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Infinite Powers: How Calculus Reveals the Secrets of the Universe

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Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not abo Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS. As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.


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Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not abo Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS. As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.

30 review for Infinite Powers: How Calculus Reveals the Secrets of the Universe

  1. 5 out of 5

    Peter Mcloughlin

    I feel bad for kids who do ordinary arithmetic in grade school. For me, the math doesn't get interesting until you get above the Calculus line. Calculus with its dealings with the continuum is the first real taste of the infinite. If you stick around for Calculus I bet you would want more. Strogatz in this book shares some of the excitement we have once the math gets weird. In some places, math can be hallucinogenic. I encourage people to look beyond grade school stuff which is way more interest I feel bad for kids who do ordinary arithmetic in grade school. For me, the math doesn't get interesting until you get above the Calculus line. Calculus with its dealings with the continuum is the first real taste of the infinite. If you stick around for Calculus I bet you would want more. Strogatz in this book shares some of the excitement we have once the math gets weird. In some places, math can be hallucinogenic. I encourage people to look beyond grade school stuff which is way more interesting especially if you have been disappointed by math. and totally unrelated a Pink Floyd mandelbrot zoom for Hippies of a Mathematical platonist bent. https://www.youtube.com/watch?v=b7JbJ... oops accidentally inserted the wrong video on AI which is also interesting but not intentional. https://www.youtube.com/watch?v=1wAgB...

  2. 4 out of 5

    Ryan Boissonneault

    Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a sub Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a subject every educated person should understand at least conceptually. Fortunately, you don’t have to trudge through a thousand-page textbook to appreciate the story and power of calculus. Steven Strogatz, in his latest book Infinite Powers, has provided a clear, concise, and fascinating tour of the subject. In fact, if you don’t understand what calculus is all about after reading this book, then the prospects are not great that you ever will. There is simply no better, clearer presentation of the ideas available. Strogatz uses metaphors, illustrations, stories, and examples to guide the reader through the most difficult concepts. While this is not an easy read, it is as reader-friendly as possible; remember, you’re tackling the most sophisticated branch of mathematics, the underlying logic of all science, and a subject that the sharpest mathematical minds in history had to grapple with for thousands of years. As Strogatz explains, calculus is difficult because it’s tackling the most difficult problems humans encounter, problems that necessitate complex equations, notation, and mathematical manipulation. But behind this computational complexity lies an obsession with simplicity, with breaking down hard problems into easier parts. The special innovation of calculus, as Strogatz explains, is that problems are broken down into infinitely small and manageable parts and then recombined back into the whole. So what is calculus, exactly? It’s easier to describe calculus by the types of problems it solves than by standard mathematical definitions. When most people hear terms like “infinite series,” “limits,” “derivatives,” and “integrals,” they lose sight of the bigger picture of what calculus is trying to accomplish. One type of problem calculus can solve is the area under a curved surface. Area is typically quite easy to solve for shapes with straight lines. For rectangles, for example, the area is simply length times width. But what about for shapes with curves where the slope is constantly changing? There is no simple formula to calculate the area in this situation. You could approximate the area by overlaying rectangular objects over the curved shape (as shown below), but this would only be an approximation as the rectangles would not fit exactly in the curved shape. However, as you made the rectangles smaller (and increased their number) the fit inside the curved shape would keep improving and the approximation would keep getting more accurate. Since you can always keep dividing a number in half (you can always make a number larger or smaller), you can add an infinite number of smaller rectangles into the curved shape. You can never complete this process (which is why the concept of “completed infinity” is logically incoherent), but you could potentially keep adding rectangles forever, which is logically coherent and shows the difference between “completed infinity” and “potential infinity.” As you increase the number of rectangles, you get closer and closer to the area, which is the limit of the infinite series. The area of the curved shape becomes the sum of the infinite series of rectangles. Calculus is the set of equations and procedures to carry out this calculation precisely. Calculus can also solve problems of motion. Straight-line motion at constant velocity is easy. If you know the speed of an object, then the distance traveled is simply speed times time. But how can you calculate the trajectory of, say, a planet, that not only continuously changes direction in orbit around the sun but that also speeds up or slows down depending on its distance from the sun? This is not so easy, but is solved in a similar way by breaking down the trajectory into infinitely