Calculus of variations has a long history. Its fundamentals were laid down by icons of mathematics like Euler and Lagrange. It was once heralded as the panacea for all engineering optimization problems by suggesting that all one needed to do was to state a variational problem, apply the appropriate Euler-Lagrange equation and solve the resulting differential equation. This, as most all encompassing solutions, turned out to be not always true and the resulting differential equations are not necessarily easy to solve. On the other hand, many of the differential equations commonly used in various fields of engineering are derived from a variational problem. Hence it is an extremely important topic justifying the new edition of this book. This third edition extends the focus of the book to academia and supports both variational calculus and mathematical modeling classes. The newly added sections, extended explanations, numerous examples and exercises aid the students in learning, the professors in teaching, and the engineers in applying variational concepts.

Primary care practitioners are often the first medical professionals to see patients after an injury, making it critical for them to stay up to date on the latest developments in sports medicine. Musculoskeletal and Sports Medicine for the Primary Care Practitioner contains the most current information on major topics in sports science and clinical medicine. It is a valuable resource for primary care physicians and allied health professionals who practice, teach, and hold specialty certifications in sports medicine and related fields. The book discusses key concepts related to the diagnosis, treatment, and prevention of sports injuries. This edition adds new sections on pro-inflammatory treatments, field-side acupuncture, and brief musculoskeletal ultrasound as well as a new chapter on wellness and video illustrations of important musculoskeletal maneuvers at www.crcpress.com/9781482220117. The book follows the Strength of Recommendation Taxonomy (SORT), which addresses the quality, quantity, and consistency of evidence. It recommends levels of patient-oriented evidence to assist physicians in their diagnoses. Also included is a link to videos that demonstrate important musculoskeletal maneuvers used in sports medicine. As exercise and sports move beyond the realm of leisurely activity to a necessary component of good health, this book has become an important resource for all those involved in sports medicine.

This second edition includes eleven new sections based on the approximation of matrix functions, deflating the solution space and improving the accuracy of approximate solutions, iterative solution of initial value problems of systems of ordinary differential equations, and the method of trial functions for boundary value problems. The topics of the two new chapters are integral equations and mathematical optimization. The book provides alternative solutions to software tools amenable to hand computations to validate the results obtained by "black box" solvers. It also offers an insight into the mathematics behind many CAD, CAE tools of the industry. The book aims to provide a working knowledge of the various approximation techniques for engineering practice.

For more than a century, we have had a firm grasp on rotor dynamics involving rigid bodies with regular shapes, such as cylinders and shafts. However, to achieve an equally solid understanding of the rotational behavior of flexible bodies-especially those with irregular shapes, such as propeller and turbine blades-we require more modern tools and methods. Computational Techniques of Rotor Dynamics with the Finite Element Method explores the application of practical finite element method (FEM)-based computational techniques and state-of-the-art engineering software. These are used to simulate behavior of rotational structures that enable the function of various types of machinery-from generators and wind turbines to airplane engines and propellers. The book's first section focuses on the theoretical foundation of rotor dynamics, and the second concentrates on the engineering analysis of rotating structures. The authors explain techniques used in the modeling and computation of the forces involved in the rotational phenomenon. They then demonstrate how to interpret and apply the results to improve fidelity and performance. Coverage includes: Use of FEM to achieve the most accurate computational simulation of all gyroscopic forces occurring in rotational structures Details of highly efficient and accurate computational and numerical techniques for dynamic simulations Interpretation of computational results, which is instrumental to developing stable rotating machinery Practical application examples of rotational structures' dynamic response to external and internal excitations An FEM case study that illustrates the computational complexities associated with modeling and computation of forces of rotor dynamics Assessment of propellers and turbines that are critical to the transportation and energy industries Useful to practicing engineers and graduate-level students alike, this self-contained volume also serves as an invaluable reference for researchers and instructors in this field. CRC Press Authors Speak Louis Komzsik introduces you to two books that share a common mathematical foundation, the finite element analysis technique. Watch the video.

Finite element analysis (FEA) has become the dominant tool of analysis in many industrial fields of engineering, particularly in mechanical and aerospace engineering. This process requires significant computational work divided into several distinct phases. What Every Engineer Should Know About Computational Techniques of Finite Element Analysis offers a concise, self-contained treatment of FEA and all of the tools needed for efficient use and practical implementation.

This book provides you with a walk-through of the process from the physical model to the computed solution. Based on the author's thirty years of practical experience in finite element analysis in the shipbuilding, aerospace, and automobile industries, it describes the transformation of the physical problem into a mathematical model, reduction of the model to a more efficient, numerically solvable form, and the solution of the problem using specific computational techniques. The author discusses time and frequency domain solutions as used in practice, as well as the representation of the computed results.

What Every Engineer Should Know About Computational Techniques of Finite Element Analysis serves as a to-the-point guide to using or implementing FEA for both beginners and everyday users who must apply the finite element method to your daily work. The techniques can be easily executed in most available FEA software packages.

CRC Press Authors Speak

Louis Komzsik introduces you to two books that share a common mathematical foundation, the finite element analysis technique. Watch the video.

