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Example 1.
First initialize various jet spaces of one or two independent variables and one dependent variable and prolong them to order 4.
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Case 1. Projectable transformations from to .
K >
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| (2.1) |
When a transformation is first defined, it is not given a type.
E >
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Now assign the transformation a type.
E >
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| (2.3) |
E >
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| (2.4) |
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:
E >
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| (2.5) |
E >
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| (2.6) |
E >
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Case 3. Contact transformations:
E >
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| (2.8) |
E >
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| (2.9) |
E >
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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.
E >
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| (2.11) |
F >
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| (2.12) |
F >
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Case 4. Differential Substitutions:
F >
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| (2.14) |
E >
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| (2.15) |
E >
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E >
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| (2.16) |
Case 5. Generalized Differential Substitutions:
E >
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| (2.17) |
E >
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E >
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| (2.18) |
Case 6. Generic:
E >
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| (2.19) |
E >
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F >
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