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This graph, besides showing naively in a rather simplistic way the structure of the theory of differential geometry (and most of what I have implemented in the diffgeom module), brings attention to the one non-trivial part of the module on which I have spent most of my time lately. Namely …

I have written already a few posts about the theory behind the module, the structure of the module, etc. However, besides some rare examples, I have not described in much details how the work progresses. So here is a short summary (check the git log for more details):

• The basics …

Form fields or vector fields over a manifold (as opposed to forms and vectors) do not form a vector space. They form a module.

The difference is that the scalars of a vector space form an algebraic field while the scalars of a module form a ring. For us humans …

This post provides a summary of all mathematical types of expression implemented in the diffgeom module. I have chosen not to mention any python classes or other implementation details at all. This table shows how an object expected by the user to be of certain mathematical type operates on another …

An “easy” solution to the Einstein equation (in free space) is the spherically symmetric Schwarzschild solution. The pdf bellow shows how one can use the diffgeom module in order to get the equations describing this solution.

One starts with the most general spherically symmetrical metric and by using Einstein equation …

In most of my posts that discuss the SymPy diffgeom module I do not try to make a distinction between a tensor and a tensor field, as it is usually obvious from the context. However, it would be nice to spell it out at least once.

I have two favorite …

The Math

The notion of “a tensor” is commonly defined in two different ways. The first definition goes roughly like this (“roughly” means “do not tell this to your math teacher”):

A tensor is a geometrical object that can be represented in some coordinate system as an n-dimensional array (it …

This week I was doing some interesting refactoring, that brings quite a bit of new possibilities, however I will write about this in the coming days. For now… printing. Most importantly, any suggestions for improvements are very welcomed.

Printing in SymPy is done really easily. You just add a _print_Whatever …

A week or two ago I implemented some basic functionality for work with integral curves of vector fields. However, I needed to make additional changes in other parts of SymPy in order for the ODE solver to work with systems of equations and with initial conditions. I also wanted to …

The work on the differential geometry module has not progressed much this week. I have fixed some minor issues, docstrings and naming conventions, however I have not done much with respect to the implementation of form fields as there are still some questions about the design to be ironed out …

The Differential Geometry module for SymPy already supports some interesting basic operations. However, it would be appropriate to describe its structure before giving any examples.

First of all, there are the Manifold and Patch classes which are just placeholders. They contain all the coordinate charts that are defined on the …

The next few moths will be interesting. I got accepted in the Google Summer of Code program and I am already starting to worry (irrationally) about the project and the schedule. I will be working on a differential geometry module for SymPy (and time permitting, some more advanced tensor algebra …