  This article is the second of a trilogy that addresses the perturbative
response of general quantum systems, with possibly nontrivial ground state
geometry, beyond linear order. Here, we establish concise, general formulae for
second order response to a spatially uniform, time-varying electric field in
the velocity gauge that are $\textit{manifestly}$ free of static limit spurious
divergences. We first discuss general quantum evolution in a curved space, then
detail how such a situation is a natural byproduct of Hilbert space truncation,
and point out crucial subtleties associated with the resulting finite
curvatures. We then present a geometric perspective of the two popular gauges
often used in quantum transport theories, the velocity gauge and the length
gauge, and discuss how they, taking truncation-induced curvature effects into
account, naturally lead to the same results in spite of the truncation. We
highlight subtle formal discrepancies in the literature. Finally, we provide a
general scheme for removing static limit spurious divergences in the velocity
gauge $\textit{without}$ frequency expansions and present concise and
comprehensive Green's function formulae for responses up to second order. As an
application of specific aspects of our theory, second order charge current
responses in selected cases are analyzed in parts I and III.
