The main aim of this paper is to offer a few commonplace notions and results on partial variational analysis so that others may come up with valuable development on partial variational analysis based on partial subdifferentials in a general direction set. Given a bounded closed set $\Theta $ (as a direction set) in a Banach space $X$ , this paper introduces and studies partial subdifferentials of proper lower semicontinuous extended-real functions on $X$ in $\Theta $ and partial normal cones to closed subsets of $X$ in $\Theta $ . Under mild smoothness assumptions on a direction set $\Theta $ , the paper establishes calculus rules of partial subdifferentials and geometric properties of partial normal cones in $\Theta $ . Moreover it is proved that the smoothness assumption on a direction set can be dropped in the finite dimensional case.
