This text focuses on a variety of topics in mathematics in common usage in graduate engineering programs including vector calculus, linear and nonlinear ordinary differential equations, approximation methods, vector spaces, linear algebra, integral equations and dynamical systems. The book is designed for engineering graduate students who wonder how much of their basic mathematics will be of use in practice. Following development of the underlying analysis, the book takes students through a large number of examples that have been worked in detail. Students can choose to go through each step or to skip ahead if they so desire. After seeing all the intermediate steps, they will be in a better position to know what is expected of them when solving assignments, examination problems, and when on the job. Chapters conclude with exercises for the student that reinforce the chapter content and help connect the subject matter to a variety of engineering problems. Students have grown up with computer-based tools including numerical calculations and computer graphics; the worked-out examples as well as the end-of-chapter exercises often use computers for numerical and symbolic computations and for graphical display of the results.

Combustion Thermodynamics and Dynamics builds on a foundation of thermal science, chemistry, and applied mathematics that will be familiar to most undergraduate aerospace, mechanical, and chemical engineers to give a first-year graduate-level exposition of the thermodynamics, physical chemistry, and dynamics of advection-reaction-diffusion. Special effort is made to link notions of time-independent classical thermodynamics with time-dependent reactive fluid dynamics. In particular, concepts of classical thermochemical equilibrium and stability are discussed in the context of modern nonlinear dynamical systems theory. The first half focuses on time-dependent spatially homogeneous reaction, while the second half considers effects of spatially inhomogeneous advection and diffusion on the reaction dynamics. Attention is focused on systems with realistic detailed chemical kinetics as well as simplified kinetics. Many mathematical details are presented, and several quantitative examples are given. Topics include foundations of thermochemistry, reduced kinetics, reactive Navier-Stokes equations, reaction-diffusion systems, laminar flame, oscillatory combustion, and detonation.