Updates to Physics in Maple 16
The Physics package of Maple 16 includes 17 new commands, extended its functionality in Vector and Tensor Analysis, General Relativity and Quantum Fields, and brings with it the largest number of developments since its introduction in Maple 11, underscoring Maple's goal of having a state-of-the-art environment for algebraic computations in Physics.
In addition, a vast number of changes were introduced towards making the computational experience as natural as possible, resembling the paper-and-pencil way of doing computations and with textbook-quality display of results.
Tensors in Special and General Relativity (Examples)
Maple's Physics introduces contravariant indices in a simple way: in your input, prefix contravariant indices with ~; in the output they are displayed as superscripts, without the ~.
In a tensorial expression, you can enter any pair of contracted indices as one covariant, the other contravariant, not being relevant which one is of which kind, so you can also enter both covariant or both contravariant and the system will automatically rewrite them as one covariant and the other contravariant.
A complete set of General Relativity tensors got added to Physics, mainly D_, representing the covariant derivative, Christoffel, LeviCivita in curvilinear coordinates, Einstein, Ricci, Riemann and Weyl, all displayed on the screen with standard physics textbook notation.
You can compute with these tensors algebraically, as done with paper and pencil, automatically taking into account their symmetry properties, and Checking tensor indices in expressions, differentiating or Simplifying them using Einstein's sum rule for repeated indices.
A set of working tools for General Relativity got added, including TransformCoordinates, and TensorArray, to respectively perform transformations of coordinates in an arbitrary tensorial expression, and create a tensor array to compute each of its tensorial components, SumOverRepeatedIndices, to expand the summation in expressions written using Einstein's sum rule for repeated indices and SubstituteTensorIndices to substitute covariant and/or contravariant indices in tensorial expressions.
You can now use the Physics `.` product operator instead of * in order to "multiply and simplify" - in one step - contracted products of tensors. This permits simplifying only the product sub-expressions you want while keeping the remaining products expressed using *.
A set of convert routines permits rewriting expressions in terms of Christoffel symbols, or derivatives of the metric or expand covariant derivatives.
The d_ indexed differential operator got extended to work also without indices, the same way as the new covariant D_, in order to compute differentials.
When working in curvilinear coordinates, the LeviCivita tensor can optionally be set to galilean or nongalilean, with its display automatically changing from ε to Ε
