Calculus History Differential Calculus Integral Calculus

Calculus is a branch of mathematics, developed from algebra and geometry, involving two major complementary ideas: The first, called differential calculus is a theory about rates of change, and involves the method of differentiation; in terms of mathematical functions, velocity, acceleration, and slopes of curves at a given point can all be discussed on a common symbolic basis. The second, called integral calculus, involves the idea of integration, and uses a general idea of area bounded by the graph of a function, to include related concepts such as volume.

The two concepts define inverse operations, in a sense made quite precise by the fundamental theorem of calculusThe fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. This means that if a continuous function is first integrated and then differentiated, the origin. This means that either may in fact be given priority, but the usual educational approach is to introduce differential calculus first.

1 History

See main article History of calculusAlthough Archimedes and others have used integral methods throughout history, and a great many (Barrow, Fermat, Pascal, Wallis and others) had previously invented the idea of a derivative, Gottfried Wilhelm Leibniz and Isaac Newton are usually credited wi

The development of calculus is credited to ArchimedesSee also Archimedes computer, Archimedes (disambiguation). Archimedes of Syracuse (circa 287 BC 212 BC), was a Greek mathematician, astronomer, philosopher, physicist and engineer. He was killed by a Roman soldier during the sack of the city, despite orde, LeibnizGottfried Wilhelm von Leibniz ( July 1, 1646 in Leipzig November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. Leibniz is credited with the term " function" ( 1694), which he use and NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis. One of the primary motives for the development of the differential branch was the solution of the so-called " tangentIn mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Geometry In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the poi line problem". When calculus was first being developed, there was a significant controversy to who came up with the idea first - Leibniz and Newton being the contenders for the crown. The truth of the matter will likely never be known, and in any case is unimportant to anyone alive today. Leibniz' great contribution to calculus was his notationMathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. The complexity of such notation ranges from relatively simple symbolic representations, such as one and two; to conceptual symbols, such as + a, and this is beyond doubt purely of Leibniz's invention. The controversy was unfortunate however in that it divided english-speaking mathematicians from those in Europe for many years. This set back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than that of Leibniz, yet it was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now thought that Newton had discovered several ideas related to calculus earlier than Leibniz had, however Leibniz was the first to publish. Today, both Leibniz and Newton are considered to have discovered calculus independently.

Lesser credit for the development of calculus is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. Specifically, de Fermat is sometimes described as the "father" of differential calculus. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus, though this was not known in the West at the time, and he had no contact with Western scholars. [1]

2 Differential calculus

Main article derivative

Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes of the function's arguments. This idea lies at the heart of most of the physical sciences. For example basic theory of electrical circuits is formulated in terms of differential equations, to describe the cases when there is oscillation.

The derivative of a function is directly relevant to finding its maxima and minima — because those are points at which the graph is (expected to be) flat. Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the function by its tangents. These are just some of a large number of ways in which calculus is applied in questions that at first sight are not formulated in calculus terms.

3 Integral calculus

Main article integral

Integral calculus studies methods for finding the integral of a function; which may be defined as the limit of a sum of terms, each of which corresponds to a small strip of area under the graph of a function. Considered as such, integration provides effective ways to calculate the area under a curve, and the surface area and volume of solids such as spheres and cones.

4 Foundations

The conceptual foundations of calculus include the notions of functions, limits, infinite sequences, infinite series, and continuity. Its tools include the symbol manipulation techniques associated with elementary algebra, and mathematical induction. The modern version of calculus is known as real analysis; this consists of a rigorous derivation of the results of calculus as well as generalisations such as measure theory and functional analysis.

5 Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. It was this realization by Newton and Leibniz that was the key to the explosion of analytic results after their work became known. This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many definite integrals algebraically, without actually performing the limit processes, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivatives. Differential equations are ubiquitous in the sciences.

6 Applications

The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, and especially physics. Almost all modern developments such as building techniques, aviation, and nearly all other technologies make fundamental use of calculus.

Calculus has been extended to differential equations, vector calculus, calculus of variations, complex analysis, time scale calculus infinitesimal calculus, and differential topology.

7 See also

8 Further reading

Tom M. Apostol. (1967) BooksEnthsiast.com and BooksEnthsiast.com Calculus, 2nd Ed. Wiley.
Robert A. Adams. (1999) BooksEnthsiast.com Calculus: A complete course.
Spivak, Michael. (Sept 1994) BooksEnthsiast.com Calculus. Publish or Perish publishing.
Cliff Pickover. (2003) BooksEnthsiast.com Calculus and Pizza: A Math Cookbook for the Hungry Mind.
Silvanus P. Thompson and Martin Gardner. (1998) BooksEnthsiast.com Calculus Made Easy.
Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey, Mathematical Association of America No. 7, 1986.
Calculus for a New Century; A Pump, Not a Filter. Mathematical Association of America, The Association, Stony Brook, NY. 1988. ED 300 252.
Elementary Calculus: An Approach Using Infinitesimals http://www.math.wisc.edu/~keisler/calc.html

9 External links

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