the divergence of a current
and admits a constant integrating factor:
When combined with the rest of the Maple library, the Euler operator can serve varied purposes. Consider for example deriving the most general form of the divergence of a current that is also a first order PDE in two variables. The starting point is a generic expression, , so it depends only on the first order derivatives.
The conditions that Delta must satisfy in order to be a divergence are:
These conditions can be integrated.
Verify that these conditions are sufficient by applying Euler's operator to this result. First convert the result from jet notation to function notation.
So the above is the most general form of a divergence that is also a first order PDE. The following verifies that this form is correct.
The most general form of a second order linear PDE in two independent variables that is also a divergence of a current can be derived in a similar way, starting with the following definition.
The conditions for divergence, in the form of equations satisfied by the A[j]( x, t ) are obtained by applying Euler's operator.
Note that in the above calculation, the dependent variable of the problem must be specified, otherwise A[j]( x, t ) for all j would be also picked up as dependent variables. The result above is a single expression from which A[j]( x, t ) for one j can be isolated; the simplest form is achieved by isolating A[0].
This result can be verified applying Euler's operator to it.