  When immersed into a fluid of active Brownian particles, passive bodies might
start to undergo linear or angular directed motion depending on their shape.
Here we exploit the divergence theorem to relate the forces responsible for
this motion to the density and current induced by--but far away from--the body.
In general, the force is composed of two contributions: due to the strength of
the dipolar field component and due to particles leaving the boundary,
generating a non-vanishing vorticity of the polarization. We derive and
numerically corroborate results for periodic systems, which are fundamentally
different from unbounded systems with forces that scale with the area of the
system. We demonstrate that vorticity is localized close to the body and to
points at which the local curvature changes, enabling the rational design of
particle shapes with desired propulsion properties.
