Let f : 0 1 R n be a continuous function which obeys. Theorem 1 (Fundamental Lemma of the Calculus of Variations). This result is fundamental to the calculus of variations. Typical Problem: Consider a definite integral that depends on an unknown function \(y(x)\), as well as its derivative \(y'(x)=\frac \right]. The fundamental lemma of the calculus of variations In this section we prove an easy result from analysis which was used above to go from equation (2) to equation (3). The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations. 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.) Many problems involve finding a function that maximizes or minimizes an integral expression.
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