Transformation(M, N, Eq)

Transformation(M, N, A)

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.

DifferentialGeometry, Tools, ApplyTransformation, ComposeTransformations, DGinfo, InverseTransformation, Pullback, Pushforward, PushPullTensor