smaller parts and then summing the series. Calculus provides the procedures and notation to carry this out in the most efficient way. You’ll notice that both examples above solve for problems where some quantity is continuously changing. That means that calculus can solve any problem that involves a quantity that is continuously changing, like the spread of a virus, population growth, or continuously compounding interest in finance. Even without understanding the specific calculations, it’s amazing to contemplate the fact that we can harness the power of infinity to calculate with precision the area under any curved surface, the dynamics of any continuously changing variable, and the trajectory of any object anywhere in the universe! Of course, this brief sketch is only a description of the subject in its simplest terms; there is much more to the subject and the mechanics of the calculations gets incredibly complex. If you’re interested in diving deeper into the subject, with examples and proofs, Strogatz delivers a nice mixture of pure mathematics, practical examples, and a history of the personalities behind the development of calculus. Of particular interest for me was Strogatz’s solution of Zeno’s Achilles and the Tortoise paradox, a solution that finally made sense to me (in brief, the solution is that an infinite amount of steps can be completed in a finite amount of time). The Power of Human Cooperation If you find calculus near impossible to learn, you won’t be happy to know that Isaac Newton invented the subject before he turned 25. But you might find some solace in the fact that Newton did little else; he had few friendships and no romantic relationships, so he had all the time in the world to devote to numbers and experiments. Newton also couldn’t have done it alone. He was exactly right when he said that he was able to see further by “standing on the shoulders of giants.” As Strogatz explained: “But again, he [Newton] couldn’t have done any of this without standing on the shoulders of giants. He unified, synthesized, and generalized ideas from his great predecessors: He inherited the Infinity Principle from Archimedes. He learned his tangent lines from Fermat. His decimals came from India. His variables came from Arabic algebra. His representation of curves as equations in the xy plane came from his reading of Descartes. His freewheeling shenanigans with infinity, his spirit of experimentation, and his openness to guesswork and induction came from Wallis. He mashed all of this together to create something fresh, something we’re still using today to solve calculus problems: the versatile method of power series.” There are at least two lessons here; first, knowledge grows exponentially, not linearly, and there is no limit to what can be discovered. By standing on the shoulders of giants, each generation can build on the developments of the past, as Einstein was able to do by rejecting Newton’s ideas of space and time as absolute. Holding a person, idea, or generation in complete reverence inhibits progress, as when we followed Aristotle for 1,500+ years and maintained the belief that the earth was stationary. The best book I’ve read that elaborates on this point is The Beginning of Infinity: Explanations That Transform the World by the physicist David Deutsch. Second, calculus demonstrates the power of human cooperation. No single mind could have developed calculus from scratch. People of diverse origin and circumstance collaborated to find solutions to common, tangible problems, because they didn’t waste their time thinking about arbitrary human divisions and other products of pure imagination, like religion. Newton borrowed from ancient Greek geometry, French analytic geometry, the Indian decimal system, and Arabic algebra. As a result, he discovered the mathematical logic and underlying laws of nature that applied equally to objects anywhere in the universe, thus uniting the entire cosmos. This universality, as Strogatz recognized, sparked the beginning of the Enlightenment. A final point: in the concluding chapter, Strogatz describes Richard Feynman’s quantum electrodynamics (QED) theory, which, by using calculus, describes the quantum interaction of light and matter. Physicists use the theory to make predictions about the properties of electrons and other particles. As Strogatz said, “by comparing those predictions to extremely precise experimental measurements, they’ve shown that the theory agrees with reality to eight decimal places, better than one part in a hundred million.” This means that QED is the most accurate theory anyone has ever devised about anything. A prediction with an accuracy of 8 decimal places is like, using Strogatz’s example, planning to snap your fingers exactly 3.17 years from now down to the second, without the help of a clock or alarm. As Strogatz further explains: “I think this is worth mentioning because it puts the lie to the line you sometimes hear, that science is like faith and other belief systems, that is has no special claim on truth. Come on. Any theory that agrees to one part in a hundred million is not just a matter of faith or somebody’s opinion. It didn’t have to match to eight decimal places.” You will also often hear that science can’t determine right and wrong actions, which in some sense is correct, but misses the point. The moral element of science does not lie in any particular factual claim; it lies in the orientation to forming beliefs. The scientific mindset is not about clinging on to and forming your identity around a set of unalterable beliefs. The scientific mindset is about curiosity, orientation to discovering truth, intellectual integrity, and revising beliefs in the face of new evidence. It’s also, as I believe calculus shows, about the recognition of the power of human cooperation and the pursuit of knowledge as a collective human endeavor.

