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Calculus

Calculus

:For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry. The origin of the word 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 operations are inverses of one another, as explained 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 = \frac

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 controling an otherwise uncontrolable 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 with respect to x is usually expressed as either y′ (read "y-prime") or as :

\frac

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 captial 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 = \mathrm \cdot \mathrm

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 coordiantes, 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 fuction 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_^ f(x)\,dx = F(b) - F(a).

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

\frac\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 can now be seen to be forerunners of calculus, including the idea now known as "Rolle's theorem". He was the first to conceive of differential calculus. 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 calculus text, the Yuktibhasa, which explores methods and ideas of calculus repeated in Europe only by 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 result equivalent to the 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 and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit. There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of calculus. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. The mathematical proofs of much of what they did came later, with Cauchy and others. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, setting back British analysis (calculus-based mathematics) for a long time. 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. 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; but Leibniz published first. Today, both are given equal credit. Lesser credit for ideas that led to the development of calculus is given to Descartes, Barrow, de Fermat, Huygens, and Wallis.

External links

- [http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm A Brief Introduction to Infinitesimal Calculus] by Keith Duncan Stroyan of the University of Iowa.
- [http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Approach Using Infinitesimals] by H. Jerome Keisler, an out-of-print book available on the web.
- [http://mathworld.wolfram.com/Calculus.html MathWorld general article on calculus]
- [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_3.html Madhava of Sangamagramma ]
- [http://integrals.wolfram.com/ Online Integrator by Mathematica]
- [http://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematics]
- [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html Work of Bhaskaracharya II]
- ja:微分積分学 ko:미적분학 simple:Calculus th:แคลคูลัส

Calculus (disambiguation)

Calculus is Latin for pebble, and has a number of meanings in English.

Mathematical meanings

Calculus in its most general sense can mean any method or system of calculation.
- In mathematics, calculus most commonly refers to elementary mathematical analysis, which investigates motion and rates of change. The denotation "the calculus" is sometimes used to distinguish this from other mathematical meanings.
- In symbolic logic: the predicate calculus is the rules of inference governing the logic of predicates; and a proof calculus is a framework for expressing systems of logical inference.
- 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 University.
- Pi-calculus, a formulation of the theory of concurrent, communicating processes, invented by Robin Milner.
- Join calculus, a theoretical model for distributed programming.
- Tuple calculus, a calculus for the relational data model, inspired the SQL language.
- the calculus of sums and differences, also called the finite-difference calculus.
- in mathematics education, Precalculus is a family of mathematical topics that prepare students to begin to study differential and integral calculus.

Other meanings

- In dentistry, calculus consists of deposits of calcium phosphate salts on teeth, also known as tartar. See calculus (dental)
- In medicine a calculus is a stone formed in the body such as a gall stone or kidney stone. See calculus (medicine)
- In The Adventures of Tintin, Cuthbert Calculus is a hard-of-hearing professor and inventor and is a dear friend of Tintin, Captain Haddock, and others.

Algebra

:This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation). Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world.

History

The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
- Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
- Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
- Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
- Circa 200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
- 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to modern one. Bhaskara II (1114 AD), who wrote the text Bijaganita (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
- 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion".
- 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided into the following categories:
- elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
- abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
- linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- universal algebra, in which properties common to all algebraic structures are studied. In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups

Algebras

The word algebra is also used for various algebraic structures:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory

References

- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).

External links

- [http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics]
- [http://www.sparknotes.com/math/#algebra1 Sparknotes' Review of Algebra I and II]
- [http://www.jamesbrennan.org/algebra/ Understanding Algebra.] An online algebra text by James W. Brennan. Category:Algebra Category:Arabic words ko:대수학 ms:Algebra ja:代数学 simple:Algebra

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.

Differential calculus

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Slope

In mathematics, the slope or the gradient of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of the line relative to the horizontal axis. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of the tangent to a curve at a point. The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Definition of slope

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: :

m = \frac

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".) Given two points (x₁, y₁) and (x₂, y₂), the change in x from one to the other is x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following: :

m = \frac

Since the y-axis is vertical and the x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where Δy is the "rise" and Δx is the "run". Therefore, by convention, m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is, m is the ratio of the changes. This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus. Note that the points chosen and the order in which they are used is irrelevant; the same line will always have the same slope. Other curves have "accelerating" slopes and one can use calculus to determine such slopes.

Example 1

Suppose a line runs through two points: P(13,8) and Q(1,2). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line: :

m = \frac = \frac = \frac = \frac = \frac

The slope is 1/2 = 0.5.

