This book is concerned with partial differential equations applied to fluids problems in science and engineering. This work is designed for two potential audiences. First, this book can function as a text for a course in mathematical methods in fluid mechanics in non-mathematics departments or in mathematics service courses. The authors have taught both. Second, this book is designed to help provide serious readers of journals (professionals, researchers, and graduate students) in analytical science and engineering with tools to explore and extend the missing steps in an analysis. The topics chosen for the book are those that the authors have found to be of considerable use in their own research careers. These topics are applicable in many areas, such as aeronautics and astronautics; biomechanics; chemical, civil, and mechanical engineering; fluid mechanics; and geophysical flows. Continuum ideas arise in other contexts, and the techniques included have applications there as well.