  3. 4 out of 5

    Eric

    TL;DR In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended. Review cross-posted at Primmlife.com Review When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation mov TL;DR In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended. Review cross-posted at Primmlife.com Review When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation moving I agree, and while I do like math, I didn’t always. Until I started studying calculus, math bored me. Algebra existed as a set of rules; geometry, though my introduction to proofs, seemed too abstract. But when I first solved a derivative, my indifference turned to frustration and intrigue. My plan to take only enough math to get an engineering degree changed to a serious contemplation of switching career paths to applied mathematics (with an eye towards physics grad degrees). Ultimately, I stuck with the engineering curriculum but ended up studying higher level mathematics, and to this day, I still read about and love math. Part of my studies now involves going back and filling in what I missed during previous years. One of the voices to which I turn is Dr. Steven Strogatz, and his latest book, Infinite Powers, fills in details about calculus that I lacked. His explanations don’t rely on the familiar equations but, instead, root themselves in history, in logic, and in excellent prose. Infinite Powers transforms calculus from equations into meaning. The Story In Infinite Powers, Dr. Strogatz starts with Archimedes from ancient Greece and carries on to some of today’s most unique challenges. It is the story of calculus told as a continuum of human learning. Often, the public thinks that scientific breakthroughs happen when lone geniuses discover something new, but in reality discoveries occur when people improve upon the work of others. In Infinite Powers Dr. Strogatz traces the methods Archimedes used to Newton and Leibniz, who are the inventors or discoverers of calculus. Along the way, we learn about contributions from Fermat, Galileo, Descartes, Arabic, and Chinese mathematicians. But we don’t stop at the discovery era. Infinite Powers continues on to Fourier and Sophie Germain. We even get to see how calculus is being used today to treat HIV patients, to create microwaves, and, near to my own heart and pocketbook, help the 787 fly. The Writing Math possesses a strong language of its own rooted in symbols and logic. While I view this as a strength, I also know others view the equations, Greek letters, and symbols to be inscrutable. Others have said that math texts tend to be dry reads. For anyone who thinks this, Infinite Powers is the book for you. While equations do exists, they are few. Dr. Strogatz takes the time to explain, in detail, what each of the symbols means. But the majority of the book reads more like a history text than a mathematical treatise. While it doesn’t spoon feed the reader, it doesn’t bog down in jargon. Clarity and simplicity are the descriptors I have already used talking about this book with friends. Dr. Strogatz does an excellent job describing what the math is actually doing. The reader will NOT be able to do any calculus after reading it, but he/she will understand how powerful a tool it is. There are graphs and pictures throughout the book. In my advanced reader’s copy (ARC), the graphs didn’t show up. So, I cannot speak to their quality; however, with my background and the detail of Dr. Strogatz’s descriptions, I could picture what his intent was with the graphs. That should be an indicator of success for the prose of this book. Ugh, Math, Really? Bear with me here as I get on my soapbox for a minute. One of the other responses that I get when I’m introduced as an engineer is, “You must be really good at math.” And compared to most people, yes, I am good at math. But I’m good at math for one reason only, I’ve been practicing it in one form or another for the last 23 years. In the martial arts, there’s a saying that a black belt is simply a white belt who didn’t quit. To me, that’s all that math is. I’m good at math and calculus because I didn’t quit doing math. The general public often thinks that math requires a certain mindset or, even, a certain person. No, it requires practices and tenacity. The reason that I stuck with math is because of teachers in high school that showed the same enthusiasm that Dr. Strogatz shows in this book. Teachers and professors who care that students understand a subject make this world a better place. After reading Infinite Powers, I have no doubt Dr. Strogatz is a teacher than inspires students. I can’t help but wonder what could happen if a book like this gets into the hands of someone who thinks they have to be good at math to understand it. Math as Art Though we use math in the sciences, I’ve come to view it more as an art. The mathematician, engineer, chemist, or whoever must know and understand the tools math gives us in order to solve problems, and like a painter picks and chooses the right brush to add to the painting, the problem solver picks and chooses the correct mathematical tool. It’s a creative process that, instead of being hung in a gallery or museum, zips down the road, flows through our veins, or launches a satellite into space. Dr. Strogatz demonstrates the versatility and creativity that we are capable of when using calculus. Whether putting satellites in space or determining how viruses spread, calculus is a tool for delving into nature’s mysteries. Infinite Powers stirred that creative sense, that feeling of awe at being able to see into the universes internal mechanisms. At the same time, it reminded me of the ingenuity of the human animal to seek out and explore the world around us. Dr. Strogatz conveys the beauty that one can find in math, and I felt that thrill of discovery again as I read this book. Infinity Originally, I requested this book because I thought it was about infinity. That mathematical concept that looks like an 8 fell asleep, ∞. Instead, it was about calculus; so, I went into the first few chapters with the wrong expectations. Dr. Strogatz discusses infinity but not enough to satisfy me. And throughout the book, he does reference back to the topic of infinity, but it feels more like a forced attempt to tie the later chapters to the theme. I’m still hoping that Dr. Strogatz gives us a book about infinity in the same detail and manner that he gave us a book about calculus. Conclusion Dr. Steven Strogatz’s Infinite Powers details the history and development of calculus. Dr. Strogatz’s ability to relate complex mathematical concepts in clear and precise language is at peak form in this book. For anyone curious about calculus, this book provides answers in delightful, easy to understand prose that will awaken your curiousity.

  4. 5 out of 5

    linhtalinhtinh

    I certainly wish I had read this book while in high school or college. We grilled all the basic technical parts of calculus and yet unsure what was the point. Well certainly you don't need to know it if you do not work in science and research. Life can go on just as well. But being able to appreciate the beauty of it is an added bonus. And then who knows, seeing that beauty could change the path you take in life. Strogatz takes the same approach with his earlier pop science book, The Joy of x, i. I certainly wish I had read this book while in high school or college. We grilled all the basic technical parts of calculus and yet unsure what was the point. Well certainly you don't need to know it if you do not work in science and research. Life can go on just as well. But being able to appreciate the beauty of it is an added bonus. And then who knows, seeing that beauty could change the path you take in life. Strogatz takes the same approach with his earlier pop science book, The Joy of x, i.e. trying his best to explain calculus via intuition, allegories, and graphs. It works up to some extent. Some concepts require a lot more, but as the author tries to avoid discussing the technicals, the discussion becomes a bit shallow. That is still fine, this is not a textbook. What I'm a bit uneasy about is Strogatz's reverence for calculus to the point of being religious. Maybe I have taken it wrong, taken his comparison/allegory too much at face value when he cites Feymann over and over again that it is the language of God. Just as the title says, Strogatz argues that calculus is so special, nature acts according to the rules of calculus, everything can be described using these types of equations. Perhaps I've taken calculus for granted. Of course it is behind everything. Modern science is built upon math and ultimately one of the most powerful tools ever invented. Yes, calculus is a tool, a language. It is a lens of looking at the world and of course whatever picture one sees would be described in that medium. Is it so surprising? Could there be another language that provides a different lens to see the world? I'm an illiterate here, I have no idea. My take is that calculus has been so successful to deal with so many problems of interest, and there are so many things to extend and develop, that one isn't incentivized to go through all the pain to develop another language to compete with it. But I would prefer to think that there are possible options out there, so that calculus is not necessary the one (and only) language of "God." Strogatz also emphasizes that calculus is behind everything, behind all sorts of important equations and inventions in human history, driving progress and success. It is true that calculus is used, but I feel that the writing is tilted towards overstressing the importance of calculus and downplaying all other types of ingenious ideas and inventions that work together. Perhaps one needs to exaggerate to attract attention. It is the way of writing not the idea. And of course any language has its limitation. Strogatz gives a lot of praises and does not so much tell readers what are the stuffs that calculus cannot deal with. Sure, towards the end he does say there is a limit to what mathematicians can solve, such as nonlinearity. He hints at the need for another type of method to approach such problems, such as that from Poincare. The discussion is not as comprehensive as I would like. Don't get me wrong, I love calculus. I'm more of an abstract thinker and do not need much to be convinced that calculus a wonderful and elegant tool. Yet I have problems with the writing. I prefer writings with more nuance. I ended up looking forward to the bits talking about mathematicians and their private lives. But Professor Strogatz is not a historian, so I was naturally also not satisfied either.

