This unique reference/text presents a great deal of material that is completely new to the field of flow invariance, offering fresh insights for experienced mathematicians and rigorous training for students new to the specialty. Its four useful appendices supply the methods used throughout the book, making it a totally self-referential and self-contained unit. Containing over 1900 helpful equations, Tangency, Flow Invariance for Differential Equations, and Optimization Problems is an exceptionally thorough reference for pure and applied mathematicians, nonlinear analysts, physicists, astronomers, and mechanical engineers and an excellent text for upper-level undergraduate and graduate students in these disciplines.