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Example 1.
Create the 1st order jet space of 2 independent variables and 2 dependent variables .
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Define a vector and compute its total and evolutionary parts totand evol. Check that = totevol
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| (2.2) |
J22 >
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| (2.3) |
Define a vector and compute its total and evolutionary parts tot and evol. Check that = totevol
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J22 >
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Define a vector and compute its total and evolutionary parts tot and evol. Check that = tot evol
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| (2.9) |
J22 >
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| (2.10) |
J22 >
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| (2.11) |
| (2.12) |
Example 2.
In this example we illustrate the geometric interpretation of the evolutionary part of a projectable vector field. First define a 3-dimensional bundle over a two dimensional base. Define the base space separately.
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Define a vector field and compute its evolutionary part evolDefine the projection of the vector field onto the base manifold
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| (2.13) |
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| (2.14) |
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Calculate the flow of and the flow of .
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| (2.17) |
| (2.18) |
Define a section of sending .
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| (2.19) |
Calculate the induced flow on the space of sections.
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| (2.20) |
M >
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| (2.21) |
E >
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| (2.22) |
Compare with the components of evol
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| (2.23) |