We study the variational formulas for the normalized Abelian differentials and matrix of b-periods on Hurwitz spaces, the moduli spaces of holomorphic Abelian differentials and quadratic differentials over compact Riemann surfaces. As the main result of the thesis, we find a complete set of local vector fields on the non-hyperelliptic connected component of the principal stratum of the moduli space of holomorphic quadratic differentials preserving the moduli of the base Riemann surface.