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This book covers the principles and applications of differential algebra, a powerful new mathematical tool. The authors discuss the uses for the computation of transfer maps for all kinds of particle accelerators or any weakly nonlinear dynamical system, such as planetary orbits. The book is of interest to graduate students and researchers working in a broad range of disciplines, including applied mathematics, beam physics (accelerator physics, particle optics, geometric light optics), astronomy, and electrical engineering. Topics covered include transfer matrices, mechanics and electrodynamics, nonlinear motion, differential algebra, the structure of the classes, computer implementations, nonlinear maps, one pass systems, and repetitive systems.