  5. 5 out of 5

    Athan Tolis

    I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here. So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here. So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to be in a mood, basically. That’s the task. So I did the sensible thing and went on a bit of a binge and bought a whole bunch of popular math books in one go to read in the tube. “Infinite Powers” I read first, because it looked like it would not challenge me at all and it gets good writeups. It’s bloody awesome! It’s more than an anthology of results and it’s more than a series of mini-portraits of mathematicians, it’s almost got a plot. Surprisingly often, even the obligatory corny applications of the math are (somewhat) related to what the author’s talking about. Huge caveat: I knew both the math and even many of the stories upfront, so perhaps it’s not very well explained. I have no way of knowing. But I bet you it is. Perhaps not well enough that you could hope to learn calculus from here, Jordan Ellenberg’s praise on the back cover notwithstanding. (For that I can refer you to “Quick Calculus” by twin gods Kleppner and Ramsey.) But probably well enough to be a companion to anybody taking calculus for the first time. Steven Strogatz had me from “hello,” of course, because he starts with the Greeks, on whom he lavishes immense praise. He could have left it there and I’d still be basking in the warm glow of my ancestors’ work. Needless to say, it does not stop there, he takes you from them to Fermat and Descartes, before introducing you to Newton and Leibniz, a couple words on Fourier and from him straight to Einstein, taking special care to erase all traces of evil men like the unspeakable inventor of delta-epsilon proofs. You won’t find the C-word here. So there’s a massive hole in the nineteenth century, somewhere, but I’m sure you can buy another book to find out about that. Here you’ll discover a decent definition of e, an intuitive explanation of general relativity, the common cause of death of Leibniz and Newton, a fun game to play with your microwave oven, the first and second derivative of the sine wave, the dimension of the three-body problem, a strong defense of infinitesimals, WHAT’S NOT TO LIKE? Enough from me, I’ll now go buy some extra copies for a few boys and girls I know. If one of them likes it, my job is done. Oh, sorry, one more thing. About the plot: it’s a history of how mathematicians throughout time have sliced hard problems into infinite infinitely-thin slices where the problem has a clear answer and then dealt with infinity to sum up the solutions to the easy problem in order to come up with an answer to the hard problem. Whenever you do that, you’re doing calculus, you’re putting together the answer granule by granule.

  6. 4 out of 5

    Rama

    This book does not make calculus interesting Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is This book does not make calculus interesting Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is not a recipe book and at the same time it is not overwhelming. But in the absence of clear mathematical methods or its applications, this is a slapdash story that does not make calculus interesting.

  7. 5 out of 5

    Lemar

    "Dividing by zero summons infinity in the same way that a Ouia board supposedly summons spirits from another realm. It's risky. Don't go there." This sentence gives a good idea of the fun and rigor that Steven Strogatz brings to this book that explains what the big deal is to people who, let's face it, are unlikely to learn calculus. "The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five hundred year story of calculus." By weaving exam "Dividing by zero summons infinity in the same way that a Ouia board supposedly summons spirits from another realm. It's risky. Don't go there." This sentence gives a good idea of the fun and rigor that Steven Strogatz brings to this book that explains what the big deal is to people who, let's face it, are unlikely to learn calculus. "The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five hundred year story of calculus." By weaving examples of what's so damn useful about calculus with stories of great minds and the problems they overcame personally and mathematically, Strogatz wrote a readable, yes, fun book about math. His is not a big ego, he gives pioneers their due and busts some myths along the way, "the Pythagorean theorem did not originate with Pythagoras, it was known to the Babylonians for at least a thousand years before him." We learn about the Chinese genius Liu Hui who improved on Archimedes's method for calculating pi as well as Zu Chongzhi who pushed the study of polygons further than anyone before him. Strogatz has a special affinity for the Sicilian Greek Archimedes. The Hindu contribution is enormous and seems worth an entire book on its own. The Arabic scholar Al-Hasan Ibn al-Haytham gets recognized as one of the many giants upon whose shoulders Newton and Leibniz stood. We don't stop there but in several great chapters we learn about the absolute latest breakthroughs and applications of calculus, to Quantum Electrodynamics and Chaos theory, medicine and many other non-linear problems. Stretching my inelastic brain to the the snapping point, he discusses Eintstein's partial derivative theories. "matter tells space-time how to curve, while curvature tells matter how to move. The dance between them makes the theory nonlinear." I love that sentence! Strogatz is a true writer, an artist with words who, happily for me, applies that skill to explaining his profession, and love, calculus.

  8. 4 out of 5

    Bookjazzer2010

    4++ Wow! I wish I had had this book back in college 50+ years ago. Who knew calculus could be this interesting to read about. I only endured my class with a boring professor. No joy then, but I thoroughly enjoyed this book. All the applications of calculus through the years were fascinating. Amazing stories about the scientists and mathematicians. Very easy to read and understand. I only needed to back up to reread a few paragraphs.

  9. 5 out of 5

    Martynas Petkevičius

    Infinite Powers won't teach you calculus but it'll gently familiarise you with the subject without oversimplifying it. You'll finish the book wanting to dive into the actual maths.

