Vector Analysis |
Quantum state vector calculus (Dirac’s notation) |
- Projected vectors
- Non-projected abstract vectors
- Cartesian, cylindrical, and spherical algebraic unit vectors
- Projected vectorial differential operators (e.g. Nabla)
- Abstract vectorial differential operators
- Scalar and cross product operators for vectors that handle projected and abstract vectors
- Differentiation taking into account relationships between geometrical coordinates
- Supporting vectorial routines for changing basis, representing components of abstract vectors, etc.
|
- Anticommutative and noncommutative variables and functions
- Bras and Kets
- Projectors
- Annihilation/Creation operators
- Orthonormal discrete and continuous bases
- Hermitian and non-Hermitian operators, possibly tensorial
- Pauli and Dirac matrices including their commutator algebras and the computation of traces of products
- Commutators and anticommutators
- Customizable commutator and anticommutator algebra rules
- Simplification taking everything into account (fixed and customizable algebra rules, Einstein's sum rule, etc.)
|
Tensor Analysis (n-dimensional Geometry, Special and General Relativity) |
Field Theories (Classical and Quantum) |
- User defined tensors (under addition, multiplication, differentiation, simplification, and transformation of coordinates)
- Distinction between covariant and contravariant indices
- User defined systems of coordinates, possibly many at the same time
- Coordinate transformation of tensorial expressions taking into account covariant and contravariant indices
- Computation of components of arbitrary tensorial expressions
- Determination of free and repeated indices, simplification and differentiation using Einstein's sum rule
- Complete set of general relativity tensors
- Searchable database of solutions to Einstein's equations
|
- Ability to represent Lagrangians and Hamiltonians involving anticommutative and noncommutative variables, functions and operators
- Ability to compute field equations using standard and functional differentiation
- Ability to compute the analytic structure of the Feynman Diagrams of a model
|
Product Operators |
Differentiation Operators |
- Standard multiplication operator (*) handling commutative, anticommutative and noncommutative variables in equal footing
- Automatic normal form for products involving commutative, anticommutative, and noncommutative variables
- Scalar and cross product operators for 3-D vectors and vectorial functions or vectorial differential operators
- Scalar product operator for Bras and Kets and related quantum state (Dirac notation) vector calculus
|
- Differentiation with respect to anticommutative variables, scalar and tensor functions
- Functional differentiation with respect to commutative and anticommutative scalar and tensor functions
- The d’Alembertian
, the galilean
and covariant
tensorial differentiation operators
|
Differential Equations Support |
Differential Geometry Support |
- Computation of Poincare sections for dynamical systems
- Reducing, triangularizing, and/or solving DE systems, possibly involving anticommutative variables and functions
- Symmetry analysis for ODEs and PDEs, possibly including anticommutative variables and functions
|
- Calculus on manifolds (vector fields, differential forms and transformations)
- Tensor analysis in advanced general relativity
- Calculus on jet spaces
- Lie algebras
- Lie and transformation groups
- Computations can be performed in user-specified frames
- Computations can be performed with abstract differential forms (coordinate-free)
|