Define 4 different manifolds:
Example 1.
Define a transformation from R to Q, that is, the parametric representation of a space-curve.
Example 2.
Define a transformation presenting the mapping in the complex plane z -> w = z^3.
We use the real and imaginary parts of w to define the components of the transformation. We can set up the transformation as a map from M to N or as a map from M to itself.
Example 3.
Define a transformation encoding the change from polar to Cartesian coordinates.
Example 4.
Define a transformation from P to Q which parameterizes the hyperboloid of revolution z^2 = 1 + x^2 + y^2.
Example 5.
Define the canonical projection map [x, y, z] -> [x, y] from Q to M.
Example 6.
The command DGinfo can be used to access various attributes of a transformation.
Example 7.
Where an adapted frame is used, the Jacobian is computed relative to that frame. Here is a simple example:
Example 8.
Here we use the second calling sequence to define a Lie algebra homomorphism between two Lie algebras. See the LieAlgebraData help page for information on creating Lie algebras with Maple.
Initialize a Lie algebra Alg1 which will serve as the domain for the Lie algebra homomorphism.
Initialize a Lie algebra Alg2 which will serve as the range for the Lie algebra homomorphism.
Define a matrix which will determine the linear transformation from Alg1 to Alg2.
The output indicates that phi sends e1 to 0f1, e2 to f3, e3 to f2, and e4 to f1 - f3. The LieAlgebras Query command allows us to check that phi is a Lie algebra homomorphism.
