By Serge Lang

This 5th version of Lang's e-book covers the entire issues routinely taught within the first-year calculus series. Divided into 5 elements, each one part of a primary direction IN CALCULUS comprises examples and functions on the subject of the subject coated. furthermore, the rear of the ebook includes exact ideas to various the workouts, permitting them to be used as worked-out examples -- one of many major advancements over past variants.

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**Example text**

For example, f ∗ is a convex function even if f is not convex. 35). Also, for convex functions the Legendre transformation is involutive: if f is convex, then f ∗∗ = f . 29). We claim that it can be obtained by applying the Legendre transformation to the Lagrangian L. More precisely, for arbitrary fixed x and y let us consider L(x, y, y ) as a function of ξ = y . 28) of the momentum p. 29) of the Hamiltonian H. 38). In other words, the Legendre transform of L(x, y, y ) as a function of y (with x, y fixed) is H(x, y, p), which is a function of p (with x, y fixed) and no longer has y as an argument.

We already know that in order to satisfy the Euler-Lagrange equation, the path must be a straight line. Thus the only path satisfying the necessary conditions is a horizontal line, which is of course the optimal solution. 6 Consider a more general version of the above variable-terminal-point problem, with the vertical line replaced by a curve: xf J(y) = L(x, y(x), y (x))dx a where y(a) = y0 is fixed, xf is unspecified, and y(xf ) = ϕ(xf ) for a given C 1 function ϕ : R → R. Derive a necessary condition for a weak extremum.

We have L z = z/ 1 + z 2 , thus Lz (x, y(x), y (x)) = y (x) 1 + (y (x))2 . , the path must be a straight line. The unique straight line connecting two given points is clearly the shortest path between them. Note that we d Lz (x, y(x), y (x)). did not need to compute dx The functional J to be minimized is given by the integral of the Lagrangian L along a path, while the Euler-Lagrange equation involves derivatives of L and must hold for every point on the optimal path; observe that the integral has disappeared.