Example 2

If a line runs through the points (4, 15) and (3, 21) then: :

m = \frac = \frac = -6

Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞). The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function: :

m = \tan\,\theta

and :

\theta = \arctan\,m

(see trigonometry). Two lines are parallel if and only if their slopes are equal or if they both are vertical and therefore undefined; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1 or one has a slope of 0 and the other is vertical and undefined.

Slope of a road, etc.

perpendicular There are two common ways to describe how steep a road is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. The formula for converting a slope in percentage to degrees is: :

\theta = \arctan\frac.

Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form :

y = mx + b \,

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the slope m of a line and a point (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula: :

y - y_0 = m(x - x_0) \,

For example, consider a line running through the points (2, 8) and (3, 20). This line has a slope, m, of (20 - 8) / (3 - 2) = 12. One can then write the line's equation, in point-slope form: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16. The slope of a linear equation in the general form: :

Ax + By + C = 0 \,

is given by the formula: −A/B.

Calculus

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

Why calculus is necessary

tangent If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, :

m = \frac

, is the slope of a secant line to a curve. For a line, the secant between any two points is identical to the line itself; however, this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1.5). However, by moving the points used in the above formula closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve. It follows that the secant line is identical to the tangent line when Δy and Δx equal zero; however, this results in a slope of 0/0, which is an indeterminate form (see also division by zero). The concept of a limit is necessary to calculate this slope; the slope is the limit of Δy / Δx as Δy and Δx approach zero. However, Δx and Δy are interrelated such that it is sufficient to take the limit where only Δx approaches zero. This limit is the derivative of y with respect to x. It may be written (in calculus notation) as dy/dx.

Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

Definitions

In mathematics, a (topological) curve is defined as follows. Let

I

be an interval of real numbers (i.e. a non-empty connected subset of

\mathbb

). Then a curve

\!\,\gamma

is a continuous mapping

\,\!\gamma : I \rightarrow X

, where

X

is a topological space. The curve

\!\,\gamma

is said to be simple if it is injective, i.e. if for all

x

y

I

, we have

\,\!\gamma(x) = \gamma(y) \rightarrow x = y

. If

I

is a closed bounded interval

\,\![a, b]

, we also allow the possibility

\,\!\gamma(a) = \gamma(b)

(this convention makes it possible to talk about closed simple curve). If

\gamma(x)=\gamma(y)

for some

x\ne y

(other than the extremities of

I

), then

\gamma(x)

is called a double (or: multiple) point of the curve. A curve

\!\,\gamma

is said to be closed or a loop if

\,\!I = [a, b]

and if

\!\,\gamma(a) = \gamma(b)

. A closed curve is thus a continuous mapping of the circle

S^1

; a simple closed curve is also called a Jordan curve. A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example.

Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Length of curves

X

is a metric space with metric

d

, then we can define the length of a curve

\!\,\gamma : [a, b] \rightarrow X

by :

\mbox (\gamma)=\sup \left\

A rectifiable curve is a curve with finite length. A parametrization of

\!\,\gamma

is called natural (or unit speed or parametrised by arc length) if for any

t_1

t_2

[a, b]

, we have :

\mbox (\gamma|_)=|t_2-t_1|

\!\,\gamma

is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of

\!\,\gamma

t_0

as :

\mbox(t_0)=\limsup_

and then :

\mbox(\gamma)=\int_a^b \mbox(t) \, dt

In particular, if

X = \mathbb^n

is Euclidean space and

\gamma : [a, b] \rightarrow \mathbb^n

is differentiable then :

\mbox(\gamma)=\int_a^b \left| \,  \, \right| \, dt

Differential geometry

Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If

X

is a differentiable manifold, then we can define the notion of differentiable curve in

X

. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take

X

to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to

X

by means of this notion of curve. If

X

is a smooth manifold, a smooth curve in

X

is a smooth map :

\!\,\gamma : I \rightarrow X.

