Applied Mathematics: Body and Soul is a new foundational course in mathematics education for first-year students in the natural sciences, engineering, and mathematics, developed at Chalmers University of Technology in Gothenburg. It consists of three volumes as well as computer software. The project is rooted in the computer revolution, which has opened up entirely new possibilities for scientific computing in mathematics, the natural sciences, and engineering. It represents a synthesis of mathematical analysis (Soul) with numerical computation (Body) and its applications. Volumes I-III provide a modern version of analysis and linear algebra, including constructive numerical techniques and applications tailored for introductory lectures in mechanical engineering and the natural sciences. This volume covers analysis in m.