Advanced Engineering Analysis the Calculus of Variations and Functional Analysis with Applications in Mechanics by Leonid P. Lebedev, Michael J. Cloud, and Victor A. Eremeyev
Contents Advanced Engineering Analysis
- Basic Calculus of Variations
- Applications of the Calculus of Variations in Mechanics
- Elements of Optimal Control Theory
- Functional Analysis
- Applications of Functional Analysis in Mechanics
Preface to Advanced Engineering Analysis
A little over half a century ago, it was said that even an ingenious person could not be an engineer unless he had nearly perfect skills with the logarithmic slide rule.
The advent of the computer changed this situation crucially; at present, many young engineers have never heard of the slide rule.
The computer has profoundly changed the mathematical side of the engineering profession.
Symbolic manipulation programs can calculate integrals and solve ordinary differential equations better and faster than professional mathematicians can.
Computers also provide solutions to differential equations in numerical form.
The easy availability of modern graphics packages means that many engineers prefer such approximate solutions even when exact analytical solutions are available.
Because engineering courses must provide an understanding of the fundamentals, they continue to focus on simple equations and formulas that are easy to explain and understand.
Moreover, it is still true that students must develop some analytical abilities.
But the practicing engineer, armed with a powerful computer and sophisticated canned programs, employs models of processes and objects that are mathematically well beyond the traditional engineering background.
The mathematical methods used by engineers have become quite sophisticated.
With insufficient base knowledge to understand these methods, engineers may come to believe that the computer is capable of solving any problem.
Worse yet, they may decide to accept nearly any formal result provided by a computer as long as it was generated by a program of a known trademark.
But mathematical methods are restricted. Certain problems may appear to fall within the nominal solution capabilities of a computer program and yet lie well beyond those capabilities.
Nowadays, the properties of sophisticated models and numerical methods are explained using terminology from functional analysis and the modern theory of differential equations.
Without understanding terms such as “weak solution” and “Sobolev space”, one cannot grasp a modern convergence proof or follow a rigorous discussion of the restrictions placed on a mathematical model.
Unfortunately, the mathematical portion of the engineering curriculum remains preoccupied with 19th century topics, even omitting the calculus of variations and other classical subjects.
It is, nevertheless, increasingly more important for the engineer to understand the theoretical underpinning of his instrumentation than to have an ability to calculate integrals or generate series solutions of differential equations.
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