The fundamental theorem of calculus in differential algebra
Sprache des Vortragstitels:
DART VII (Differential algebra and related topics)
Sprache des Tagungstitel:
For incorporating the fundamental theorem of calculus in differential algebras, we consider a linear right-inverse of the derivation. In these algebras, we automatically have an induced evaluation operation corresponding to the second part of the fundamental theorem. This setting generalizes the notion of integro-differential algebras where in addition the evaluation has to be multiplicative.
We construct the corresponding algebra of linear operators as a quotient of a tensor algebra. Based on a completion process for tensor reduction systems, we find all linear consequences (like integration by parts) of the defining relations. This, in turn, enables us to determine normal forms for these generalized integro-differential operators. Normal are needed for effective computations and are implemented as a Mathematica package. We also illustrate how analogs of the Taylor formula or variation of constants can be proven in this operator framework.