  10. 4 out of 5

    Roberto Rigolin F Lopes

    A few centuries ago some clever people noticed that nature is in an ever-changing state, notably Galileo (1564-1642) studying objects in free fall and Kepler (1571-1630) studying the motion of planets around our sun. Then Newton (1643-1727) and Leibniz (1646-1716) invented a mathematical tool to get closer and closer to the changing system at hand. Steven did a great job explaining how Calculus uses divide-and-conquer to the extreme taming infinity to describe the universe. It changed civilizati A few centuries ago some clever people noticed that nature is in an ever-changing state, notably Galileo (1564-1642) studying objects in free fall and Kepler (1571-1630) studying the motion of planets around our sun. Then Newton (1643-1727) and Leibniz (1646-1716) invented a mathematical tool to get closer and closer to the changing system at hand. Steven did a great job explaining how Calculus uses divide-and-conquer to the extreme taming infinity to describe the universe. It changed civilization; this book travels from Archimedes (-212) computing pi to today’s design of airplanes. And Calculus is still evolving like a living organism after an explosion of diversity to explain CHANGE everywhere. For example, Einstein (1879-1955) used Calculus to play with space (say x, y, z) and time, at least four things changing at the same time.

  11. 5 out of 5

    Andrii Zakharov

    An absolute joy to read, this book just might make you fall in love with calculus. Yes, Steven Strogatz really is that good.

  12. 4 out of 5

    Jason Furman

    A fantastic book about calculus. A blend of the history of the development of calculus, its applications, and intuitive explanations of its power filled with nicely intuitive explanations that will either provide a refresher or a different way of understanding what you have already learned. Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back together aga A fantastic book about calculus. A blend of the history of the development of calculus, its applications, and intuitive explanations of its power filled with nicely intuitive explanations that will either provide a refresher or a different way of understanding what you have already learned. Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back together again in order to solve problems. Rather than describing an immaculate conception of calculus by Leibniz and Newton, Strogatz starts with Archimedes, shows several geometric applications, and even spends a lot of time on Descartes and Fermat before even getting to what we consider calculus today. In all of these he shows how a combination of abstract ideas but also in many cases practical problems led to the development of calculus. The chronological order is interrupted (in a good way) by Strogatz’s many descriptions of the applications of calculus to different practical problems, most of which are in the analytically relevant chapter. These include GPS, AIDS drugs, rocketry, and more. In all of these cases Strogatz shows his pedigree as an applied mathematician, going into significant but highly readable detail about the models and discoveries underlying these areas. Overall, the book is very nicely written and highly recommended.

  13. 5 out of 5

    David Schwartz

    Well written and entertaining look at the development of calculus - surely one of mankind's most impressive intellectual achievements - and how it comes into play in all sorts of important problem solving. You meet Archimedes, Descartes, Fermat, Newton, Leibniz, and sundry other geniuses along the way - even the women who figured in the Hidden Figures story. Once you've finished this, you owe it to yourself, if you have not done so, to take a full course or work through a good textbook on your o Well written and entertaining look at the development of calculus - surely one of mankind's most impressive intellectual achievements - and how it comes into play in all sorts of important problem solving. You meet Archimedes, Descartes, Fermat, Newton, Leibniz, and sundry other geniuses along the way - even the women who figured in the Hidden Figures story. Once you've finished this, you owe it to yourself, if you have not done so, to take a full course or work through a good textbook on your own. I would recommend some of the older, less expensive textbooks - I have one from my father-in-law written by Lyman Kells, and it's great. Also check out this website for on line texts: http://onlinebooks.library.upenn.edu/...

  14. 4 out of 5

    Melissa Dee

    Just to set the stage, I’m a math geek, and I do speak calculus, if somewhat imperfectly. I was thrilled by this book. Strogatz explores “the calculus” from its beginnings to the outer reaches of its applications and unsolved problems of today. He gives a very accessible explanation of both Newton’s and Leibniz’s approach to the development of the mathematics now called calculus. For the reader who doesn’t want to follow along with the math, “Infinite Powers” is still a very readable glimpse of t Just to set the stage, I’m a math geek, and I do speak calculus, if somewhat imperfectly. I was thrilled by this book. Strogatz explores “the calculus” from its beginnings to the outer reaches of its applications and unsolved problems of today. He gives a very accessible explanation of both Newton’s and Leibniz’s approach to the development of the mathematics now called calculus. For the reader who doesn’t want to follow along with the math, “Infinite Powers” is still a very readable glimpse of the strength and beauty of the mathematical concepts. "Infinite Powers” will be tremendously useful in explaining to the high school students I teach why it is that calculus should make their hearts sing! I voluntarily read and reviewed an advanced copy of this book. All thoughts and opinions are my own.

  15. 5 out of 5

    Eugenio Gomez-acebo

    Interesting and accessible history of calculus, from integral to differential equations. It is a bit extreme to describe calculus as the "language of God" and how calculus becomes modern, advances and changes our universe as a self-conscious entity. There is certain magic in calculus, but there is more in how humans have used this tool to uncover the secrets of our universe. But I guess you need to add some drama to sell a book. In any case, it is fascinating to discover the way Archimedes estim Interesting and accessible history of calculus, from integral to differential equations. It is a bit extreme to describe calculus as the "language of God" and how calculus becomes modern, advances and changes our universe as a self-conscious entity. There is certain magic in calculus, but there is more in how humans have used this tool to uncover the secrets of our universe. But I guess you need to add some drama to sell a book. In any case, it is fascinating to discover the way Archimedes estimated the area of a circle, the three Zenon paradoxes, how to estimate the value of pi, the area of a parable sector, the retrograd movement of Mars, the laws of a pendulum, how longitude is estimated, the three Kepler laws, how Fermat invented the cartesian coordinates (and not Descartes) and the optimization techniques that we use today to compress signals...