For more than a century, we have had a firm grasp on rotor dynamics involving rigid bodies with regular shapes, such as cylinders and shafts. However, to achieve an equally solid understanding of the rotational behavior of flexible bodies-especially those with irregular shapes, such as propeller and turbine blades-we require more modern tools and methods. Computational Techniques of Rotor Dynamics with the Finite Element Method explores the application of practical finite element method (FEM)-based computational techniques and state-of-the-art engineering software. These are used to simulate behavior of rotational structures that enable the function of various types of machinery-from generators and wind turbines to airplane engines and propellers. The book's first section focuses on the theoretical foundation of rotor dynamics, and the second concentrates on the engineering analysis of rotating structures. The authors explain techniques used in the modeling and computation of the forces involved in the rotational phenomenon. They then demonstrate how to interpret and apply the results to improve fidelity and performance. Coverage includes: Use of FEM to achieve the most accurate computational simulation of all gyroscopic forces occurring in rotational structures Details of highly efficient and accurate computational and numerical techniques for dynamic simulations Interpretation of computational results, which is instrumental to developing stable rotating machinery Practical application examples of rotational structures' dynamic response to external and internal excitations An FEM case study that illustrates the computational complexities associated with modeling and computation of forces of rotor dynamics Assessment of propellers and turbines that are critical to the transportation and energy industries Useful to practicing engineers and graduate-level students alike, this self-contained volume also serves as an invaluable reference for researchers and instructors in this field. CRC Press Authors Speak Louis Komzsik introduces you to two books that share a common mathematical foundation, the finite element analysis technique. Watch the video.

Calculus of variations has a long history. Its fundamentals were laid down by icons of mathematics like Euler and Lagrange. It was once heralded as the panacea for all engineering optimization problems by suggesting that all one needed to do was to state a variational problem, apply the appropriate Euler-Lagrange equation and solve the resulting differential equation. This, as most all encompassing solutions, turned out to be not always true and the resulting differential equations are not necessarily easy to solve. On the other hand, many of the differential equations commonly used in various fields of engineering are derived from a variational problem. Hence it is an extremely important topic justifying the new edition of this book. This third edition extends the focus of the book to academia and supports both variational calculus and mathematical modeling classes. The newly added sections, extended explanations, numerous examples and exercises aid the students in learning, the professors in teaching, and the engineers in applying variational concepts.

If you have ever wondered how the universe physically works, then this book focusing on the central role that rotational motion plays in our daily lives will lead you to new discoveries, knowledge, and wisdom.

Rotation is found everywhere but often overlooked. There are rotating wheels on cars, wheels that spin in games of chance, Ferris wheels at city carnivals, and other manifestations of rotation that play an essential role in virtually everything we do. Even more important is how everyday phenomena driven by rotation make up life's intrinsic components.

Supported by historical information about the lives of individuals who have contributed to the understanding of rotation, this book delves into the physical principles that explain hurricanes, trade winds, gravity measurements, and much more.

Rotation continues to play a large role in the world, and it's possible that the universe itself rotates. Take a journey defined by new ideas and little-known facts in Wheels in the Sky, a must read for anyone interested in the origin and workings of the universe.

In this easily readable volume, the author explains some of the reasons for our preference toward the number seven. He reviews its occurrences in our cultural, biological, chemical, physical, algebraic and geometrical world in an attempt to find the reasons behind them.

In this easily readable volume, the author explains some of the reasons for our preference toward the number seven. He reviews its occurrences in our cultural, biological, chemical, physical, algebraic and geometrical world in an attempt to find the reasons behind them.

In this enjoyable and lightheaded volume, he gathers a plethora of cultural, biological, geometrical, algebraic, and planetary phenomena of our lives related to the number five. He investigates these occurrences in various facets of life on earth and seeks plausible explanations for some of them and hypothesizes about some others while widening your horizon.

In this enjoyable and lightheaded volume, he gathers a plethora of cultural, biological, geometrical, algebraic, and planetary phenomena of our lives related to the number five. He investigates these occurrences in various facets of life on earth and seeks plausible explanations for some of them and hypothesizes about some others while widening your horizon.

Gravity is the most detectable physical phenomenon of the universe. Nothing or nobody can escape its grasp, even in very large, astronomical distances. It is manifested in many aspects of our lives. Yet we are still not really sure what carries it. The gradually increasing understanding of gravity throughout history and its observed effects in various physical fields are the subjects of this book. This is an enlightening read about one of the most misterious phenomena of our lives that is still far from being fully understood, presenting a historical journey through several millennia of encountering and understanding gravity's mysteries.

Gravity is the most detectable physical phenomenon of the universe. Nothing or nobody can escape its grasp, even in very large, astronomical distances. It is manifested in many aspects of our lives. Yet we are still not really sure what carries it. The gradually increasing understanding of gravity throughout history and its observed effects in various physical fields are the subjects of this book. This is an enlightening read about one of the most misterious phenomena of our lives that is still far from being fully understood, presenting a historical journey through several millennia of encountering and understanding gravity's mysteries.

