  We investigate the coordinate dependence of noncommutative theory by studying
the solutions of noncommutative $U(1,1)\times U(1,1)$ Chern-Simons theory on
$AdS_3$ in the polar and rectangular coordinates. We assume that only the space
coordinates are noncommuting. The two coordinate systems are equivalent only up
to first order in the noncommutativity parameter $\theta$. We investigate the
effect of this non-exact equivalence between the two coordinate systems in two
cases, a conical solution and a BTZ black hole solution, using the
Seiberg-Witten map. In each case, the noncommutative solutions in the two
coordinate systems obtained from the corresponding same commutative solution
turn out to be different even in the first order in $\theta$.
