Math & Science
More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. Students of literature read Shakespeare; students of music listen to Bach, he writes. But this tradition of studying the major works of the masters is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.
Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original category theorem (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.
Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery, we can see how it all came to be.
The Calculus of Friendship is the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough.
Like calculus itself, The Calculus of Friendship is an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics, The Calculus of Friendship is warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters. For anyone whose life has been changed by a mentor, The Calculus of Friendship will be an unforgettable journey.
An accessible, streamlined, and user-friendly approach to calculus
Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a "Goldilocks approach" to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure. Fernandez begins by offering an intuitive introduction to the three key ideas in calculus--limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources--including video tutorials and interactive graphs--are available on the book's website. Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions--these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics--chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus. Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence. - An accessible, intuitive introduction to first-semester calculus - Nearly 200 solved problems and more than 300 exercises (all with answers) - No prior knowledge of exponential, logarithmic, or trigonometric functions required - Additional online resources--video tutorials and supplementary exercises--provided
When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus.
Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.
The internationally bestselling authors of The Cartoon Introduction to Economics return to make calculus fun
The award-winning illustrator Grady Klein has teamed up once again with the world's only stand-up economist, Yoram Bauman, Ph.D., to take on the daunting subject of calculus. A supplement to traditional textbooks, The Cartoon Introduction to Calculus focuses on the big ideas rather than all the formulas you have to memorize. With Klein and Bauman as our guides, we scale the dual peaks of Mount Derivative and Mount Integral, and from their summits, we see how calculus relates to the rest of mathematics. Beginning with the problems of speed and area, Klein and Bauman show how the discipline is unified by a fundamental theorem. We meet geniuses like Archimedes, Liu Hui, and Bonaventura Cavalieri, who survived the slopes on intuition but prepared us for the avalanche-like dangers posed by mathematical rigor. Then we trek onward and scramble through limits and extreme values, optimization and integration, and learn how calculus can be applied to economics, physics, and so much more. We discover that calculus isn't the pinnacle of mathematics after all, but its tools are foundational to everything that follows. Klein and Bauman round out the book with a handy glossary of symbols and terms, so you don't have to worry about mixing up constants and constraints. With a witty and engaging narrative full of jokes and insights, The Cartoon Introduction to Calculus is an essential primer for students or for anyone who is curious about math.
A personal exploration of wildness, territory, and the elusive nature of our lives
In this concise, richly contemplative book, Gary Thorp records his singularquest to see a mountain lion, or cougar--the "cat of one color"-- in the wild hills and mountains of northern California, where he lives. Using the traditional form of Japanese writing known as nikki bungaku (literary diary), Thorp recounts his meditations and adventures, from taking a one-day class on tracking animals, to visiting a mountain lion in the zoo, to his numerous forays into the hills during the day and night. The pursuit of one thing invariably leads him to discover many others: The tracks of a solitary mountain lion, for example, evoke a marvelous world of photographic imagery, literary events, dancing foxes, ocean voyages, and blind poets, all gathered together just beyond the limits of human vision. Thorp explores what it means to seek something you might not find and ponders the difference between seeing only darkness and being blind, offering as well bright glimpses into the Zen tradition. Combining an elusive and challenging pursuit with a centuries-old way of uncovering life's ultimate answers, Caught in Fading Light will give readers a new way of seeing, and will captivate nature lovers and Zen practitioners alike.
Wade's theory concerns the development of consciousness per se--not merely its derivatives, such as cognition, social development, and affect--and its neurological bases, something no other developmental theory has taken into account. Using data from a wide range of empirical studies and neurological research, Wade shows that awareness considerably predates birth--probably even conception--and lasts after death, supporting the idea that the self exists outside the boundaries of linear time and a physical body. This book represents a major leap forward in psychological theory and a groundbreaking approach to human perception and being in the world.
N. Katherine Hayles here investigates parallels between contemporary literature and critical theory and the science of chaos. She finds in both scientific and literary discourse new interpretations of chaos, which is seen no longer as disorder but as a locus of maximum information and complexity. She examines structures and themes of disorder in The Education of Henry Adams, Doris Lessing's Golden Notebook, and works by Stanislaw Lem. Hayles shows how the writings of poststructuralist theorists including Barthes, Lyotard, Derrida, Serres, and de Man incorporate central features of chaos theory.