In Cycles of Time, author Louis Komzsik offers a comprehensive study of time, its relation to astronomy, and its relative nature. He reviews interesting tidbits of the historical evolution of our understanding of time and discusses some philosophical ideas about the topic. He explores the history of time, its origins in the celestial cycles observed by ancient cultures, and the expansion of our time horizon into cosmic cycles. He examines and debunks the doomsday predictions about the year 2012, and discusses the large scale measuring of time, gradually refined by humankind with medium scale instruments of calendars bringing time to the horizon of a human lifetime. While investigating the physical aspects of time, Cycles of Time examines time's relativity and ponders the possibility of time travel. Influencing one's past, an attractive idea to all of us and the potential cause of the grandfather paradox, may not be theoretically possible, but also merits speculation. Cycles of Time explains time's very strong trichotomy of past, present, and future in their simplest definitions: the past we remember, the present we experience now, and the future we anticipate.

If you have ever wondered how the universe physically works, then this book focusing on the central role that rotational motion plays in our daily lives will lead you to new discoveries, knowledge, and wisdom.

Rotation is found everywhere but often overlooked. There are rotating wheels on cars, wheels that spin in games of chance, Ferris wheels at city carnivals, and other manifestations of rotation that play an essential role in virtually everything we do. Even more important is how everyday phenomena driven by rotation make up life's intrinsic components.

Supported by historical information about the lives of individuals who have contributed to the understanding of rotation, this book delves into the physical principles that explain hurricanes, trade winds, gravity measurements, and much more.

Rotation continues to play a large role in the world, and it's possible that the universe itself rotates. Take a journey defined by new ideas and little-known facts in Wheels in the Sky, a must read for anyone interested in the origin and workings of the universe.

We eat three meals a day and use three utensils. There are three branches of the military. We use three letter acronyms such as NBC, FDR, JFK, FBI, and IRS. We eat BLTs, and we learn the ABCs. The number "three" permeates our lives. In Three of Life, author Louis Komzsik examines the unique role that this number plays in life.

Three of Life focuses on the diverse mathematical roles of the number as a geometric and arithmetic building block as well as a gatekeeper of, or a gateway to, infinity. It also discusses appearances of the number in places ranging from historical occurrences, to archaeological finds, to religious symbols, and to biology and physics.

In this light-hearted narrative, Komzsik describes the spectacular role that the number three plays in our physical lives-from biology to the mechanical world to the planetary motions. The number three has occupied a prominent role in human history for thousands of years and has demonstrated an intrinsic presence in our physical life.

In Cycles of Time, author Louis Komzsik offers a comprehensive study of time, its relation to astronomy, and its relative nature. He reviews interesting tidbits of the historical evolution of our understanding of time and discusses some philosophical ideas about the topic. He explores the history of time, its origins in the celestial cycles observed by ancient cultures, and the expansion of our time horizon into cosmic cycles. He examines and debunks the doomsday predictions about the year 2012, and discusses the large scale measuring of time, gradually refined by humankind with medium scale instruments of calendars bringing time to the horizon of a human lifetime. While investigating the physical aspects of time, Cycles of Time examines time's relativity and ponders the possibility of time travel. Influencing one's past, an attractive idea to all of us and the potential cause of the grandfather paradox, may not be theoretically possible, but also merits speculation. Cycles of Time explains time's very strong trichotomy of past, present, and future in their simplest definitions: the past we remember, the present we experience now, and the future we anticipate.

We eat three meals a day and use three utensils. There are three branches of the military. We use three letter acronyms such as NBC, FDR, JFK, FBI, and IRS. We eat BLTs, and we learn the ABCs. The number "three" permeates our lives. In Three of Life, author Louis Komzsik examines the unique role that this number plays in life.

Three of Life focuses on the diverse mathematical roles of the number as a geometric and arithmetic building block as well as a gatekeeper of, or a gateway to, infinity. It also discusses appearances of the number in places ranging from historical occurrences, to archaeological finds, to religious symbols, and to biology and physics.

In this light-hearted narrative, Komzsik describes the spectacular role that the number three plays in our physical lives-from biology to the mechanical world to the planetary motions. The number three has occupied a prominent role in human history for thousands of years and has demonstrated an intrinsic presence in our physical life.

This second edition includes eleven new sections based on the approximation of matrix functions, deflating the solution space and improving the accuracy of approximate solutions, iterative solution of initial value problems of systems of ordinary differential equations, and the method of trial functions for boundary value problems. The topics of the two new chapters are integral equations and mathematical optimization. The book provides alternative solutions to software tools amenable to hand computations to validate the results obtained by "black box" solvers. It also offers an insight into the mathematics behind many CAD, CAE tools of the industry. The book aims to provide a working knowledge of the various approximation techniques for engineering practice.

The Lanczos Method: Evolution and Application is divided into two distinct parts. The first part reviews the evolution of one of the most widely used numerical techniques in the industry. The development of the method, as it became more robust, is demonstrated through easy-to-understand algorithms. The second part contains industrial applications drawn from the author's experience. These chapters provide a unique interaction between the numerical algorithms and their engineering applications. This is a valuable reference for numerical analysts and engineers and can be used as a supplementary or reference text at the graduate level. Readers should be familiar with basic linear algebra and numerical analysis.