Partial Differential Equations Quotes

She knew that in the end you really cant know. You cant get hold of the world. You can only draw a picture. Whether it’s a bull on the wall of a cave or a partial differential equation it’s all the same thing.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

Even there, something inside me (and, I suspect, inside many other computer scientists!) is suspicious of those parts of mathematics that bear the obvious imprint of physics, such as partial differential equations, differential geometry, Lie groups, or anything else that's “too continuous.

Von Neumann was in many ways a traditional mathematician, who (like Turing) believed he needed to turn to partial differential equations in describing natural systems.

mass times its acceleration—is a differential equation because acceleration is a second derivative with respect to time. Equations involving derivatives with respect to time and space are examples of partial differential equations and can be used to describe elasticity, heat, and sound, among other things.

In electrodynamics the continuous field appears side by side with the material particle as the representative of physical reality. This dualism, though disturbing to any systematic mind, has today not yet disappeared...The successful physical systems that have been set up since then represent rather a compromise between these two programs, and it is precisely this character of compromise that stamps them as temporary and logically incomplete...I incline to the belief that physicists will...be brought back to the attempt to realize that program which may suitably be called Maxwell's: the description of physical reality by fields which satisfy...a set of partial differential equations.

You cant get hold of the world. You can only draw a picture. Whether it’s a bull on the wall of a cave or a partial differential equation it’s all the same thing. Jesus.

In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations.

Bah! Do you know,' the Devil confided, 'not even the best mathematicians on other planets - all far ahead than yours - have solved it? Why, there's a chap on Saturn - he looks something like a mushroom on stilts - who solves partial differential equations mentally; and even he's given up.

it was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator’s brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike.

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

[My disappointment with mathematics] was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator's brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike.

Well. In this case it was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator’s brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike. Something like that. You

How did Biot arrive at the partial differential equation? [the heat conduction equation] . . . Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals.

One of the ideas of this book is to give the reader a possibility to develop problem-solving skills using both systems, to solve various nonlinear PDEs in both systems. To achieve equal results in both systems, it is not sufficient simply “to translate” one code to another code. There are numerous examples, where there exists some predefined function in one system and does not exist in another. Therefore, to get equal results in both systems, it is necessary to define new functions knowing the method or algorithm of calculation.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

Instead of the principle of maximal generality that is usual in mathematical books the author has attempted to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation; only then can the method developed be extended to more complicated cases. Although it is usually simpler to prove a general fact than to prove numerous special cases of it, for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood.

Duration” tells you how risky a bond is. The greater the duration, the greater the risk. For example, a ten-year bond has greater duration—and greater risk—than a one-year bond. That’s it. Mathematically, of course, duration is more complicated than this. It’s the length of time until you receive the average present value-weighted cash flow, and is itself a derivative (in calculus terms), of the partial differential equation that describes the price behavior of a bond.

IN THE SCHOOLS Memorizing multiplication tables may be a seminal school experience, among the few that kids today share with their grandparents. But a Stanford University professor says rapid-fire math drills are also the reason so many children fear and despise the subject. Moreover, the traditional approach to math instruction — memorization, timed testing and the pressure to speedily arrive at answers — may actually damage advanced-level skills by undermining the development of a deeper understanding about the ways numbers work. “There is a common and damaging misconception in mathematics — the idea that strong math students are fast math students,” says Jo Boaler, who teaches math education at the California university and has authored a new paper, “Fluency Without Fear.” In fact, many mathematicians are not speedy calculators, Boaler says. Laurent Schwartz, the French mathematician whose work is considered key to the theory of partial differential equations, wrote that as a student he often felt stupid because he was among the slowest math-thinkers in class.

This allows us, for example, to make sense of the plucked string, where the initial displacement is continuous, but not even once differentiable. This is a common phenomenon when solving partial differential equations. A technique which is very often used is to rewrite the equation as an integral equation, meaning an equation involving integrals rather than derivatives. Integrable functions are much more general than differentiable functions, so one should expect a more general class of solutions.

By the time he was in high school, his family had moved to Miami. Bezos was a straight-A student, somewhat nerdy, and still completely obsessed with space exploration. He was chosen as the valedictorian of his class, and his speech was about space: how to colonize planets, build space hotels, and save our fragile planet by finding other places to do manufacturing. “Space, the final frontier, meet me there!” he concluded. He went to Princeton with the goal of studying physics. It sounded like a smart plan until he smashed into a course on quantum mechanics. One day he and his roommate were trying to solve a particularly difficult partial differential equation, and they went to the room of another person in the class for help. He stared at it for a moment, then gave them the answer. Bezos was amazed that the student had done the calculation—which took three pages of detailed algebra to explain—in his head. “That was the very moment when I realized I was never going to be a great theoretical physicist,” Bezos says. “I saw the writing on the wall, and I changed my major very quickly to electrical engineering and computer science.” It was a difficult realization. His heart had been set on becoming a physicist, but finally he had confronted his own limits.

A hole in a hole in a hole—Numberphile Around the World in a Tea Daze—Shpongle But what is a partial differential equation?—Grant Sanderson, who owns the 3Blue1Brown YouTube channel Closer to You—Kaisaku Fourier Series Animation (Square Wave)—Brek Martin Fourier Series Animation (Saw Wave)—Brek Martin Great Demo on Fibonacci Sequence Spirals in Nature—The Golden Ratio—Wise Wanderer gyroscope nutation—CGS How Earth Moves—vsauce I am a soul—Nibana

Good afternoon,” Fletcher said to the young library aide who was working at the desk. “I am Dr. Duncan Fletcher, Fitzhugh Senior Fellow of applied mathematics, director for the Oxford Centre for Nonlinear Partial Differential Equations, and executive liaison between the university and the Alan Turing Institute. And you are…?

Even at the cutting edge of modern physics, partial differential equations still provide the mathematical infrastructure. Consider Einstein’s general theory of relativity. It reimagines gravity as a manifestation of curvature in the four-dimensional fabric of space-time. The standard metaphor invites us to picture space-time as a stretchy, deformable fabric, like the surface of a trampoline. Normally the fabric is pulled taut, but it can curve under the weight of something heavy placed on it, say a massive bowling ball sitting at its center. In much the same way, a massive celestial body like the sun can curve the fabric of space-time around it. Now imagine something much smaller, say a tiny marble (which represents a planet), rolling on the trampoline’s curved surface. Because the surface sags under the bowling ball’s weight, it deflects the marble’s trajectory. Instead of traveling in a straight line, the marble follows the contours of the curved surface and orbits around the bowling ball repeatedly. That, says Einstein, is why the planets go around the sun. They’re not feeling a force; they’re just following the paths of least resistance in the curved fabric of space-time.

The equations of fluid flow are nonlinear partial differential equations, unsolvable except in special cases. Yet Ruelle worked out an abstract alternative to Landau’s picture, couched in the language of Smale, with images of space as a pliable material to be squeezed, stretched, and folded into shapes like horseshoes.