CONTENTS Introduction.....................................................................................................................................................5 Notation........................................................................................................................................................10 Prologue. Partial differential equations of first order. Maxwell's equations....................................................11 §1. Partial differential equations of first order...............................................................................................11 §2. Maxwell's equations................................................................................................................................14 Chapter I. Small solutions of the relativistic Vlasov-Maxwell system..............................................................20 §3. Definition of the problem.........................................................................................................................21 §4. A representation theorem for Q..............................................................................................................26 §5. Representation of the derivatives of QK.................................................................................................32 §6. The existence and uniqueness theorem for small solutions....................................................................36 Chapter II. Global solutions of the modified relativistic and nonrelativistic Vlasov-Maxwell system................41 §7. An existence and uniqueness theorem...................................................................................................41 §8. The initial-value problem for the modified relativistic (and nonrelativistic) Vlasov-Maxwell system..........47 Appendix A...................................................................................................................................................56 Appendix B...................................................................................................................................................57 References...................................................................................................................................................59