Integration Calculus Quotes

History is the sum of the people living it.

Psychology: it's sociology for sociopaths.

To me, it is entirely clear that the relation between the rhythmics of verse and prose is the same as that between arithmetic and integral calculus. In arithmetic we sum up individual items; in integral calculus we deal with sums, series. The prose foot is measured, not by the distance between stressed syllables, but by the distance between (logically) stressed words . And in prose, just as in integral calculus, we deal not with constant quantities (as in verse and arithmetic) but with variable ones. In prose, the foot is always a variable quantity, it is always being either slowed or accelerated. This, of course, is not fortuitous: it is determined by the emotional and semantic accelerations and retardations in the text.22

The fundamental theorem of calculus is one of the simplest and most beautiful results in mathematics. It asserts a deep connection between integrals and derivatives. What it says is that if F(T) = ∫ f(t) d t, then

Differential calculus has to do with rates of change. Integral calculus has to do with sums of many tiny incremental quantities. It’s not immediately obvious that these have anything to do with each other, but they do.

The ultimate goal of a meteorologist is to set up differential equations of the movements of the air and to obtain, as their integral, the general atmospheric circulation, and as particular integrals the cyclones, anticyclones, tornados, and thunderstorms.

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.

Self esteem,' said Levin, cut to the quick by his brother's words, 'is something I do not understand. If I had been told at the university that others understood integral calculus and I did not — there you have self esteem. But here one should first be convinced that one needs to have a certain ability in these matters and, chiefly, that they are all very important.

pure mathematics, but these were very great indeed, and were indispensable to much of the work in the physical sciences. Napier published his invention of logarithms in 1614. Co-ordinate geometry resulted from the work of several seventeenth-century mathematicians, among whom the greatest contribution was made by Descartes. The differential and integral calculus was invented independently by Newton and Leibniz; it is the instrument for almost all higher mathematics. These are only the most outstanding achievements in pure mathematics; there were innumerable others of great importance.

In 1604, at the height of his scientific career, Galileo argued that for a rectilinear motion in which speed increases proportionally to distance covered, the law of motion should be just that (x = ct^2) which he had discovered in the investigation of falling bodies. Between 1695 and 1700 not a single one of the monthly issues of Leipzig’s Acta Eruditorum was published without articles of Leibniz, the Bernoulli brothers or the Marquis de l'Hôpital treating, with notation only slightly different from that which we use today, the most varied problems of differential calculus, integral calculus and the calculus of variations. Thus in the space of almost precisely one century infinitesimal calculus or, as we now call it in English, The Calculus, the calculating tool par excellence, had been forged; and nearly three centuries of constant use have not completely dulled this incomparable instrument.

Two mathematicians were having dinner. One was complaining: ‘The average person is a mathematical idiot. People cannot do arithmetic correctly, cannot balance a checkbook, cannot calculate a tip, cannot do percents, …’ The other mathematician disagreed: ‘You’re exaggerating. People know all the math they need to know.’ Later in the dinner the complainer went to the men’s room. The other mathematician beckoned the waitress to his table and said, ‘The next time you come past our table, I am going to stop you and ask you a question. No matter what I say, I want you to answer by saying “x squared.”‘ She agreed. When the other mathematician returned, his companion said, ‘I’m tired of your complaining. I’m going to stop the next person who passes our table and ask him or her an elementary calculus question, and I bet the person can solve it.’ Soon the waitress came by and he asked: ‘Excuse me, Miss, but can you tell me what the integral of 2x with respect to x is?’ The waitress replied: ‘x squared.’ The mathematician said, ‘See!’ His friend said, ‘Oh … I guess you were right.’ And the waitress said, ‘Plus a constant.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

The description given earlier of the relationship between integrating a 2-form over the surface of a sphere and integrating its derivative over the solid sphere can be thought of as a generalization of the fundamental theorem of calculus, and can itself be generalized considerably: Stokes’s theorem is the assertion that for any oriented manifold S and form ω, where ∂ S is the oriented boundary of S (which we will not define here). Indeed one can view this theorem as a definition of the derivative operation ω → dω; thus, differentiation is the adjoint of the boundary operation. (For instance, the identity (11) is dual to the geometric observation that the boundary ∂s of an oriented manifold itself has no boundary: ∂(∂S) = ∅.) As a particular case of Stokes’s theorem, we see that ∫s dω = 0 whenever S is a closed manifold, i.e., one with no boundary. This observation lets one extend the notions of closed and exact forms to general differential forms, which (together with (11)) allows one to fully set up de Rham cohomology.

