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- 1. Sir Isaac Newton was an English physicist, mathematician, astrono mer, natural philosopher, alchemist, and theologian, who has been considered by many to be the greatest and most influential scientist who ever lived. Newton described universal gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries.
- 2. Isaac Newton was born on what is retroactively considered 4 January 1643 at Woolsthorpe Manor in Woolsthorpe-by- Colsterworth, a hamlet in the county of Lincolnshire. At the time of Newtons birth, England had not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, 25 December 1642. Newton was born three months after the death of his father, a prosperous farmer also named Isaac Newton.
- 3. Newton was educated at The Kings School, Grantham , and in October 1659 he was removed from school. In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role. In 1665, he discovered the generalized binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague.
- 4. Newton received a bachelor’s degree at Trinity College, Cambridge in 1665 The next two years Newton returned home where he came up with most of his discoveries. He returned to Trinity College in 1667, where he became a professor of mathematics in 1669.
- 5. King school Trinity College
- 6. Newtons work has been said "to distinctly advance every branch of mathematics then studied". His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers.
- 7. Calculus was invented by sir Isaac Newton Isaac Newton developed the use of calculus in his laws of motion and gravitation.
- 8. Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
- 9. Calculus has historically been called "the calculusof infinitesimals", or "infinitesimal calculus". More generally, calculus refers to any method or system ofcalculation guided by the symbolic manipulation ofexpressions.Some examples of other well-known calculiare propositional calculus, variational calculus, lambdacalculus, pi calculus, and join calculus .
- 10. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
- 11. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are mere instructions, with no indication as to method, and some of them are wrong. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resembles the methods of integral calculus.
- 12. The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere.
- 13. In the 14th Century Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated many components of calculus such as the Taylor series, infinite series approximations, an integral test for convergence, early forms of differentiation, term by term integration, iterative methods for solutions of non- linear equations, and the theory that the area under a curve is its integral. Some consider the Yuktibhāṣā to be the first text on calculus.
- 14. In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
- 15. Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and 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 Newtons time, the fundamental theorem of calculus was known.
- 16. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental 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. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".
- 17. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on finite and infinitesimal analysis was written in 1748 by Maria Gaetana Agnesi
- 18. Sir Isaac Newton portrait
- 19. While some of the ideas of calculus had been developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
- 20. In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. 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 are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
- 21. differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.
- 22. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.
- 23. In calculus, Leibnizs notation, named in honor of the 17th- century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" (or infinitesimal) increments of x and y, just as Δx and Δy represent finite increments of x and y.
- 24. Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators.
- 25. The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
- 26. In geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point—that is, coincides with the curve at that point without crossing to the other side of the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point(c, f(c)) on the curve and has slope f(c) where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
- 27. Tangent Tangent Tangentgraph circle line
- 28. the maximum and minimum of a function, known collectively as extrema are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety More generally, the maximum and minimum of a set are the greatest and least element in the set. Unbounded infinite sets such as the set of real numbers have no minimum and maximum.
- 29. In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. The Newton polynomial is sometimes called Newtons divided differences interpolation polynomial because the coefficients of the polynomial are calculated using divided differences. For any given set of data points, there is only one polynomial (of least possible degree) that passes through all of them. Thus, it is more appropriate to speak of "the Newton form of the interpolation polynomial" rather than of "the Newton interpolation polynomial". Like the Lagrange form, it is merely another way to write the same polynomial.
- 30. Newtons formula is of interest because it is the straightforward rate of change of its rate of change, etc. at one particular x value. Newtons formula is Taylors polynomial based on finite differences instead of instantaneous rates of change.
- 31. Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below the x axis subtract from the total.
- 32. There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lévesque integrals.
- 33. A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.
- 34. The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ(x, y) or ƒ(x, y, z). Integrals of a function of two variables over a region in ℝ2 are called double integrals.
- 35. The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
- 36. A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
- 37. A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.
- 38. In the 1690s, Newton wrote a number of religious tracts dealing with the literal interpretation of the Bible. Henry Moores belief in the Universe and rejection of Cartesian dualism may have influenced Newtons religious ideas. A manuscript he sent to John Locke in which he disputed the existence of the Trinity was never published. Later works – The Chronology of Ancient Kingdoms Amended (1728) and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John (1733) – were published after his death. He also devoted a great deal of time to alchemy (see above). Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed.
- 39. Newton died in his sleep in London on 31 March 1727 and was buried in Westminster Abbey. Newton, a bachelor, had divested much of his estate to relatives during his last years, and died intestate. After his death, Newtons hair was examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newtons eccentricity in late life.
- 40. French mathematician Joseph-Louis Lagrange often said that Newton was the greatest genius who ever lived, and once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to establish." English poet Alexander Pope was moved by Newtons accomplishments to write the famous epitaph: Nature and natures laws lay hid in night; God said "Let Newton be" and all was light. Newton himself had been rather more modest of his own achievements, famously writing in a letter to Robert Hooke in February 1676: If I have seen further it is by standing on the shoulders of giants.
- 41. Newtons monument (1731) can be seen in Westminster Abbey, at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor Michael Rysbrack(1694–1770) in white and grey marble with design by the architect William Kent. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts putti using instruments such as a telescope and prism.[71] The Latin inscription on the base translates as:
- 42. Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners.

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