  16. 5 out of 5

    Daniel

    This book put me off part way through the first chapter as the author waxed poetic about calculus as the center of the universe and "God's language" (according to Feynman with whom I previously took issue). Since my dad was reading it though, I read further, and was rewarded - at least initially. At its best, this book tells some fascinating history surrounding major mathematical figures and their circumstances, which is a joy to read. I particularly enjoyed reading about contemporaries Descarte This book put me off part way through the first chapter as the author waxed poetic about calculus as the center of the universe and "God's language" (according to Feynman with whom I previously took issue). Since my dad was reading it though, I read further, and was rewarded - at least initially. At its best, this book tells some fascinating history surrounding major mathematical figures and their circumstances, which is a joy to read. I particularly enjoyed reading about contemporaries Descartes and Fermat, and their resulting rivalry. More typically, it provides a gentle and approachable introduction by way of analogies to the core concepts of calculus. At its worst, and sadly far too often, it bludgeons the reader with proclamations of calculus as the most important thing simply because it has been used to solve a lot of real world problems (so has writing). Combined with a writing style reminiscent of the first edict provided to a child on essay writing: "tell them what you're going to tell them, tell them, and finally tell them what you told them," this book became mightily cumbersome by the end. Despite some interesting tidbits and stories, it was work to finish, and not a particularly rewarding type either.

  17. 4 out of 5

    Chen Xin

    Shear elegance of writing! Clarity, simplicity, rigor, warmth...... Wow

  18. 5 out of 5

    Mike

    I found the beginning of this book peomising as the author made calculus sounds incredibly intriguing. However, I found this book to not make calculus that much more interesting than I’d thought befure due to how shallowly the author penetrates the subject. Additionally, I found the ending of this book to be a drag due to the author forgoing any mathematical explainations and merely describing complicated processes that calculus was magically the answer to. This effectively sucked any of the int I found the beginning of this book peomising as the author made calculus sounds incredibly intriguing. However, I found this book to not make calculus that much more interesting than I’d thought befure due to how shallowly the author penetrates the subject. Additionally, I found the ending of this book to be a drag due to the author forgoing any mathematical explainations and merely describing complicated processes that calculus was magically the answer to. This effectively sucked any of the intrigue out of these otherwise fascinating situations. Sprinkle in a bunch of half-baked analogies and the end of this book is fairly boring and tedious.

  19. 5 out of 5

    Hans Trivedi

    This book provides a neat perspective on the historical development of calculus and how it has been useful, largely by breaking up challenging problems to ‘infinitely’ many smaller pieces and then stitching them back together again! The almost story like exploration of mathematicians throughout history was enjoyable and refreshing way to read this book. The key takeaway being that their perseverance with these problems and obsessive focus took calculus from its humble beginnings to its formidable This book provides a neat perspective on the historical development of calculus and how it has been useful, largely by breaking up challenging problems to ‘infinitely’ many smaller pieces and then stitching them back together again! The almost story like exploration of mathematicians throughout history was enjoyable and refreshing way to read this book. The key takeaway being that their perseverance with these problems and obsessive focus took calculus from its humble beginnings to its formidable stature today. Many of the explanations are not too rigorous and arguably rely on the reader possessing some pre-existing knowledge on the subject matter. I can imagine it’s a very different read depending on how familiar you are with the key concepts. Some knowledge makes a better read. One thing that I found that set this book apart was that it successfully focuses on giving a deeper intuitive perspective on understanding how mathematics is successfully applied. Providing a clear connection between calculus’ development and how it is applied to solving real world problems, leads to a clear understanding of how calculus truly works. This shines light on the fundamental “credo” of calculus; “To solve a hard problem about anything continuous, slice it into infinitely many parts and solve them. Then by putting the answers back together, you can make sense of the original whole.” Which the author aptly names ‘The Infinity Principle’. Roll credits. A key idea I found particularly illuminating stemmed from the carefully discussion of the difference between analysis and synthesis, with a focus on why (or how) one is much harder than the other. There seems to be a deep connection with this throughout the book, and between differentiation and integration themselves. More analysis and more depth across more problems would have been nice. Highly recommended read with a slightly bittersweet ending as you are left wishing the author wrote a longer book!

  20. 5 out of 5

    Robert Yokoyama

    Steven Strogatz points out the practical applications of calculus. This is what makes this book enjoyable for me because no teacher ever explained how math can used to me. I enjoy watching Olympic track and field. I learned that it is possible to find out how fast a runner is going at a specific time in a race by using graphing and calculus. I love the example of Usain Bolt that the author uses, because I admire the way Usain runs. He runs slowly in the beginning of a race and always runs faste Steven Strogatz points out the practical applications of calculus. This is what makes this book enjoyable for me because no teacher ever explained how math can used to me. I enjoy watching Olympic track and field. I learned that it is possible to find out how fast a runner is going at a specific time in a race by using graphing and calculus. I love the example of Usain Bolt that the author uses, because I admire the way Usain runs. He runs slowly in the beginning of a race and always runs faster in the middle than in the end. The author makes calculus sound like fun especially when he explains how differential equations can used in treated people who have the HIV virus. A differential equation can used to show the rate of how HIV cells die in the body with treatment and grow without treatment. I love to listen to music, and I learned that calculus can be used in data compression. Data compression is used when I listen to the many songs on my MP3 player. I found it fascinating that calculus can also be used to predict the outcome of facial surgery for people who need them. Calculus and geometry can determine how the person will look using shapes and graphing. Calculus can also be used to locate tumors in the brain and determine well blood flows in my body. This is greater reinforcement for me that math can tell me how healthy I am. I love watching animated movies. The author sheds some light about how animated characters come to life by using triangles. I learned that animated characters can look more detailed by using triangles. An animated character can look more life like if the animator uses triangles in different positions. I am motivated to improve my math skills and discover more applications of math for myself because of this insightful book.

