Given a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of these lists or operators, possibly representing an n-dimensional symmetry group, Invariants returns a sequence of  mathematical expressions  simultaneously invariant under all the  symmetry generators () related to the given  infinitesimals. The  are a complete set of differential invariants, so they are functions of the independent and dependent variables of the problem (DepVars), as well as of their partial derivatives up to order (default is 1), and satisfy  for all  and , where the number k depends on the problem.

Note that in the multidimensional case, depending on the form of the infinitesimals given, the problem may have no solution. Also the  are all independent of each other and their number is unique, but one can still rewrite the  in different ways, by combining invariants to construct equivalent but algebraically different invariants of the same order.

Optionally, a simplifier can be specified to be used instead of the default which is simplify/size. For that purpose use the optional argument simplifier = .... By default, the output of Invariants is in jet notation. You can change that also by passing the optional argument jetnotation = ... where the right-hand-side can be false or any of the jet notations available.

The invariants returned by Invariants can also be used as optional arguments for the group InvariantSolutions command when computing solutions to PDE systems.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.