  INTRODUCTION
  This papers deals with partial differential equations of second order,
linear, with constant and not constant coefficients, in two variables, which
admit real characteristics. I face the study of PDEs with the mentality of the
applied physicist, but with a weakness for formalization: look inside the black
box of the formulas, try to compact them (for example, proceeding from an
inverse transformation of coordinates) and make them smart (in the context,
reformulating the theory by means of differential operators and related
invariants), applying them with awareness and then connecting them to geometry
or to spatial categories, which are in mathematics what is closest to the
sensible reality. Finally, proposing examples that are exercise and
corroborating for theory.
  TOPICS
  The geometric meaning of invariant to a differential operator. Operator
Principal Part and its factorization: commutativity and product with and
without residues(first order terms). Related conditions by operators and
invariants derivatives. Coordinate transformation by invariants and expression
of the hyperbolic and parabolic operators in the new coordinates. Properties of
the Jacobian Matrix and relations between invariants derivatives and inverse
coordinates transformation or the initial variables derivatives. Commutativity
conditions and product without residues in terms of inverse coordinate
transformations that allow to build commutative differential operators or whose
product is without residues (or both). Diffeomorphisms and plane
transformations: new operators and invariants in the new coordinate space which
lead to the chain rule in compact form. Conclusive considerations and examples
who compares different methods of solution.
