![]() This module and Analytic number theory 1 (M823) are entry-level modules for the MSc in Mathematics (F04), and normally you should have studied one of them before progressing to the intermediate and advanced intermediate modules in the degree. ![]() Successful study of this module should enhance your skills in understanding complex mathematical texts, communicating solutions to problems clearly and interpreting mathematical results in real-world terms. Applications will be discussed but you are not expected to have a detailed understanding of the underlying physical ideas. Throughout, the emphasis is on the mathematical ideas and one aim is to illustrate the need for mathematical rigour. The module also contains some more advanced material, such as an analysis of the second variation and of discontinuous solutions it ends with a discussion of the general properties of the solutions of an important class of linear differential equations, namely Sturm–Liouville systems. ![]() Many of the simple applications of calculus of variations are described and, where possible, the historical context of these problems is discussed. This module provides an introduction to the central ideas of variational problems, as well as some of the mathematical background necessary for the subject. The calculus of variations also provides useful methods of approximating solutions of linear differential equations furthermore, variational principles also provide the theoretical underpinning for the coordinate-free formulations of many laws of nature. Problems such as the determination of the shortest curve between two points on a given smooth surface and the shapes of soap films are most easily formulated using ideas from the calculus of variations.
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