This textbook provides a comprehensive introduction to the classical and modern calculus of
variations serving as a useful reference to advanced undergraduate and graduate students as
well as researchers in the field. Starting from ten motivational examples the book begins with
the most important aspects of the classical theory including the Direct Method the
Euler-Lagrange equation Lagrange multipliers Noether's Theorem and some regularity theory.
Based on the efficient Young measure approach the author then discusses the vectorial theory
of integral functionals including quasiconvexity polyconvexity and relaxation. In the second
part more recent material such as rigidity in differential inclusions microstructure convex
integration singularities in measures functionals defined on functions of bounded variation
(BV) and -convergence for phase transitions and homogenization are explored. While
predominantly designed as a textbook for lecture courses on the calculus of variations this
book can also serve as the basis for a reading seminar or as a companion for self-study. The
reader is assumed to be familiar with basic vector analysis functional analysis Sobolev
spaces and measure theory though most of the preliminaries are also recalled in the appendix.