  21. 4 out of 5

    Kenneth Bachmann

    Without a doubt the historical background provided by Strogatz, particularly insofar as it linked the development and implementation of calculus to all manner of scientific discoveries, was the most satisfying feature of the book. Though the book is intended for the layman, I found some of the more technical examples quite difficult, though interesting, even though I had both taken some college calculus and used some in my career as a biological scientist. Of course college for me was a half-cen Without a doubt the historical background provided by Strogatz, particularly insofar as it linked the development and implementation of calculus to all manner of scientific discoveries, was the most satisfying feature of the book. Though the book is intended for the layman, I found some of the more technical examples quite difficult, though interesting, even though I had both taken some college calculus and used some in my career as a biological scientist. Of course college for me was a half-century ago, and my academic career ended about a dozen years ago. So. Nevertheless, Strogatz tells a good story, and is a decidedly excellent writer. I'll bet his classes would be wonderful to attend. I would imagine that scientists with both more recent and more extensive training in calculus would have far more "aha" moments reading this book than I did. I guess at this stage of life I'm much more akin to a laymen than a scientist. Whether Infinite Powers would be either interesting or fun for someone with no scientific or mathematical training I couldn't say, but as a former experimentalist I can't wait to get downstairs and measure the speed of light in microwave oven (Chapter 10 pp 63).

  22. 4 out of 5

    Derek Davis

    OK, the 3 stars is unfair, since I didn't get beyond the Kindle sample (probably 10% of the whole). I have a general reservation about science/math topics that try to personalize the forces of nature (or our interpretation of them). Here, Strogatz refers to calculus as though it were his buddy who's really nice to know. What grabs me about science or math is that they're not personal forces or wanglers of reality, but the imbedded essence of existence. Calculus is, in one sense, the invention of OK, the 3 stars is unfair, since I didn't get beyond the Kindle sample (probably 10% of the whole). I have a general reservation about science/math topics that try to personalize the forces of nature (or our interpretation of them). Here, Strogatz refers to calculus as though it were his buddy who's really nice to know. What grabs me about science or math is that they're not personal forces or wanglers of reality, but the imbedded essence of existence. Calculus is, in one sense, the invention of humanity, but in another it's the outgiven explanation of how everything works. And I don't much like the implication that inventing it (if that's the case) made all of recent, often negative, social life possible. There's the suggestion that it might never have happened. Well, of course it would have, later perhaps, but probably not much later. Social evolution progresses in its own weird way at its own weird pace – that the calculus we know was invented simultaneously by Newton and Leibniz shows this. I need to finish at least one or two other books that I've started so I can add a reasonable review of something.

  23. 5 out of 5

    JQAdams

    This is a history/popularization of calculus, building everything from the idea of "the Infinity Principle": many tasks are easier if you break them into an infinite number of pieces and then reassemble them, and that's just what calculus does. Pretty much whenever Strogatz was talking about the mathematics qua mathematics, I found it pretty engrossing; I'd probably have given the book four stars had it just been that material. Unfortunately, the author wants to convince the reader of how broadl This is a history/popularization of calculus, building everything from the idea of "the Infinity Principle": many tasks are easier if you break them into an infinite number of pieces and then reassemble them, and that's just what calculus does. Pretty much whenever Strogatz was talking about the mathematics qua mathematics, I found it pretty engrossing; I'd probably have given the book four stars had it just been that material. Unfortunately, the author wants to convince the reader of how broadly and magically useful calculus is, so several of the chapters include example applications that calculus plays some part in, ranging from the epidemiology of AIDS to the Global Positioning System to reconstructive surgery. And pretty much every time, I felt like things screeched to a halt once he started talking about the application, with the examples seemingly arbitrarily chosen and involving enough non-calculus innovation that they could equally have been part of a book about some other field—materials science, say, or language.

  24. 4 out of 5

    Saurabh Jha

    This is a pop maths book and I don't think there are many in this genre. This is mainly about Calculus and related subjects. Just for some perspective, I had studied some calculus, single and multivariable, right before reading this book. The big lesson I got from this book was fruitful informal and intuitive thinking can be. I always presumed that problems in calculus ought to be considered as formally as possible. However, after reading this book, I realised that both formal and informal thinki This is a pop maths book and I don't think there are many in this genre. This is mainly about Calculus and related subjects. Just for some perspective, I had studied some calculus, single and multivariable, right before reading this book. The big lesson I got from this book was fruitful informal and intuitive thinking can be. I always presumed that problems in calculus ought to be considered as formally as possible. However, after reading this book, I realised that both formal and informal thinking go hand in hand as you consider a problem or study some new topic. In fact, as shown in this book. Calculus started in a "hand-wavvy" manner two thousands year back and got more and more formalised with time. It turned from being an intuitive, concrete, real-world picture into an abstract and mathematical realm. This book is great to get an overall perspective of mathematics. It's written in a friendly tone and the style is conversational.

  25. 5 out of 5

    Stuart

    What an integrative look at calculus. When I was taught calculus, I knew it was special, but I didn't see where it came from. Steven Strogatz takes us through history to reveal that calculus has been evolving with us for thousands of years. The journey has characters who will surprise you. After hearing so much about Pierre de Fermat and Gottfried Wilhelm Leibniz, it felt like I finally met them at a party, where I learned that these two are doing math on the side while balancing their lives and What an integrative look at calculus. When I was taught calculus, I knew it was special, but I didn't see where it came from. Steven Strogatz takes us through history to reveal that calculus has been evolving with us for thousands of years. The journey has characters who will surprise you. After hearing so much about Pierre de Fermat and Gottfried Wilhelm Leibniz, it felt like I finally met them at a party, where I learned that these two are doing math on the side while balancing their lives and the rest of their intellectual pursuits. When I first studied calculus, I felt lost without a why. I learned limits, derivatives, and integrals, but I struggled to understand why I was working problems with these tools. I didn't feel like I understood the principles, even though I was helping tutor my peers (for ref, I got a 4 on Calc AB). After reading Infinite Powers, I feel motivated and better equipped as I prepare to study Calculus again.

