  We introduce the concept of Type-I/II generating functionals defined on the
space of boundary data of a Lagrangian field theory. On the Lagrangian side, we
define an analogue of Jacobi's solution to the Hamilton-Jacobi equation for
field theories, and we show that by taking variational derivatives of this
functional, we obtain an isotropic submanifold of the space of Cauchy data,
described by the so-called multisymplectic form formula. We also define a
Hamiltonian analogue of Jacobi's solution, and we show that this functional is
a Type-II generating functional. We finish the paper by defining a similar
framework of generating functions for discrete field theories, and we show that
for the linear wave equation, we recover the multisymplectic conservation law
of Bridges.
