  We consider the derivative $D\pi$ of the projection $\pi$ from a stratum of
Abelian or quadratic differentials to Teichm\"uller space. A closed one-form
$\eta$ determines a relative cohomology class $[\eta]_\Sigma$, which is a
tangent vector to the stratum. We give an integral formula for the pairing of
of $D\pi([\eta]_\Sigma)$ with a cotangent vector to Teichm\"uller space (a
quadratic differential).
  We derive from this a comparison between Hodge and Teichm\"uller norms, which
has been used in the work of Arana-Herrera on effective dynamics of mapping
class groups, and which may clarify the relationship between dynamical and
geometric hyperbolicity results in Teichm\"uller theory.
