  We provide a detailed derivation of a recently developed first-principles
approach to calculating averages in systems of interacting, spherical Brownian
particles under time-dependent flow. Although we restrict ourselves to flows
which are both homogeneous and incompressible, the time-dependence and geometry
(e.g. shear, extension) are arbitrary. The approximations formulated within
mode-coupling theory are particularly suited to dense colloidal suspensions and
capture the slow relaxation arising from particle interactions and the
resulting glass transition to an amorphous solid. The delicate interplay
between slow structural relaxation and time-dependent external flow in
colloidal suspensions may thus be studied within a fully tensorial theory.
