  We give a new method for constructing vector-valued modular forms from
singular scalar-valued ones. As an application we prove the identity between
two remarkable spaces of vector-valued modular forms which seem to be unrelated
at a first look, since they are constructed in two very different ways. If
$V_{grad}$ is the vector space generated by vector-valued modular forms
constructed with gradients of odd theta functions and $V_\Theta$ is the one
generated by vector-valued modular forms arising from second order theta
constants with our new construction, we will prove that $V_{grad}=V_\Theta$.
This result could also be proven as a consequence of the "heat equation" for
theta functions.
