There are several methods differential calculus book pdf defining infinitesimals rigorously, but it is sufficient to say that an infinitesimal number is smaller in absolute value than any positive real number, just as an infinitely large number is larger than any real number. This page was last edited on 15 December 2017, at 05:09.
This article is about the branch of mathematics. Greece, then in China and the Middle East, and still later again in medieval Europe and in India. 3rd century AD in order to find the area of a circle. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking. 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri’s work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. Europe at around the same time. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton’s time, the fundamental theorem of calculus was known. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name.
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today. The foundations of differential and integral calculus had been laid. Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called “infinitesimal calculus”.
The reach of calculus has also been greatly extended. Limits are not the only rigorous approach to the foundation of calculus. Leibniz-like development of the usual rules of calculus. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves. Calculus is also used to gain a more precise understanding of the nature of space, time, and motion.
Calculus is usually developed by working with very small quantities. These are objects which can be treated like real numbers but which are, in some sense, “infinitely small”. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In this treatment, calculus is a collection of techniques for manipulating certain limits.
Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine.
The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by deriving the squaring function turns out to be the doubling function. If the input of the function represents time, then the derivative represents change with respect to time. This gives an exact value for the slope of a straight line. Derivatives give an exact meaning to the notion of change in output with respect to change in input.