We consider the notion of shift tangent vector introduced in  for
real valued BV functions and introduced in  for vector valued BV functions.
These tangent vectors act on a function $u\in L^1$ shifting horizontally the points
of its graph at different rates, generating in such a way a continuous path in $L^1$. The main result of  is that
if the semigroup $\mathcal S$ generated by a scalar
strictly convex conservation law is shift differentiable, i.e. paths generated by
shift tangent vectors at $u_0$ are mapped in paths generated by shift tangent
vectors at $\mathcal S_t u_0$ for almost every $t\geq 0$.
This leads to the introduction of a sort
of differential, the "shift differential",
of the map $u_0 \to \mathcal S_t u_0$.
In this paper, using a simple decomposition of
$u\in $BV in terms of its
derivative, we extend the results of  and we give a unified definition of shift
tangent vector, valid both in the scalar and vector case. This extension allows
us to study the shift differentiability of the flow generated by a hyperbolic
system of conservation laws.