  We construct and study differential and integral calculus on the space of
states of a C*-algebra by equipping it with a formal smooth structure. To
achieve this goal we first concentrate on the space of nonpure states of a
commutative C*-algebra as a guideline for the noncommutative case. In
particular, we prove Stokes' theorem over the both commutative and
noncommutative smooth Wasserstein space.
