  We study the field dependence of the quasi-particle density of states, the
thermodynamics and the transport properties in the vortex state of d-wave
superconductors when a magnetic field is applied perpendicular to the
conducting plane, specially for the low field and the low temperature compared
to the upper critical field and transition temperature, respectively, $H/H_{c2}
\ll 1$ and $T/T_c \ll 1$. Both the superfluid density and the spin
susceptibility exhibit the characteristic $\sqrt{H}$-field dependence, while
the nuclear spin lattice relaxation rate T$_1^{-1}$ and the thermal
conductivity are linear in field $H$. With increasing temperature, these
quantities exhibit the scaling behavior in $T/\sqrt{H}$. The present theory
applies to 2D $f$-wave superconductor as well; a possible candidate of the
superconductivity in Sr$_2$RuO$_4$.
