Example 1.

Create an abstract manifold  with a function 1-forms  and a 2-form .

The command DGinfo gives the names all scalars and forms which are defined.  

Scalar products, wedge products and sums of abstract forms can be defined.

The command DGinfo can also be used to extract information about the form .

New forms can be defined on M.

We can use the DGzip and GetComponents commands with abstract forms.

We can take the exterior derivative of a form.

The 2-form dalpha has been added to list of defined forms and is now available for subsequent computations.

Exterior derivatives of defined forms can be specified.

Example 2.

In this example we illustrate calculations using the second calling sequence for working with abstract forms. The 1-forms defining the co-frame are enclosed in separate list (the degrees of the forms defining the co-frame need not be given).

All the functionality of Example 1  is retained but now the manifold  is taken to have dimension 3. The 1-forms  define a co-frame on  and the dual vector fields { have been initialized.   

We can define vector fields on .

We can calculate the interior products of vectors and forms.

The interior products of {} with the 2-form alpha are automatically defined as new forms on  

Iterated interior products are known to be skew-symmetric:

The forms are taken to be independent so the commands such as Annihilator and DGbasis will work in this setting.

The Lie derivative of forms are computed from the Cartan formula.

Here both terms in this equation are new forms which are added to the list of defined forms on .

Equations for both exterior derivatives and interior products can be specified.

The Lie bracket can also be computed.

DifferentialGeometry, Annihilator, DGbasis, DGinfo, DGsetup, ExteriorDerivative, Hook, LieBracket, LieDerivative