![]() Typical Problem: Consider a definite integral that depends on an unknown function \(y(x)\), as well as its derivative \(y'(x)=\frac \right]. ![]() ![]() The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations. 1 Form the two possible compositions of f ( x) x and g ( x) 625 x 2 and compute the derivatives. One example is finding the curve giving the shortest distance between two points - a straight line, of course, in Cartesian geometry (but can you prove it?) but less obvious if the two points lie on a curved surface (the problem of finding geodesics.) In general, if f ( x) and g ( x) are functions, we can compute the derivatives of f ( g ( x)) and g ( f ( x)) in terms of f ( x) and g ( x). Many problems involve finding a function that maximizes or minimizes an integral expression. It is also easy to identify the differential of an expression from the chain rule, as the format is eye catching. But the main use of it is to be able to differentiate a function within a function (within a function.). multivariable functions, find critical points and extreme values. MATH0043 Handout: Fundamental lemma of the calculus of variations What is the statement of the Chain Rule In addition to learning how to differentiate a variety of basic functions, we have also been developing our ability to use rules to differentiate certain algebraic combinations of them. Ideas form the chain rule are used throughout, deeper integration is fundamentally based on it. Calculus is the branch of Mathematics that deals with differentiation and integration.The Euler-Lagrange Equation, or Euler’s Equation.MATH0043 §2: Calculus of Variations MATH0043 §2: Calculus of Variations
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