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.
Here are the metrics of the form (4.2) with vanishing Einstein tensor.
Here are the metrics of the form (4.2) with vanishing Bach tensor.
Example 2.
In this example we define a 2-form which depends upon parameters . We find those values of the parameters for which
Example 3.
We define a connection and calculate the parallel transport of a vector along a curve .
We can use the keyword argument auxiliaryequations to specify an initial position for the vector
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