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V SEMESTER B.Sc., , MATHEMATICS PAPER — VI, , , , CALCULUS OF VARIATIONS, , Dr.SHIVASHARANAPPA SIGARKANTI, , ASSOCIATE PROFESSOR AND HOD, DEPARTMENT OF MATHEMATICS, GFGC K.R.PURAM, BANGALORE 560036

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GFGC K.R.PURAM V SEM B.Sc. PAPER — VI CALCULUS OF VARIATIONS, , V SEMESTER B.Sc. MATHEMATICS PAPER - VI, MATHEMATICAL METHODS - II, , Calculus of variations, , Functionals:, A functional is a mapping from functions to real numbers for example the integral, X2, I(y) = | f(y, y')dx is a functional. Here for each y, I(y) takes areal number ., xy, The aim of calculus of variation is to find a function y(x) satisfying y(x,) = y, and, y(x2) = y2 such that the functional I(y) has optimum value (Maximum or minimum), , 1.Euler’s equation:, , Theorem 1. Necessary condition for, x2, 1) = | f(x,y,y"dx tobe extremum is that, x1, of da (4), ay dx \ay', , y, , B(x2V2), , F(x) = v@)+ ae(x), , , , Proof : Let y = f(x) be some curve which optimizes the given functional, , Dr. SHIVASHARANAPPA SIGARKANTI HOD OF MATHEMATICS Page 1 of 25