This is a basic notion. There are less and more restricted ideas, too. If

X

is a

C^k

manifold (i.e., a manifold whose charts are

k

times continuously differentiable), then a

C^k

curve in

X

is such a curve which is only assumed to be

C^k

(i.e.

k

times continuously differentiable). If

X

is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and

\!\,\gamma

is an analytic map, then

\!\,\gamma

is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two

C^k

differentiable curves :

\!\,\gamma_1 :I \rightarrow X

and :

\!\,\gamma_2 : J \rightarrow X

are said to be equivalent if there is a bijective

C^k

map :

\!\,p : J \rightarrow I

such that the inverse map :

\!\,p^ : I \rightarrow J

is also

C^k

, and :

\!\,\gamma_(t) = \gamma_(p(t))

for all

t

. The map

\!\,\gamma_2

is called a reparametrisation of

\!\,\gamma_1

; and this makes an equivalence relation on the set of all

C^k

differentiable curves in

X

. A

C^k

arc is an equivalence class of

C^k

curves under the relation of reparametrisation.

Algebraic curve

Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.

History

A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html List of famous curves] Category:Curves Category:Metric geometry Category:Topology Category:General topology ko:곡선 ja:曲線

Volume

Volume, also called capacity, is a quantification of how much space an object occupies. The international unit for volume is the cubic meter. The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the three-dimensional space. Mathematically, volumes are defined by means of integral calculus, by approximating the given body with a large amount of small cubes, and adding the volumes of those cubes. The generalization of volume to arbitrarily many dimensions is called content. In differential geometry, volume is expressed by means of the volume form. Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units). Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure.

Volume formula

Common equations for volume: :A cube:

s^3 = s \cdot s \cdot s

(where s is the length of a side) : :A rectangular prism:

l \cdot w \cdot h

(length, width, height) : :A cylinder:

\pi \cdot r^2 h

(r = radius of circular face, h = distance between faces) : :A sphere:

\frac \pi r^3

(r = radius of sphere) : :An ellipsoid:

\frac \pi abc

(a, b, c = semi-axes of ellipsoid) : :A pyramid:

\frac A h

(A = area of base, h = height from base to apex) : :A cone (circular-based pyramid):

\frac \pi r^2 h

(r = radius of circle at base, h = distance from base to tip) : :Any prism that has a constant cross sectional area along the height
- :

A \cdot h

(A = area of the base, h = height) : :Any figure (calculus required) :

\int A(h) dh

where h is any dimension of the figure, and A(h) is the area of the cross-sections perpendicular to h described as a function of the position along h; this will work for any figure (no matter if the prism is slanted or the cross-sections change shape). The volume of a parallelepiped is the absolute value of the scalar triple product of the subtending vectors, or equivalently the absolute value of the determinant of the corresponding matrix. The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.

Volume measures: other metric units

A commonly used metric unit for volume is the litre (American spelling liter), and one thousand litres is the volume of a cubic metre (American spelling cubic meter), which was formerly termed a stere and often called a "cube" in engineering slang. A cubic centimetre (American spelling cubic centimeter) is the same volume as a millilitre.

Volume measures: USA

U.S. customary units of volume:
- U.S. fluid ounce, about 29.6 mL
- U.S. liquid pint = 16 fluid ounces, or about 473 mL
- U.S. dry pint = 1/64 U.S. bushel, or about 551 mL (used for things such as blueberries)
- U.S. liquid quart = 32 fluid ounces or two U.S. pints, or about 946 mL
- U.S. dry quart = 1/32 U.S. bushel, or about 1.101 L
- U.S. gallon = 128 fluid ounces or four U.S. quarts, about 3.785 L
- U.S. dry gallon = 1/8 U.S. bushel, or about 4.405 L
- U.S. (dry level) bushel = 2150.42 cubic inches, or about 35.239 L The acre foot is often used in measuring the volume of water in a reservoir or an aquifer. It is the volume of water that would cover an area of one acre to a depth of one foot. It is equivalent to 43,560 cubic feet or exactly 1233.481 837 547 52 m³.
- cubic inch = 16.387 064 cm³
- cubic foot = 1,728 in³ ≈ 28.317 dm³
- cubic yard = 27 ft³ ≈ 0.7646 m³
- cubic mile = 5,451,776,000 yd³ = 3,379,200 acre-feet ≈ 4.168 km³

Volume measures: UK

Imperial units of volume:
- UK fluid ounce, about 28.4 mL (this equals the volume of an avoirdupois ounce of water under certain conditions)
- UK pint = 20 fluid ounces, or about 568 mL
- UK quart = 40 ounces or two pints, or about 1.137 L
- UK gallon = 160 ounces or four quarts, or exactly 4.546 09 L May it be noted that due to metrication within the UK, the quart is now obsolete and the fluid ounce extremely rare. The gallon is only used for transportation uses, (it is illegal for petrol & diesel to be sold by the gallon). The pint is the only Imperial unit that is in everyday use, for the sale of draught beer & cider (bottled & canned beer is sold in SI units) and for milk (this too is increasingly being sold in SI units).