-- "Times Literary Supplement" In these "Messenger Lectures," originally delivered at Cornell University and recorded for television by the BBC, Richard Feynman offers an overview of selected physical laws and gathers their common features into one broad principle of invariance. He maintains at the outset that the importance of a physical law is not "how clever we are to have found it out, but . . . how clever nature is to pay attention to it, " and tends his discussions toward a final exposition of the elegance and simplicity of all scientific laws. Rather than an essay on the most significant achievements in modern science, "The Character of Physical Law" is a statement of what is most remarkable in nature. Feynman's enlightened approach, his wit, and his enthusiasm make this a memorable exposition of the scientist's craft.
The Law of Gravitation is the author's principal example. Relating the details of its discovery and stressing its mathematical character, he uses it to demonstrate the essential interaction of mathematics and physics. He views mathematics as the key to any system of scientific laws, suggesting that if it were possible to fill out the structure of scientific theory completely, the result would be an integrated set of mathematical axioms. The principles of conservation, symmetry, and time-irreversibility are then considered in relation to developments in classical and modern physics, and in his final lecture Feynman develops his own analysis of the process and future of scientific discovery.
Like any set of oral reflections, "TheCharacter of Physical Law" has special value as a demonstration of the mind in action. The reader is particularly lucky in Richard Feynman. One of the most eminent and imaginative modern physicists, he was Professor of Theoretical Physics at the California Institute of Technology until his death in 1988. He is best known for his work on the quantum theory of the electromagnetic field, as well as for his later research in the field of low-temperature physics. In 1954 he received the Albert Einstein Award for his "outstanding contribution to knowledge in mathematical and physical sciences"; in 1965 he was appointed to Foreign Membership in the Royal Society and was awarded the Nobel Prize.
We all played tag when we were kids. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are also at play in military strategy, high-seas chases by the Coast Guard, and even romantic pursuits. In Chases and Escapes, Paul Nahin gives us the first complete history of this fascinating area of mathematics, from its classical analytical beginnings to the present day.
Drawing on game theory, geometry, linear algebra, target-tracking algorithms, and much more, Nahin also offers an array of challenging puzzles with their historical background and broader applications. Chases and Escapes includes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis.
Now with a gripping new preface on how the Enola Gay escaped the shock wave from the atomic bomb dropped on Hiroshima, this book will appeal to anyone interested in the mathematics that underlie pursuit and evasion.
its grubby concepts, spells, recipes, and rules. Peter Atkins wants to change all that. In this Very Short Introduction to Chemistry, he encourages us to look at chemistry anew, through a chemist's eyes, in order to understand its central concepts and to see how it contributes not only towards our material comfort, but also to human culture. Atkins shows how chemistry provides the
infrastructure of our world, through the chemical industry, the fuels of heating, power generation, and transport, as well as the fabrics of our clothing and furnishings. By considering the remarkable achievements that chemistry has made, and examining its place between both physics and biology, Atkins presents a fascinating, clear, and rigorous exploration of the world of chemistry - its structure, core concepts, and exciting contributions to new cutting-edge
technologies. ABOUT THE SERIES:
The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make
interesting and challenging topics highly readable.
In the 1930s C. G. Jung embarked upon a bold investigation into childhood dreams as remembered by adults to better understand their significance to the lives of the dreamers. Jung presented his findings in a four-year seminar series at the Swiss Federal Institute of Technology in Zurich. Children's Dreams marks their first publication in English, and fills a critical gap in Jung's collected works.
Here we witness Jung the clinician more vividly than ever before--and he is witty, impatient, sometimes authoritarian, always wise and intellectually daring, but also a teacher who, though brilliant, could be vulnerable, uncertain, and humbled by life's great mysteries. These seminars represent the most penetrating account of Jung's insights into children's dreams and the psychology of childhood. At the same time they offer the best example of group supervision by Jung, presenting his most detailed and thorough exposition of Jungian dream analysis and providing a picture of how he taught others to interpret dreams. Presented here in an inspired English translation commissioned by the Philemon Foundation, these seminars reveal Jung as an impassioned educator in dialogue with his students and developing the practice of analytical psychology.
