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.