Variational Methods with Applications to Science and Engineering 1st Edition by Kevin W. Cassel (PDF)

Description

There is a resurgence of applications in which the calculus of variations has direct relevance. In addition to application to solid mechanics and dynamics, it is now being applied in a variety of numerical methods, numerical grid generation, modern physics, various optimization settings and fluid dynamics. Many applications, such as nonlinear optimal control theory applied to continuous systems, have only recently become tractable computationally, with the advent of advanced algorithms and large computer systems. This book reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. The mathematical fundamentals of calculus of variations (at least those necessary to pursue applications) is rather compact and is contained in a single chapter of the book. The majority of the text consists of applications of variational calculus for a variety of fields.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a well-written textbook that provides both rigor and a very good logical sequence of explanations. The explanations are very clear and the examples are very helpful. The author invests book space to develop foundational principles that are absolutely necessary later in the book. The book reads very well and has proven to be an excellent resource for introductory concepts as well as intermediate ones that allow the reader to move to other resources that are more specific. This author writes very well and provides good explanations for fairly abstract ideas. If I were using this textbook for a course (as an instructor), I would augment it with graphical explanations that build on what the author develops.

⭐I admit bias in that I had Prof. Cassel for this class when I was in grad school.His teaching is polished and impecable. His text reflects his teaching. He is perfect for teaching math to engineers. He doesn’t take what is often a pretentious assumption when discussing the math background for the material.

⭐I would say it is one of the best well written technical books I’ve ever read. The topics are pretty complex so I gotta give the props.

⭐Good very good!

⭐I was a senior in high school when I first tried to learn Calculus of Variations (CV). A. R. Forsyth’s classic 1926 book of that title had just been republished by Dover Publications (and is now available for free on the internet). I asked for a copy for Christmas. That’s not as unusual as you might think. My senior science fair project was an experiment to test Bernoulli’s principle, and I wanted to present also the mathematics of fluid flow. I found Forsyth inscrutable. I have returned to it several times throughout my mathematics career, and still find it inscrutable–or rather magic. It seems as though the problems which CV claimed to solve were impossibly under-determined. Other texts reinforced the mystery.Cassel cracks open the mystery when he says “use trial functions.” (91) Why does he take 91 pages to get to that point? Because he’s an award-winning teacher. He begins with a review of integration by parts, which seems innocuous enough until he springs on us using integration by parts “to shift derivatives from the trial to the weight functions.” (101) That did it for me! Coupled with the rest of this book, I understand CV for the first time in 50 years.The typical undergraduate curriculum in mathematics never addresses CV. Applied mathematics courses are more likely to use its results than to teach its methods. As a mathematics student at MIT looking for an applied course, I took Norbert Weiner’s Fourier Analysis (using his Dover Publications text), but no mathematics or engineering courses were available in CV.Yet Cassel offers CV as a beautiful synthesis of all manner of problems that seem unrelated until brought under the umbrella of CV: a breathtaking breadth of applied topics. Here are are three paragraphs of examples that Cassel considers.F=ma is a special case of Hamilton’s principle. (118) Einstein’s field equation is the Euler-Lagrange Equation for the Einstein-Hilbert action. (248) Feynman’s path integral “involve[s] the Lagrangian, and it reduces to Hamilton’s variational principle in the classical limit.” (258) Cassel even discusses Emmy Noether’s beautiful theorem relating Hamilton’s principle and conservation laws. (143-146)Bernoulli’s equation of my science fair project relies on the assumption of a conservative system, where “the Hamiltonian is a constant.” (263) Schrödinger’s wave equation is the Euler-Lagrange equation for electron waves. (253) Navier-Stokes equations for incompressible, viscous Newtonian fluids can be derived from CV principles. (266-274)CV can be used for optimal control for both discrete and continuous systems, whether physical or economic; and for computer graphics applications such as image processing, complementing Fourier methods (363), and automatic best-fitting of grid generation for image rendering.Cassel brings the historical context into play at every opportunity, from Newton’s calculations for an airfoil (306-310) to MRI scanning. (363) He also brings philosophy of science into play, indirectly in his chapter epigraphs and directly in a whole section emphasizing the beauty and simplicity of creation as a reason why physical laws are discoverable. (148-152)Each chapter ends with a summary (as does Forsyth’s text). The end-of-chapter problems are routine applications of the text. (Forsyth’s text has no end of chapter problems.) I found no errata. The index is sparse, but a keyword search on Amazon.com compensates. Cassel chooses a path through his topics that explicitly lists additional topics that he could have chosen, with suggested references for those who want to pursue those byways. As an engineer, he gives a good deal of attention to fluid flow, including a lengthy section (281-297) on stability results, although generally he assumes–rather than establishes–stability results (so there is no mention of Lyapunov functions for example).Prerequisites for a course using this textbook are vector calculus, partial differential equations, and matrices, with specific additional prerequisites for individual examples related to those examples.

⭐This is a great text to learn variational methods. I had zero experience with calculus of variations when I took a course with Professor Cassel (using this text and a few others). But I found this to be well-written and easy to read and comprehend. The example problems were explained clearly. Above all, this was actually really interesting, and I was amazed at how many different types of problems can be solved more easily with variational methods.

⭐I’ve been looking for a while for a good introductory book for the math needed to really understand finite elements and i think this is a great book to learn from at this level. I am using as a complement Gelfhand and Formin’s Calculus of Variations for some proofs.

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