  26. 5 out of 5

    Martin

    As a late bloomer in mathematics, I was very intrigued with the reviews of Steve Strogatz's book. Reviews described it as calculus for those who hate math. That sounded very interesting to me - I had been a math "dummy" for most of my life, and only tried again to master algebra and calculus when I was in my 30s. It was a huge revelation when I discovered that all that was missing was motivation. I ended up with degree in computer science degree and a math minor. This is a beautifully written bo As a late bloomer in mathematics, I was very intrigued with the reviews of Steve Strogatz's book. Reviews described it as calculus for those who hate math. That sounded very interesting to me - I had been a math "dummy" for most of my life, and only tried again to master algebra and calculus when I was in my 30s. It was a huge revelation when I discovered that all that was missing was motivation. I ended up with degree in computer science degree and a math minor. This is a beautifully written book, with lucid, clear descriptions of calculus, including many examples taken from ordinary life - how calculus forms the basis everything from astronomy and biology to medicine and to the social sciences. When I retire, I'd like to teach, say, a volunteer community math course for those who hate math or who feel it's too difficult, and I look forward to using Strogatz's book as an important part of the lesson plan.

  27. 5 out of 5

    Kitri Miller

    I really enjoyed this book, and let me tell you why. One, it broke into plain english what calculus is there for. It is basic simple math, and why it's important. Second, it gives a history of math that gives credit where it is due, which is that it wasn't all European-centric. There are places, like the Middle East, India, and Asia for the progress we were able to make to getting to the point where we have calculus. Third, it isn't all male, which is great. It also give reference to the fact tha I really enjoyed this book, and let me tell you why. One, it broke into plain english what calculus is there for. It is basic simple math, and why it's important. Second, it gives a history of math that gives credit where it is due, which is that it wasn't all European-centric. There are places, like the Middle East, India, and Asia for the progress we were able to make to getting to the point where we have calculus. Third, it isn't all male, which is great. It also give reference to the fact that social factors rather then talent is why there are fewer women in the record for doing great math (but even despite that there have been great mathematicians who have been women, here's to more). Forth, and lastly, it covers why math, and calculus in particular, are still relevant and why there might still be more discover. Just because we think we know something doesn't mean we do.

  28. 4 out of 5

    Rahul Patel

    Strogatz has a talent for writing about math for the masses! His latest book gives a history of calculus, a subject that turns many off from math. Strogatz does a great job giving a comprehensive history of calculus over time, including how ancient Greeks were tantalizingly close to its "discovery". Beyond the math, we learn about personalities - Newton vs Leibniz, Descartes vs Fermat, and other colorful personalities in the history of math and calculus. Strogatz also points out the important ro Strogatz has a talent for writing about math for the masses! His latest book gives a history of calculus, a subject that turns many off from math. Strogatz does a great job giving a comprehensive history of calculus over time, including how ancient Greeks were tantalizingly close to its "discovery". Beyond the math, we learn about personalities - Newton vs Leibniz, Descartes vs Fermat, and other colorful personalities in the history of math and calculus. Strogatz also points out the important role of women mathematicians in the history of calculus - Sophie Germain in particular, but also Virginia Johnson of "Hidden Figures" fame, in applied math and calculus. Overall, this is a good book for those interested in the history of math, and also those who want a more accessible introduction to the foundations of calculus.

  29. 4 out of 5

    Rashalia

    Really fascinating book that is sure to inspire awe even in math-haters. While you do not need college level math to understand the ideas in this book (thanks to Strogatz's brilliantly lean and intuitive explanations), a mathematics major will still enjoy this foray into the history and applications of calculus. Best of all, Strogatz avoids the patriarchal, eurocentric historical narrative of calculus presented in so many textbooks. He gives non-European and female mathematicians their due, and I Really fascinating book that is sure to inspire awe even in math-haters. While you do not need college level math to understand the ideas in this book (thanks to Strogatz's brilliantly lean and intuitive explanations), a mathematics major will still enjoy this foray into the history and applications of calculus. Best of all, Strogatz avoids the patriarchal, eurocentric historical narrative of calculus presented in so many textbooks. He gives non-European and female mathematicians their due, and I was particularly delighted to learn about a number of brilliant female mathematicians of whom I had not previously heard.

  30. 5 out of 5

    Arunkrishnan

    The book tracks the history of calculus from Archimedes to Newton in depth and gives us a glimpse of what were the ideas behind the invention of calculus. At the end, the author gives us some really good examples of the predictive power of calculus and how it's predictive powers led to some of the most intriguing discoveries and inventions of the twentieth century. The author also predicts that the next important innovations will come from the application of calculus to nonlinear and chaotic sys The book tracks the history of calculus from Archimedes to Newton in depth and gives us a glimpse of what were the ideas behind the invention of calculus. At the end, the author gives us some really good examples of the predictive power of calculus and how it's predictive powers led to some of the most intriguing discoveries and inventions of the twentieth century. The author also predicts that the next important innovations will come from the application of calculus to nonlinear and chaotic systems that are an integral part of biological and social sciences and weather systems respectively.

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