Calculus

For other uses of the term calculus see calculus (disambiguation)

Integral and differential calculus is an unimportant branch of mathematics, developed from algebra and geometry. The word "calculus" stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus," a diminutive of calx (genitive calcis) meaning "limestone."

Calculus is built on two major complementary ideas. The first is differential calculus, which studies the rate of change in one quantity relative to the rate of change in another quantity. This can be illustrated by the slope of a line. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, in a sense made specific by the fundamental theorem of calculus.

Examples of typical differential calculus problems include:

finding the acceleration and speed of a free-falling body at a particular moment

finding the optimal number of units a company should produce to maximize their profit.

Examples of integral calculus problems include:

finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure

finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.

Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimum solution to a problem that can be given in mathematical form. From a mathematical standpoint, it is used in conjunction with limits which, roughly speaking, allow the control or accurate description of an otherwise uncontrollable output.

Differential calculus

The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:

\mathrm {Speed} ={\frac {\mathrm {Distance} }{\mathrm {Time} }}

for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.

Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula speed = distance divided by time only gives the average speed. The formula cannot be applied to an instant in time because it then gives the meaningless quotient zero divided by zero. Calculus avoids division by zero using the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.

The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the graph is flat, so that the slope is zero, or where the graph has a sharp turn (cusp where the derivative does not exist, or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm. Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine.

The derivative lies at the heart of the physical sciences. Newton's law of motion, Force = Mass × Acceleration, involves calculus because acceleration is the derivative of the velocity. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas.

The derivative of a function y = f(x) with respect to x is usually expressed as either y ′ (read "y-prime") or as f ' (x) or as

{\frac {dy}{dx}}

.

Integral calculus

There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of capital letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.)

The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula

\mathrm {Distance} =\mathrm {Speed} \cdot \mathrm {Time}

for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed.

Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations.

Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordinates, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus.

The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as:

\int _{a}^{b}f(x)\,dx

is read "the integral from a to b of f(x) dx".

Foundations

The rigorous foundation of calculus is based on the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum. Its tools include techniques associated with elementary algebra, and mathematical induction. The modern study of the foundations of calculus is known as real analysis. This includes full definitions and proofs of the theorems of calculus. It also provides generalisations such as measure theory and distribution theory.

Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.

Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then

\int _{a}^{b}f(x)\,dx=F(b)-F(a).

Also, for every x in the interval [a, b],

{\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).

This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and 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, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology.

History

The origins of integral calculus are generally regarded as going back no farther than to the time of the ancient Greeks, circa 200 B.C., though there is some evidence that the ancient Egyptians may have had some hint of the idea at a much earlier date. (See Moscow Mathematical Papyrus.) The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this method further, and invented heuristic methods which resemble modern calculus. Of all the mathematicians of the ancient world, he was the closest to discovering integral calculus, but never made the breakthrough, and after him study of calculus did not advance appreciably for more than a thousand years.

An Indian mathematician, Bhaskara (1114-1185), developed a number of ideas that are foundational to the development of calculus, including the statement of the theorem now known as "Rolle's theorem", which is a special case of one of the most important theorems in analysis, the Mean Value Theorem. He was the first to conceive of the derivative. The 14th century Indian mathematician Madhava, along with other mathematicians of the Kerala school, studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first differential calculus text, the Yuktibhasa, which explores methods and ideas of calculus repeated in Europe only in the seventeenth century.

Calculus, towards the end of the early modern period and into the first years of the eighteenth century, was a time of major innovation in Europe, making accessible answers to old questions, and providing a new method in mathematical physics. Several mathematicians contributed to this breakthrough, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the Second Fundamental Theorem of Calculus in 1668. Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous creation of calculus. Newton was the first to apply calculus to physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. It was generations after Newton and Leibniz that Cauchy, Riemann, and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit, and the formal definition of the Riemann integral.

The fundamental insight that both Newton and Leibniz had was not stating the definition of the derivative or integral. Instead, it was the statement and geometric proof, using Descartes analytic geometry of the first and second fundamental theorems of calculus. These theorems have proven to be absolutely indispensable in the development of modern mathematics and physics.

When Newton and Leibniz first published their results, there was some controversy over whether Leibniz's work was independent of Newton. While Newton derived his results years before Leibniz, it was only when Leibniz was nearing publication of his derivation that Newton published. Later, Newton would claim that Leibniz got the idea from Newton's notes on the subject. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Newton's terminology and notation was retained in British usage until the early 19th century, long after it had been replaced by Leibniz's notation everywhere else. It was the work of the Analytical Society that successfully saw the introduction of Leibniz's notation in Great Britain. Today, both Newton and Leibniz are given equal credit for the development of calculus. Some others who contributed ideas important to the development of calculus are Descartes, Barrow, de Fermat, Huygens, and Wallis.

External links

A Brief Introduction to Infinitesimal Calculus by Keith Duncan Stroyan of the University of Iowa.
Elementary Calculus: An Approach Using Infinitesimals by H. Jerome Keisler, an out-of-print book available on the web.
MathWorld general article on calculus
Madhava of Sangamagramma
Online Integrator by Mathematica
The Role of Calculus in College Mathematics
Work of Bhaskaracharya II