Volume measures: cooking

Traditional cooking measures for volume also include:
- teaspoon = 1/6 U.S. fluid ounce (about 4.929 mL)
- teaspoon = 1/6 Imperial fluid ounce (about 4.736 mL) (Canada)
- teaspoon = 5 mL (metric)
- tablespoon = 1/2 U.S. fluid ounce or 3 teaspoons (about 14.79 mL)
- tablespoon = 1/2 Imperial fluid ounce or 3 teaspoons (about 14.21 mL) (Canada)
- tablespoon = 15 mL or 3 teaspoons (metric)
- tablespoon = 5 fluidrams (about 17.76 mL) (British)
- cup = 8 U.S. fluid ounces or 1/2 U.S. liquid pint (about 237 mL)
- cup = 8 Imperial fluid ounces or 1/2 fluid pint (about 227 mL) (Canada)
- cup = 250 mL (metric)

Relationship to density

The volume of an object is equal to its mass divided by its average density. This is a rearrangement of the calculation of density as mass per unit volume. The term specific volume is used for volume divided by mass. This is the reciprocal of the mass density, expressed in units such as cubic meters per kilogram (m³/kg).

Volume comparisons

To help compare different volumes, see orders of magnitude (volume)

External links

- [http://www.unitconversion.org/unit_converter/volume.html Online Volume Converter - convert between various units of volume, such as cubic meter, liter, barrel, tablespoon, cup, and so on]
- [http://www.unitconversion.org/unit_converter/volume-v.html Interactive Volume Conversion Table - convert selected unit to all other units of volume]
- [http://www.unitconversion.org/unit_converter/volume-dry-v.html Volume - Dry Online Interactive Unit Converter]
- [http://jumk.de/calc/volume.shtml Volume or capacity conversion of English and American units to metric units]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.ex.ac.uk/trol/scol/ccvol.htm Conversion Calculator for Units of Volume for many units (Cleave Books)]
- ko:부피 ja:体積 simple:Volume

Inverse operation

In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f⁻¹ is its inverse function if and only if for every

x \in X

we have: :

f^(f(x))=f(f^(x))=x.\,

For example, if the function x → 3x + 2 is given, then its inverse function is x → (x−2) / 3. This is usually written as: :

f\colon x\to 3x+2

f^\colon x\to(x-2)/3

The superscript "−1" is not an exponent. Similarly, as long as we are not in trigonometry, f ²(x) means "do f twice", that is f(f(x)), not the square of f(x). For example, if : f : x → 3x + 2, then f ² : x = 3 ((3x + 2)) + 2, or 9x + 8. However, in trigonometry, for historical reasons, sin²(x) usually does mean the square of sin(x). As such, the prefix arc is sometimes used to denote inverse trigonometric functions, e.g. arcsin x for the inverse of sin(x). If a function f has an inverse then f is said to be invertible.

Simplifying rule

Generally, if f(x) is any function, and g is its inverse, then g(f(x)) = x and f(g(x)) = x. In other words, an inverse function undoes what the original function does. In the above example, we can prove f⁻¹ is the inverse by substituting (x − 2) / 3 into f, so : 3(x − 2) / 3 + 2 = x. Similarly this can be shown for substituting f into f⁻¹. Indeed, an equivalent definition of an inverse function g of f, is to require that g o f be the identity function on the domain of f, and f o g be the identity function on the codomain of f, where "o" represents function composition.

Existence

For a function f to have a valid inverse, it must be a bijection, that is:
- (f is onto) each element in the codomain must be "hit" by f: otherwise there would be no way of defining the inverse of f for some elements.
- (f is one-to-one) each element in the codomain must be "hit" by f only once: otherwise the inverse function would have to send that element back to more than one value. If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test, that is a horizontal line

y=k

placed on the graph of f must pass through f exactly once for all real k. It is possible to work around this condition, by redefining fs codomain to be precisely its range, and by admitting a multi-valued function as an inverse. If one represents the function f graphically in an x-y coordinate system, then the graph of f⁻¹ is the reflection of the graph of f across the line y = x. Algebraically, one computes the inverse function of f by solving the equation : $y=f(x)$ for x, and then exchanging y and x to get : $y=f^(x)$ This is not always easy; if the function f(x) is analytic, the Lagrange inversion theorem may be used. The symbol f⁻¹ is also used for the (set valued) function associating to an element or a subset of the codomain, the inverse image of this subset (or element, seen as a singleton).
See also

- inverse function theorem
- implicit function theorem
- inverse image Category:Set theory

Acceleration

In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is meter/second².

