The operation of total differentiation is a fundamental one in the study of jet spaces and their application to differential equations and the calculus of variations.  Informally, total differentiation of a function on a jet space with respect to an independent variable is the same as ordinary differentiation with respect to that variable if the jet coordinates are treated temporarily as functions of the independent variables.  Here is the formal definition.  Let pi: E -> M be a bundle and J^k(E) -> M be the bundle of k-jets of local sections of E.  Let U be a coordinate neighborhood of M (which trivializes E) with local coordinates (x^1, x^2, ... x^n), let s: U -> E be a local section of E, and let j^k(s): U -> J^k(E) be the tautological lift ofs.  If f is a smooth real-valued function on J^k(E), then the total derivative of f with respect to x^i is the function D_i(f) defined on J^(k+1)(E) (or more precisely on the open set pi^(-1)(U)) by D_i(f) (j^(k + 1)(s)(x)) = partial_{x^i} f(j^k(s)(x)).

If f is a function on a jet space and v an independent variable, then TotalDiff(f, v) calculates the total derivative of f with respect to v.

Total differentiation is extended to differential forms and bi-forms by Lie derivative with respect to the corresponding total vector field. The total derivative with respect to a coordinate function x^i commutes with the exterior derivative, the horizontal exterior derivative, and the vertical exterior derivative.

The command TotalDiff is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form TotalDiff(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-TotalDiff(...).