The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics curricula. This volume should nevertheless give the undergraduate reader a sense of its great character and importance. Interesting functionals, such as area or energy, often give rise to problems whose most natural solution occurs by differentiating a one-parameter family of variations of some function. The critical points of the functional are related to the solutions of the associated Euler-Lagrange equation. These differential equations are at the heart of the calculus of variations and its applications to wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow. All are readily accessible to advanced undergraduates. This book is derived from a workshop sponsored by Rice University. It is suitable for advanced undergraduates, graduate students and research mathematicians interested in the calculus of variations and its applications to other subjects.

In this book, six representative works by musicians from major periods of Western music history, including Classical and Romanticism, are selected for study and interpretation in the form of “musical ontology”, with the aim of refining and exploring their compositional concepts and techniques, In this way the aesthetic characteristics and artistic values of the works are recognised and understood, summarised and outlined. The six works are: Beethoven’s Violin Concerto in D major, Stravinsky’s The Rite of Spring, Gershwin’s An American in Paris, Tchaikovsky’s Symphony No. 6 in B Minor “Pathétique”, Liszt’s Symphonic Poem No.3 Les Préludes, and Brahms’s Symphony No. 1 in C minor.

Variation is a fundamental musical principle, yet its most naked expression - variation form - resists all but the broadest of descriptions. This book offers listener, performer, analyst and composer an eclectic array of approaches to `Theme and Variations', including: patterns of departure and return; real versus perceived time; strategies of propulsion and closure in an intrinsically cyclic and open-ended form; the interplay of authorial voices deriving from dialogue between the `self' of variations and the `other' of their theme; critique of a theme through a set's generic references; drama and narrative achieved through textural and tonal control; and the intrinsic sound of a variation, so different from that of a freely composed work. These topics are introduced through a general survey of the form, seen through the prisms of the provenance of themes and the ideologies of sets, before being developed through close study of Brahms's variation sets and movements. Brahms was supremely aware of his place in music history and was uncommonly self-conscious in his manipulation of different techniques of composition. His variation sets - some of the most well-crafted and beloved examples - place the interplay of forms and styles at the heart of their identity. Moreover, in their stunning breadth and diversity they offer a microcosm of Brahms's entire output, a succinct revelation of his life-long concerns. Through them we marvel at his technical and poetic mastery, and journey to the heart of his creative character.

Headlam closely analyzes Berg's compositional technique and the use of symmetry and cycles throughout his oeuvre. He brings into the discussion Berg's own writings, as well as those of composer and musicologist George Perle; the techniques of Schoenberg, Webern, and other serialists; and aspects of pitch-class set and twelve-tone theory.