  We study entropies caused by the unstable part of partially hyperbolic
systems. We define unstable metric entropy and unstable topological entropy,
and establish a variational principle for partially hyperbolic diffeomorphsims,
which states that the unstable topological entropy is the supremum of the
unstable metric entropy taken over all invariant measures. The unstable metric
entropy for an invariant measure is defined as a conditional entropy along
unstable manifolds, and it turns out to be the same as that given by
Ledrappier-Young, though we do not use increasing partitions. The unstable
topological entropy is defined equivalently via separated sets, spanning sets
and open covers along a piece of unstable leaf, and it coincides with the
unstable volume growth along unstable foliation. We also obtain some properties
for the unstable metric entropy such as affineness, upper semi-continuity and a
version of Shannon-McMillan-Breiman theorem.