An invaluable document of perhaps the most important psychologist of the twentieth century at work, this splendid volume is the fullest representation of Jung's views on the interpretation of children's dreams, and signals a new wave in the publication of Jung's collected works as well as a renaissance in contemporary Jung studies.
Scientists elucidate the astounding collective sensory capacity of Earth and its evolution through time.
Chimeras and Consciousness begins the inquiry into the evolution of the collective sensitivities of life. Scientist-scholars from a range of fields--including biochemistry, cell biology, history of science, family therapy, genetics, microbial ecology, and primatology--trace the emergence and evolution of consciousness. Complex behaviors and the social imperatives of bacteria and other life forms during 3,000 million years of Earth history gave rise to mammalian cognition. Awareness and sensation led to astounding activities; millions of species incessantly interacted to form our planet's complex conscious system. Our planetmates, all of them conscious to some degree, were joined only recently by us, the aggressive modern humans.
From social bacteria to urban citizens, all living beings participate in community life. Nested inside families within communities inside ecosystems, each metabolizes, takes in matter, expends energy, and excretes. Each of the members of our own and other species, in groups with incessantly shifting alliances, receives and processes information. Mergers of radically different life forms with myriad purposes--the chimeras of the title--underlie dramatic metamorphosis and other positive evolutionary change. Since early bacteria avoided, produced, and eventually used oxygen, Earth's sensory systems have expanded and complexified. The provocative essays in this book, going far beyond science but undergirded by the finest science, serve to put sensitive, sensible life in its cosmic context.
The core of this book consists of an in-depth study of what modern mathematicians still refer to as the Chinese remainder theorem for the solution of indeterminate equations of the first degree. This was Ch'in's most original contribution to mathematics--so original that no one could correctly explain Ch'in's procedure until the early nineteenth century. This volume's concluding study unites information on artisanal, economic, administrative, and military affairs dispersed throughout Ch'in's writings, providing rare insights into thirteenth-century China.
Empirical and theoretical foundations of a cognitive neuroscience of consciousness.
This book investigates the philosophical, empirical, and theoretical bases on which a cognitive neuroscience of consciousness can be founded. The research questions reviewed include: Does perception occur without awareness? Can the neural bases of perceptual awareness be visualized with brain-imaging methods? What do unilateral neglect and extinction tell us about conscious and unconscious processing? What is the contribution of brainstem nuclei to conscious states? How can we identify mental processes uniquely associated with consciousness? An introductory chapter proposes a theoretical framework for building a cognitive neuroscience of consciousness, and two concluding chapters evaluate the progress made so far.
The Firmament of Time (1960), a lyrical and meditative tour de force, looks back at the many ways in which the sciences have been shaped by the changing cultures in which they developed. Examining the role of metaphor in scientific thought, anticipations of scientific discoveries in the works of poets and novelists, and the "unconscious conformity" of scientific theory to prevailing orthodoxies, Eiseley argues provocatively for the ongoing relevance to scientific progress of dreams, the imagination, and the irrational. In his wide-ranging collection The Unexpected Universe (1969), Eiseley turns to the theme of the voyage of discovery: accounts of the mythical and historic journeys of Odysseus, Captain Cook, and Darwin frame his own more modest wanderings in the environs of Philadelphia. Sometimes he travels no farther than the local dump: and yet, like Homer's hero or these great explorers, he continually finds a universe "not only queerer than we suppose, but queerer than we can suppose." As an added feature, this volume presents a selection of Eiseley's uncollected prose, including early autobiographical sketches, vivid and haunting entries from his private notebooks, and his 1957 lecture "Neanderthal Man and the Dawn of Human Paleontology." A companion volume presents The Invisible Pyramid (1970), The Night Country (1971), and the essays gathered after his death in The Star Thrower (1978). LIBRARY OF AMERICA is an independent nonprofit cultural organization founded in 1979 to preserve our nation's literary heritage by publishing, and keeping permanently in print, America's best and most significant writing. The Library of America series includes more than 300 volumes to date, authoritative editions that average 1,000 pages in length, feature cloth covers, sewn bindings, and ribbon markers, and are printed on premium acid-free paper that will last for centuries.