Maple differential geometry1/23/2024 ![]() ![]() ![]() If the command beta := Pullback(F, alpha) is executed, then the result, beta, is a differential form on M and the prompt changes to The name of the prompt always reflects the name of the coordinate system or manifold of the last computed object.įor 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 Within the DifferentialGeometry environment, the Maple prompt changes to the name of the current or active coordinate system. Note that the Maple prompt has changed to F. To create a rank 2 fiber bundle F over a 3 dimensional base manifold with base coordinates and fiber coordinates, use the DGsetup command as follows: 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 display the internal representation of the vector field D_x and the 1-form dy, use the Maple lprint command. 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. We will cover the basis operations involving vectors, forms and tensors in the next lesson.īefore 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. 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. The vectors D_x, D_y are assigned and protected they define the coordinate basis for the tangent space of E2 at each point. The following differential 1-forms have been defined and protected:Īt this point, the coordinate names have been protected and cannot be assigned values. The following vector fields have been defined and protected: The following coordinates have been protected: We declare to be the names of the coordinates and we name the manifold (or, more precisely, the coordinate patch for the manifold) E2. We first use DGsetup to create a coordinate system for a 2-dimensional manifold. The DGsetup command can be used many times within a given Maple session. This command is used to setup the computation environment by creating coordinate systems, frames, Lie algebras, and so on. Obtain various attributes of a defined coordinate system.Ĭontrol simplification of the output from DifferentialGeometry commands.Īll DifferentialGeometry sessions begin by executing the DGsetup command. Remove coordinate systems from the Maple session. In this lesson, you will learn to do the following: ![]()
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