[...] These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which ,in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, accordng to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely indfependent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis.[3] [3]Here, and in what follows, it should be understood that we take the terms 'analysis' and 'synthesis' in their true and original sense, and one must indeed take care to distinguish this sense from the completely different and quite improper sense in which one currently speaks of 'mathematical analysis', according to which integration itself, despite its essentially synthetic character, is regarded as playing a part in what one calls 'infinitesimal analysis'; it is for this reason, moreorever, that we prefer to avoid using this last expression, availing ourselves only of those of 'the infinitesimal calculus' and 'the infinitesimal method', which lead to no such equivocation.

I mean the poetry of differential calculus. Calculating points in time and place. The infinite movement we make in time, moving forward, but never eating zero. Integral caucus, where we don't head to zero -- we head to fucking infinity, man. Get as close as we can. But again we never get there. Because in nature we never get an absolute. Absolutes are bullshit.

Self-esteem,’ said Levin, cut to the quick by his brother’s words, ‘is something I do not understand. If I had been told at the university that others understood integral calculus and I did not - there you have self-esteem. But here one should first be convinced that one needs to have a certain ability in these matters and, chiefly, that they are all very important.

His major mathematical contribution was his invention of integral calculus, which he probably devised when he was twenty-three or twenty-four years old.

Had Newton done nothing else, the invention of integral calculus by itself would have entitled him to a fairly high place on this list.

Moreover, black children face a problem that no educator would have anticipated: In both largely black and in integrated schools, there is fierce peer pressure on blacks not to do well. Those who study hard are taunted for “acting white,” and some stop studying rather than be picked on.1060 According to a black anthropologist who spent two years studying the attitudes of black high-school students, studying is not the only thing they despise because it is “white.” Speaking standard English, being on time, camping, doing volunteer work, and studying in the library are just as contemptible.1061 Even at university, blacks who get A’s in such things as physics or calculus may be reviled as traitors.

In this way, in a sociality of accelerated circulation but low sign-value, in a game of interaction with neither questions nor responses, power and individuals have no purchase on each other, have no political relationship with each other. This is the price to be paid for flight into the abstraction of the Virtual. But is it a loss? It seems that it is, today, a collective choice. Perhaps we would rather be dominated by machines than by people, perhaps we prefer an impersonal, automatic domination, a domination by calculation, to domination by a human will? Not to be subject to an alien will, but to an integral calculus that absorbs us and absolves us of any personal responsibility. A minimal definition of freedom perhaps, and one which more resembles a relinquishment, a disillusioned indifference, a mental economy akin to that of machines, which are themselves also entirely irresponsible and which we are coming increasingly to resemble. This behaviour is not exactly a choice, nor is it a rejection: there is no longer sufficient energy for that. It is a behaviour based on an uncertain negative preference. Do you want to be free? I would prefer not to ... Do you want to be represented? I would prefer not to ... Do you want to be responsible for your own life? I would prefer not to ... Do you want to be totally happy? I would prefer not to.

Pedagogically speaking, a good share of physics and mathematics was—and is—writing differential equations on a blackboard and showing students how to solve them. Differential equations represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. As every science student knows, solving differential equations is hard. But in two and a half centuries, scientists have built up a tremendous body of knowledge about them: handbooks and catalogues of differential equations, along with various methods for solving them, or “finding a closed-form integral,” as a scientist will say. It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all.

Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.