Define manifolds with coordinates and .
Example 1.
Find all invariant functions, 1-forms, metrics and invariant type [1, 1] tensors for the infinitesimal group of rotations on
Invariant Functions:
Invariant 1-forms:
Note that the format of the answer can be improved with the assuming command.
Invariant Metrics:
Invariant [1, 1] Tensors:
Example 2.
Find the vector fields which commute with the Lie algebra of vector fields .
Give the partial differential equations which were solved to calculate the commuting vectors in the list Z.
Find the vector fields of the special form + c(x)D_z which commute with .
Example 3.
Find the second and third order differential invariants for the infinitesimal Euclidean group acting on the plane.
Find the invariant Lagrangians on the 1-jet.
Find the invariant "source" forms on the 2-jet.
Example 4.
Find the invariant 1-forms for a list of vector fields depending on a parameter alpha.
Example 5.
The command InvariantGeometricObjectFields can also be used to calculate tensors on a Lie algebra which are invariant with respect to a subalgebra.
Retrieve a Lie algebra from the DifferentialGeometry library.
Find the symmetric rank 2 tensors on alg1 which are invariant with respect to the subalgebra spanned by