Vector Analysis 
Quantum state vector calculus (Dirac’s notation) 
 Projected vectors
 Nonprojected 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 nonHermitian 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 (ndimensional 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 3D 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 userspecified frames
 Computations can be performed with abstract differential forms (coordinatefree)
