Heat equation for theta functions and vector-valued modular forms

We give a method for constructing vector-valued modular forms from singular scalar-valued ones of a suitable type. As an application, we prove that 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, are the same. More precisely, let V-grad be the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and let V-Theta be the one generated by vector-valued modular forms arising from second order theta constants with our construction. We will prove that V-grad = V-Theta.