In this lesson, you will learn to do the following:

All DifferentialGeometry sessions begin by executing the DGsetup command.  This command is used to setup the computation environment by creating coordinate systems, frames, Lie algebras, and so on.  The DGsetup command can be used many times within a given Maple session.

We first use DGsetup to create a coordinate system for a 2-dimensional manifold.  We declare [x, y] to be the names of the coordinates and we name the manifold (or, more precisely, the coordinate patch for the manifold) E2.

At this point, the coordinate names have been protected and cannot be assigned values.  The vectors D_x, D_y are assigned and protected; they define the coordinate basis for the tangent space of E2 at each point [x, y].  The differential 1-forms dx and dy are assigned and protected; they define the coordinate basis for the cotangent space of E2 at each point [x, y].

We will cover the basis operations involving vectors, forms and tensors in the next lesson.

Before proceeding, we remark that all the various geometric objects which arise in differential geometry have an internal representation within Maple which describes various attributes of the object as well as all the component values of the geometric object.  A detailed understanding of this internal representation is not required to use any of the DifferentialGeometry commands -- here we simply wish to make the user aware of its existence.  To display the internal representation of the vector field D_x and the 1-form dy, use the Maple lprint command.

Notice that the internal representation of D_x clearly marks D_x as a geometric object of type "vector" attached to the manifold E2, while the internal representation of dy shows that dy is a "form" attached to the manifold E2 and has degree 1.

To create a rank 2 fiber bundle F over a 3 dimensional base manifold with base coordinates [x, y, z] and fiber coordinates [u, v], use the DGsetup command as follows:

Note that the Maple prompt has changed to F.  Within the DifferentialGeometry environment, the Maple prompt changes to the name of the current or active coordinate system.  The name of the prompt always reflects the name of the coordinate system or manifold of the last computed object.

For example, as we shall see in Lesson 6, if F : M -> N is a transformation from a manifold M to a manifold N and a differential form alpha is defined on N, then the prompt is

N  >

If the command beta := Pullback(F, alpha) is executed, then the result, beta, is a differential form on M and the prompt changes to

M  >

To create the second-order jet space J^2(R^2, R) with independent variables [x, y] and dependent variable [u], use the DGsetup command with the following syntax:

In this context, the dependent variable u is now referred to as u[ ], the derivative of u with respect to x is u[1], the derivative of u with respect to y is u[2], the second derivative of u with respect to x is u[1,1] and so on.

To suppress the display of the protected variables, vector fields, and differential 1-forms, run DGsetup with the quiet option.

This option, together with options for controlling the format of the Maple prompt, can be set using the Preferences command.

The DGinfo command is located in the Tools package of DifferentialGeometry.  This utility program can be used to retrieve information about any object created by the DifferentialGeometry package.  Here are a few examples.

List all the coordinate systems defined in the current Maple session.

Obtain the name of the active frame/coordinate system.

Obtain the dimensions of the base manifold and of the fiber space of the fiber bundle.

Obtain the coordinate basis for the tangent space of the manifold E3.

The command ChangeFrame is used to change the active frame.

The command RemoveFrame removes a frame from Maple memory (the number of remaining frames is returned) and unassigns and unprotects the names associated to that frame.

Prior to displaying the result of a computation, most DifferentialGeometry commands simplify the result using the Maple simplify command.  The user may control this simplification using the Preferences command.

Let us multiply the vector field D_x by the scalar a.  We see that the result has been simplified using the Maple command simplify.

Let us define a new simplify procedure, NewSimplify, and use the Preferences command to dictate that NewSimplify is to be used to simplify the output of all DifferentialGeometry commands.

Let us multiply the vector field D_x by the scalar a again.  We see that the result has been simplified using the user defined command NewSimplify.

Here is another example.

The user can similarly control how DifferentialGeometry invokes dsolve, pdsolve and a number of LinearAlgebra procedures.  Before continuing, we reset the simplification to its default procedure.