A large-signal field theory of a linear version of an injection-type crossed-field amplifier is presented based on a simultaneous solution of Maxwell's equations and the equations of electron motion in the interaction region. A key step in the analysis is a representation of the field components in the interaction region as functionals of the electron-arrival time and the electron-transverse position. Substitution of this representation into the equations of electron motion, modified to allow for the interception of the beam by the slow wave structure and the sole, puts the latter into the fixed-point format for a pair of nonlinear operators in an appropriate function space. The fixed point, and hence the solution for the field components may be obtained by standard successive approximation techniques. Numerical computation of the amplifier parameters based on such a successive-approximation solution will be deferred to the second part of this contribution.