The first truly up-to-date treatment of the calculus of variations, this text is also the first to offer a simple introduction to such key concepts as optimal control and linear-quadratic control design. Suitable for junior/senior–level students of math, science, and engineering, this volume also serves as a useful reference for engineers, chemists, and forest/environmental managers. Its broad perspective features numerous exercises, hints, outlines, and comments, plus several appendixes, including a practical discussion of MATLAB. Students will appreciate the text's reader-friendly style, which features gradual advancements in difficulty and starts by developing technique rather than focusing on technical details. The examples and exercises offer many citations of engineering-based applications, and the exercises range from elementary to graduate-level projects, including longer projects and those related to classic papers.
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Yes, you can access Calculus of Variations by Charles R. MacCluer in PDF and/or ePUB format, as well as other popular books in Mathematics & Applied Mathematics. We have over one million books available in our catalogue for you to explore.
This chapter reviews basic tools from calculus that are used in the calculus of variations—directional derivatives, gradients, the chain rules, contour surfaces, sublevel sets, Lagrange multipliers, and the basic notion of convexity. All of these concepts form the basic toolset for attacking optimization problems.
1.1 Directional Derivatives and Gradients
A point in
n is denoted by x = (x1, x2, …, xn)
, which is an n × 1 column vector (the superscript
denotes transpose). Suppose a function f of x represents the profit of a commercial enterprise, where x is a vector of the parameters of the operation such as labor costs, production output levels, price of the commodity, and so on. The manager naturally desires to know in which direction from the present operating point x0 should the company move in order to obtain the maximum increase in profit. The desired direction is found by using a multi-variable notion of a derivative: For each unit vector u ∈
n, the directional derivative of f at x0 in direction u is given by
provided the limit exists. Geometrically, this limit is the slope of the line tangent to the curve above x0 obtained by cutting the hypersurface z = f(x) with the hyperplane determined by u and the z-axis. See Figure 1.1. The directional derivatives in the directions parallel to the coordin...
