  In this paper we study inverse boundary value problems with partial data for
the bi-harmonic operator with first order perturbation. We consider two types
of subsets of $\mathbb{R}^{n}(n\geq 3)$, one is an infinite slab, the other is
a bounded domain. In the case of a slab, we show that, from Dirichlet and
Neumann data given either on the different boundary hyperplanes of the slab or
on the same boundary hyperplane, one can uniquely determine the magnetic
potential and the electric potential.
  In the case of a bounded domain, we show the unique determination of the
magnetic potential and the electric potential from partial Dirichlet and
Neumann data under two different assumptions. The first assumption is that the
magnetic and electric potentials are known in a neighborhood of the boundary,
in this situation we obtain the uniqueness result when the Dirichlet and
Neumann data are only given on two arbitrary open subsets of the boundary. The
second assumption is that the Dirichlet and Neumann data are known on the same
part of the boundary whose complement is a part of a hyperplane, we also
establish the unique determination result in this local data case.
