Let
C
2
C_{2}
be a smooth and projective curve over the ring of dual numbers of a field
k
.
k.
Given non-zero rational functions
f
,
g
,
f,g,
and
h
h
on
C
2
,
C_{2},
we define an invariant
ρ
(
f
∧
g
∧
h
)
∈
k
.
\rho (f\wedge g \wedge h) \in k.
This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm of [Algebra Number Theory 3 (2009), pp. 1–34]. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture of Goncharov [J. Amer. Math. Soc. 18 (2005), pp. 1–60] with an explicit formula for the homotopy map. Also using
ρ
,
\rho ,
we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park’s construction in the case of cycles with modulus [Amer. J. Math. 131 (2009), pp. 257–276].