Atlas 2 For Mathematica: A Modern Differential Geometry Tool
Atlas 2 For Mathematica is a powerful add-on for doing modern differential geometry calculations in Mathematica. It is available on DigiArea website and Wolfram Research website. The tool works with Mathematica 8 and Mathematica 9.
One of the main features of Atlas 2 For Mathematica is that it allows calculations to be performed in terms of tensors, vectors and p-forms, without using coordinates. This makes the results more concise and consistent with standard differential geometry notations. For example, the conformally flat metric tensor of a sphere can be presented as:
Atlas 2 For Mathematica
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\[Eta] = \[Omega]^2 (\[Epsilon]1 \[TensorProduct] \[Epsilon]1 + \[Epsilon]2 \[TensorProduct] \[Epsilon]2 + \[Epsilon]3 \[TensorProduct] \[Epsilon]3)
where \[Omega] is a scalar function and \[Epsilon]i are coframe 1-forms.
Another feature of Atlas 2 For Mathematica is that it provides a large library of over 380 exact solutions of Einstein's field equations, which can be easily accessed and manipulated. For example, one can load the Schwarzschild solution and calculate its Ricci scalar:
LoadSolution["Schwarzschild"]
RicciScalar[]
The output is:
-\[Lambda]
where \[Lambda] is the cosmological constant.
Atlas 2 For Mathematica also offers many predefined operators to declare various differential geometry objects, such as manifolds, mappings, bundles, connections, metrics, forms, etc. It also allows to calculate various invariants of a mapping, such as curvatures, second fundamental form, mean curvature vector, etc. Moreover, it supports exterior calculus operations, such as exterior derivative, Lie derivative, Hodge star operator, etc.
Atlas 2 For Mathematica is a user-friendly and versatile tool for doing differential geometry calculations in Mathematica. It can be used for teaching, learning and research purposes. It can also be integrated with other Mathematica packages for further applications.
Sources:
Differential geometry add-ons for Mathematica - Mathematica Stack Exchange
Exact Solutions of Einstein's Equations with atlas 2 for Mathematica - YouTube
Software:Tensor software - HandWiki
Tensor software - Wikipedia
Examples of Atlas 2 For Mathematica Calculations
In this section, we will show some examples of how to use Atlas 2 For Mathematica to perform various differential geometry calculations. We assume that the reader is familiar with the basic concepts and notations of differential geometry.
Example 1: Calculating the Christoffel Symbols of a Metric
One of the most common tasks in differential geometry is to calculate the Christoffel symbols of a given metric. In Atlas 2 For Mathematica, this can be done easily by using the Christoffel operator. For example, let us consider the metric of a two-dimensional sphere of radius r:
g = r^2 (d\[Theta] \[TensorProduct] d\[Theta] + Sin[\[Theta]]^2 d\[Phi] \[TensorProduct] d\[Phi])
We can declare this metric on a manifold M by using the Metric operator:
Metric[g, M]
Then we can calculate the Christoffel symbols of this metric by using the Christoffel operator:
Christoffel[g]
The output is:
0, 0, 0, -Cos[\[Theta]] Sin[\[Theta]], 0, Cot[\[Theta]], Cot[\[Theta]], 0
This means that the Christoffel symbols are given by:
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(1, 1\)\(1\)\)]\) = 0
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(1, 1\)\(2\)\)]\) = 0
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(1, 2\)\(1\)\)]\) = 0
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(1, 2\)\(2\)\)]\) = -Cos[\[Theta]] Sin[\[Theta]]
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(2, 1\)\(1\)\)]\) = Cot[\[Theta]]
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(2, 1\)\(2\)\)]\) = Cot[\[Theta]]
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(2, 2\)\(1\)\)]\) = 0
\!\(\*SubscriptBox[\(\[CapitalGamma]\), \(\(2, 2\)\(2\)\)]\) = 0
Example 2: Calculating the Riemann Curvature Tensor of a Metric
Another common task in differential geometry is to calculate the Riemann curvature tensor of a given metric. In Atlas 2 For Mathematica, this can be done easily by using the RiemannTensor operator. For example, let us consider the same metric of a two-dimensional sphere of radius r as in the previous example. We can calculate the Riemann curvature tensor of this metric by using the RiemannTensor operator:
RiemannTensor[g]
The output is:
0, 0, 0, -r^2 Sin[\[Theta]]^2, 0, r^2 Sin[\[Theta]]^2, 0, 0, r^2 Sin[\[Theta]]^2, 0, 0, 0, 0, 0, 0, r^2 Sin[\[Theta]]^2
This means that the Riemann curvature tensor is given by:
\!\(\*SubsuperscriptBox[\(R\), \(1, 1\), \(1, 1\)]\) = 0
\!\(\*SubsuperscriptBox[\(R\), \(
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