Explanation

To accelerate an object is to change its velocity over a period of time. In this strict scientific sense, acceleration can have positive and negative values – respectively called acceleration and deceleration (or retardation) in common speech – as well as change of direction. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation :

\mathbf =

where :a is the acceleration vector :v is the velocity vector expressed in m/s :t is time expressed in seconds. This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared"). An alternative equation is: :

\mathbf =

where :ā is the average acceleration (m/s²) :u is the initial velocity (m/s) :v is the final velocity (m/s) :t is the time interval (s) Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have :

\mathbf = - \frac \frac = - \omega^2 \mathbf

One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth at sea level at 45° latitude (Paris), or about 9.81 m/s². Jerk is the rate of change of an object's acceleration over time. In classical mechanics, acceleration

a \

is related to force

F \

and mass

m \

(assumed to be constant) by way of Newton's second law: :

F = m \cdot a

As a result of its invariance under the Galilean transformations, acceleration is an absolute quantity in classical mechanics.

Relation to relativity

After defining his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant acceleration are indistinguishable from those in a gravitational field, and thus defined general relativity that also explained how gravity's effects could be limited by the speed of light. If you accelerate away from your friend, you could say (given your frame of reference) that it is your friend who is accelerating away from you, although only you feel any force. This is also the basis for the popular Twin paradox, which asks why only one twin ages when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving. General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In special relativity, only inertial frames of reference (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. With changing velocity, accelerated objects exist in warped space (as do those that reside in a gravitational field). Therefore, frames of reference must include a description of their local spacetime curvature to qualify as complete. Acceleration can be measured using an accelerometer.

References

-
-

External links and references

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.sandia.gov/LabNews/labs-accomplish/2005/pulp.html Experiments on Z produced a world-record peak velocity of 34 km/s] (that is about 76,000 mph)
- [http://www.sandia.gov/media/NewsRel/NR2001/flyer.htm Magnetic field shocklessly shoots pellets 20 times faster than rifle bullet] Category:Physical quantity Category:Classical mechanics ko:가속도 ja:加速度 simple:Acceleration th:ความเร่ง

Computer science

Computer science, an academic discipline (abbreviated CS or compsci), is a body of knowledge generally about computer hardware, software, computation and its theory. The discipline itself includes, but is not limited to, the fundamentals of computer languages, operating systems and mathematical foundations of computer science. The study of these fundamentals may lead to a wide variety of topics, such as algorithms, formal grammars, programming languages, program design, artificial intelligence and computer engineering. There exist a number of technical definitions of computer science. The status of computer science as a science is often challenged, typically arguing that it is more like mathematics and that it does not follow the scientific method, however these facts are not unanimously accepted. In popular language, the term computer science is often confusingly used to denominate anything related to computers.

History of computer science

Evolutionary

Before the 1920s, computers were human clerks that performed calculations. They were usually under the lead of a physicist. Many thousands of computers were employed in commerce, government, and research establishments. Most of these computers were women, and they were known to have a degree in calculus. After the 1920s, the expression computing machine refered to any machine that performed the work of a human computer, especially those in accordance with effective methods of The Church-Turing Thesis. The thesis states that a mathematical method is effective if it could be set out as a list of instructions able to be followed by a human clerk with paper and pencil, for as long as necessary, and without ingenuity or insight. Machines that computed with discrete values became known as the analog kind. They used machinery that represented discrete numeric quantities, like the angle of a shaft rotation or difference in electrical potential. Digital machinery, in contrast to analog, were able to render a state of a numeric value and store each individual digit. Digital machinery used difference engines or relays before the invention of faster memory devices. The phrase computing machine gradually gave away, after the late 1940s, to just computer as the onset of electronic digital machinery became common. These computers were able to perform the calculations that were performed by the previous human clerks. Since the values stored by digital machines were not bound to physical properties like the analog device, a logical computer, based on digital equipment, was able to do anything that could be described "purely mechanical." Alan Turing, known as the Father of Computer Science, invented such a logical computer, also known as a Turing Machine, that evolved into the modern computer from the tasks performed by the previous human clerks. These new computers were also able to perform non-numeric computations, like music. Computability, by logical computers, began a science by being able to make evident which was not explicitly defined into ordinary sense more immediate.

