  We pose a new algebraic formalism for studying differential calculus in
vector bundles. This is achieved by studying various functors of differential
calculus over arbitrary graded commutative algebras (DCGCA) and applying this
language to a particularly simple class of two-component graded objects
introduced in this work, that we call Diolic algebras. A salient feature of
this conceptual approach to calculus is that it recovers many well-known
objects and notions from ordinary differential, symplectic and Poisson geometry
but also provides some unique aspects, which are of their own independent
interest.
