Example 1.
First initialize various jet spaces of one or two independent variables and one dependent variable and prolong them to order 4.
Case 1. Projectable transformations from to .
When a transformation is first defined, it is not given a type.
Now assign the transformation a type.
This indicates that the transformation is a projectable transformation, the 0 indicates that the transformation has not been prolonged to a jet space.
Case 2. Point transformations:
Case 3. Contact transformations:
By the conventions adopted here, a contact transformation need not be a local diffeomorphism so that, in particular, the dimensions of the bundles and need not coincide.
Case 4. Differential Substitutions:
Case 5. Generalized Differential Substitutions:
Case 6. Generic: