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 calculus. This means that either may in fact be given priority, but the usual educational approach is to introduce differential calculus first.

History

See main article History of calculus

The development of calculus is credited to Archimedes, Leibniz and Newton. However, when calculus was first being developed, there was a 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 notation, 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 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. 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]

One of the primary motives for the development of differential calculus was the solution of the so-called "tangent line problem".

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 where 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.

de Fermat is sometimes described as the "father" of differential calculus.

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.

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.

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.

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 and differential topology.

External link

Other uses of the term

In mathematics and related fields, the term calculus more generally refers to a system of formal rules of inference and axioms that are used for computation.

This usage is particularly common in mathematical logic, where a calculus is applied to compute universally true statements of a certain formal logic. Examples include the calculus of natural deduction, the sequent calculus, as well as many other calculi that are deviced in proof theory.

Derived from the Latin word for "pebble", calculus in its most general sense can mean any method or system of calculation. Other topics where the term calculus is used in this sense include:

Lambda calculus (a formulation of the theory of reflexive functions with deep connections to computational theory; due in final form to Alonzo Church of Princeton)
Predicate calculus (the rules governing the logic of predicates in symbolic logic)

In medicine a calculus refers to a stone formed in the body such as a gall stone. See calculus (medicine)