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Example 1.
Let be a 4-dimensional space. We define a metric tensor depending upon an arbitrary function. We find the metrics which have vanishing Einstein tensor, and vanishing Bach tensor.
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| (4.2) |
Here are the metrics of the form (4.2) with vanishing Einstein tensor.
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| (4.3) |
Here are the metrics of the form (4.2) with vanishing Bach tensor.
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| (4.4) |
Example 2.
In this example we define a 2-form which depends upon parameters . We find those values of the parameters for which
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| (4.6) |
Example 3.
We define a connection and calculate the parallel transport of a vector along a curve .
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| (4.8) |
| (4.9) |
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| (4.10) |
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| (4.11) |
We can use the keyword argument auxiliaryequations to specify an initial position for the vector
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| (4.12) |
Example 4.
The source-free Maxwell equations may be expressed in terms of a 2-form by the equations and , where is the exterior derivative and is the Hodge star operator. In this example we define a 2-form depending on 2 functions of 4 variables and solve the Maxwell equations for
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| (4.14) |
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| (4.15) |
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| (4.16) |