Academic discipline

Computer science has roots in electrical engineering, logic, mathematics, and linguistics. In the last third of the 20th century computer science emerged as a distinct discipline and developed its own methods and terminology. Originally, CS was taught as part of mathematics or engineering departments, for instance at the University of Cambridge in England and at the Gdansk University of Technology in Poland, respectively. Cambridge claims to have the world's oldest taught qualification in computing. The first computer science department in the United States was founded at Purdue University in 1962, while the first college entirely devoted to computer science was founded at Northeastern University in 1982. Most universities today have specific departments devoted to computer science, while some conjoin it with engineering, with applied mathematics, or other disciplines. Most research in computer science has focused on von Neumann computers or Turing machines (computation models that perform one small, deterministic step at a time). These models resemble, at a basic level, most real computers in use today. Computer scientists also study other models of computation, which includes parallel machines and theoretical models such as probabilistic, oracle, and quantum computers.

Careers

Some of the potential careers for those who study computer science are listed below:

Demographics

- Nearly half of all computer programmers held a bachelor’s degree in 2002; about 1 in 5 held a graduate degree. [http://www.bls.gov/oco/ocos110.htm]
- Computer programmer employment is expected to grow much more slowly than that of other computer specialists.
- Education requirements range from a 2-year degree to a graduate degree. [http://www.bls.gov/oco/ocos042.htm]
- Employment, other than computer programmer, is expected to increase much faster than the average as organizations continue to adopt increasingly sophisticated technologies.

Sub-disciplines of computer science

Computer Science has a number of major sub-fields which can be classified by a number of means (for example the [http://www.acm.org/class/1998/overview.html ACM classification system]).

Algorithms

The study of algorithms is aimed at creating techniques that will enable a computer to perform a certain task in an efficient manner. An algorithm is a set of well-defined instructions for accomplishing some task, often explained by analogy with a culinary recipe. Algorithms are often implemented in software, and advancing the state of the art in algorithms is responsible for many of the most spectacular successes in computing. An algorithms specialist may come up with methods to accomplish new tasks, but just as often, they will work on improving the efficiency of an existing algorithm. These improvements can come in "time" (the length of time it takes for the algorithm to work) and "space" (the amount of computer memory the algorithm consumes. The field of algorithms is highly formal and many things can be proved about a given algorithm (using complexity theory), including roughly how long it will take to complete, as compared to the size of its input (the number of options it must consider). One interesting open question in algorithms concerns the complexity classes P and NP, and whether or not P = NP. If it does, then a whole range of seemingly difficult algorithms can in theory be performed quickly.

Data structures

A data structure is a way to store data so that it can be remembered and retrieved efficiently. A well-designed data structure means that an algorithm can be performed using as little memory space and time as possible. Some data structures are very simple: the simplest is an array which is simply a numbered list of items. Since the memory of a computer is usually modelled as an array (of bytes), this is also one of the most important data structures in computer science. For example, strings of text are usually modelled as arrays. But there are many other data structures such as linked lists, trees, hash tables and many others that are quite different but critical to the science. Type theory classifies data at a most basic (mathematical) level into different types, such as integers, complex numbers, strings, etc. and deals with how those types can interact. Abstract data types, at a more concrete level, deal with how types and data structures are used in software programming.

Listing of sub-disciplines

Computer science is closely related to a number of fields. These fields overlap considerably, though important differences exist:

Calculus

Differential calculus

Integral calculus

Foundations

Fundamental theorem of calculus

Applications

History

See also

Further reading

External links

Calculus (disambiguation)

Mathematical meanings

Other meanings

Algebra

History

Classification

Algebras

References

See also

External links

Arithmetic

Arithmetic operations

Number theory

See also

Differential calculus

Differentiation and differentiability

Newton's difference quotient

Notations for differentiation

Lagrange's notation

Leibniz's notation

Newton's notation

Euler's notation

Critical points

Physics

Algebraic manipulation

Using derivatives to graph functions

Generalizations

See also

External links

References

Slope

Definition of slope

Example 1

Example 2

Geometry

Slope of a road, etc.

Algebra

Calculus

Why calculus is necessary

See also

Curve

Definitions

Conventions and terminology

Length of curves

Differential geometry

Algebraic curve

History

See also

External links

Volume

Volume formula

Volume measures: other metric units

Volume measures: USA

Volume measures: UK

Volume measures: cooking

Relationship to density

Volume comparisons

See also

External links

Inverse operation

Simplifying rule

Existence

See also

Acceleration

Explanation

Relation to relativity

References

External links and references

Computer science

History of computer science