The book will be particularly useful for beginning graduate students from the physical, engineering, and mathematical sciences with a rigorous theoretical background. 183 line to curve in reality results in a definite curve which satisfies the conditions of the problem. Forexample,aclassicalprobleminthecalculusofvariationsisndingtheshortest path between two points. ![]() Moreover,thisfunctionalisdenedinterms of the line or path integral FI x b xa I(y,y ,x)dx. The calculus of variations has a wide range of applications in physics, engineering, applied and pure mathematics, and is intimately connected to partial dierential equations(PDEs). Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study. Here, the essential elements involved in the calculus of variations are briey summarized.1 Consider a functionalFdepending on a functionyof a single variablex(i.e.,y y(x))anditsrstderivativeydy/dx. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself. This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning).
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