Engineering problems differential calculus

During the next two years, as you take your calculus classes, you will learn many new concepts and rules dealing with differential calculus. Make sure you take the time to understand these concepts and rules. In your calculus classes, you may not apply the concepts to actual engineering problems, but be assured that you will use them eventually in your engineering classes. Some of these concepts and rules are summarized in Table 18.6. Examples that demonstrate how to apply these differentiation rules follow. As you study these examples, keep in mind that our intent here is to introduce some rules, not to explain them thoroi hly. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In a search problem, almost nothing is known in advance about how the criterion of effectiveness depends upon the operating variables, the only way to learn being to perform experiments. Here the obstacle to using the calculus is the complete lack of a function that can be differentiated. The objective of the search is to get as close as possible to the optimum after only a limited number of experiments. Box and Wilson, with their paper The Experimental Attainment of Optimum Conditions published in 1951, were the first to interest engineers in search problems (B4). 

Typical chemical engineering curriculums consist of the equivalent of three semesters of calculus capped off by a course in elementary ordinary differential equations. This usually occurs within the first two years of a four-year program (five years, if co-op is an option). The next two or three years are usually dedicated to solving unit operation problems using prederived formulae. The point being, with few exceptions, the use of the four semesters worth of mathematics is not applied until the first year of graduate school. 

Den Hartog, J. P. 1985. Mechanical Vibrations, 4th ed. New York Dover Publications. Reprint. Originally published 1956. New York McGraw-Hill. Covers the fundamentals of mechanical vibrations. Can be used by practicing engineers as well as for classroom instruction. An elementary knowledge of dynamics and calculus is necessary, but differential equations are explained in detail. Examples have been drawn from real-life experiences of the author and his friends. Each chapter includes problems that illustrate typical practical situations with answers included at the back. 

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