Indexed expression package for Mathematica. Designed for simplicity and flexibility. Still experimental.
This is an incomplete introduction of this package, explaining basic design ideas. A complete documentation of the package will be added in the future.
Most existing symbolic tensor computation packages follow a top-down approach, which typically defines manifold, vector bundles, and then tensors, covariant derivatives, etc. Peanotica instead follows a bottom-up approach which focus on the behaviours of the tensor expressions, i.e., what indices slots an expression has and what symmetry does these slots have. All objects, including tensors, can be defined on top of that.
The two most important functions in Peanotica are FindIndicesSlots and SymmetryOfExpression. For example, to define a rank-2 symmetric tensor T[a, b],
FindIndicesSlots[_T] ^= {{1} -> Null, {2} -> Null};
SymmetryOfExpression[_T] ^= {SCycles@{1, 2}};FindIndicesSlots[expr] returns a list of positions and types of the index slots, whereas SymmetryOfExpression[expr] gives the generators of the symmetry group of the slots. Then you can canonicalize expressions using ITensorReduce
In[1] := ITensorReduce[T[b, a]]
Out[1] = T[a, b]note that there's no need to define abstract indices.
This makes expressions in Peanotica highly flexible. Here are more examples:
- Symmetric tensor
t[a, b, ...]with an arbitary number of indices
FindIndicesSlots[expr_t] ^= Array[{#} -> Null &, Length@expr];
SymmetryOfExpression[expr_t] ^= Array[SCycles@{#, # + 1} &, Length@expr - 1];- Covariant derivative
covd[A, a], which represents$\nabla_a A$ , may be defined by
FindIndicesSlots[covd[expr_, a_]] ^:= Join[FindIndicesSlots[expr, {1}], {{2} -> Null}];
SymmetryOfExpression[covd[expr_, a_]] ^:= SymmetryOfExpression@expr;Here FindIndicesSlots[expr, pos] prefixes pos to each positions it returns. This definition is used by the differential geometry subpackage DiffGeo.
The most important algorithm of this package is double coset representative algorithm.
GNU General Public License. See LICENSE for detail.