In this article, we take up the construction of spectral triples and
associated calculus in the context of $SU_q(2)$ and $S^2_{qc}$. In
order to construct explicit spectral triples, we begin with the
computation of $K$-groups, and then from explicit generators we
construct spectral triples which induce generating elements in
$K$-homology. Using these spectral triples, we compute a modified
version of the space of Connes-de Rham forms and the associated
calculus. The space of $L^2$ forms have also been described
explicitly.