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This article is about the branch of mathematics. For other uses, see Calculus (disambiguation).
It has been suggested that infinitesimal calculus be merged into this article or section. (Discuss)
Topics in Calculus
Fundamental theoremLimits of functionsContinuityMean value theorem
Differential calculus
DerivativeChange of variablesImplicit differentiationTaylor's theoremRelated ratesIdentitiesRules:
Power rule, Product rule, Quotient rule, Chain rule
Integral calculus
Integral
Lists of integralsImproper integralsIntegration by:parts, disks, cylindricalshells, substitution,trigonometric substitution,partial fractions, changing order
Vector calculus
GradientDivergenceCurlLaplacianGradient theoremGreen's theoremStokes' theoremDivergence theorem
Multivariable calculus
Matrix calculusPartial derivativeMultiple integralLine integralSurface integralVolume integralJacobian
Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change[1], in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
Contents
1 History
1.1 Ancient
1.2 Medieval
1.3 Modern
1.4 Significance
1.5 Foundations
2 Principles
2.1 Limits and infinitesimals
2.2 Differential calculus
2.3 Leibniz notation
2.4 Integral calculus
2.5 Fundamental theorem
3 Applications
4 See also
4.1 Lists
4.2 Related topics
5 References
5.1 Notes
5.2 Books
6 Other resources
6.1 Further reading
6.2 Online books
6.3 External links
//
[edit] History
Main article: History of calculus
[edit] Ancient
Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation.
The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[2][3] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[4] The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method which would later be called Cavalieri's principle to find the volume of a sphere.[3]
[edit] Medieval
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Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that is readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration.[5] In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form of Rolle's theorem.[6] Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Indian mathematician Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[8] which are treated in the text Yuktibhasa.[9][10][11]
[edit] Modern
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In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1675.
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work (only in Britain, not in continental Europe), but is now regarded as an independent inventor of and contributor to calculus.
These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism—he often spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass (see (ε, δ)-definition of limit). It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.
Calculus is a ubiquitous topic in most modern high schools and universities around the world.[12]
[edit] Significance
While some of the ideas of calculus were developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.
Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.
[edit] Foundations
In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.
There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.
[edit] Principles
[edit] Limits and infinitesimals
Main articles: Limit of a function and Infinitesimal
Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.
In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
[edit] Differential calculus
Main article: Differential calculus
Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative, the doubling function.
If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = mx + b, where:
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:
Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.
The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.
The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.
[edit] Leibniz notation
Main article: Leibniz's notation
A common notation, introduced by Leibniz, for the derivative in the example above is
In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
In this usage, the dx in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.
[edit] Integral calculus
Main article: Integral
Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators.
The indefinite integral is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.)
The definite integral inputs a function and outputs a number, which gives the area between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.
A motivating example is the distances traveled in a given time.
If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
If f(x) in the diagram on the left represents speed as it varies over time, the distance traveled (between the times represented by a and b) is the area of the shaded region s.
To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero.
The symbol of integration is , an elongated S (the S stands for "sum"). The definite integral is written as:
and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation
is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; dx is not a number, and is not being multiplied by f(x).
The indefinite integral, or antiderivative, is written:
Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:
An undetermined constant like C in the antiderivative is known as a constant of integration.
[edit] Fundamental theorem
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
The Fundamental Theorem of Calculus states: 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
Furthermore, for every x in the interval (a, b),
This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, 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.
[edit] Applications
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.
Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are interrelated through calculus. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term "rate of change" which refers to the derivative saying The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Commonly expressed today as Force = Mass × acceleration, it involves differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.
Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or it can be used in probability theory to determine the probability of a continuous random variable from an assumed density function. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points.
Green's Theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.
In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it's used to derive dosing laws. In nuclear medicine, it's used to build models of radiation transport in targeted tumor therapies.
In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
Calculus is also used to find approximate solutions to equations; in practice it's the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.
[edit] See also
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Main article: Outline of calculus
[edit] Lists
List of differentiation identities
List of calculus topics
Publications in calculus
Table of integrals
[edit] Related topics
Calculus of finite differences
Calculus with polynomials
Complex analysis
Differential equation
Differential geometry
Elementary calculus
Fourier series
Integral equation
Mathematical analysis
Mathematics
Multivariable calculus
Non-classical analysis
Non-standard analysis
Non-standard calculus
Precalculus (mathematical education)
Product Integrals
Stochastic calculus
Taylor series
Time-scale calculus
Tuesday, September 7, 2010
MATHEMATICS
"Maths" and "Math" redirect here. For other uses of "Mathematics" or "Math", see Mathematics (disambiguation) and Math (disambiguation).
Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]
There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.[7]
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]
Contents
1 Etymology
2 History
3 Inspiration, pure and applied mathematics, and aesthetics
4 Notation, language, and rigor
5 Mathematics as science
6 Fields of mathematics
6.1 Quantity
6.2 Structure
6.3 Space
6.4 Change
6.5 Foundations and philosophy
6.6 Theoretical computer science
6.7 Applied mathematics
7 See also
8 Notes
9 References
10 External links
Etymology
The word "mathematics" comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
History
Main article: History of mathematics
Pythagoras (c.570-c.495 BC) has commonly been given credit for discovering the Pythagorean theorem. Well-known figures in Greek mathematics also include Euclid, Archimedes, and Thales.The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[11] was probably that of numbers: the realization that a collection of two apples and a collection two oranges (for example) have something in common, namely quantity of their members.
In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.[12] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.
Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed] Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]
Mayan numeralsThe earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[13] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.[14]
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[15]
Inspiration, pure and applied mathematics, and aesthetics
Main article: Mathematical beauty
Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[16] Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".[17] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[18] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[19] Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.[20][21] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
Leonhard Euler, who created and popularized much of the mathematical notation used todayMain article: Mathematical notation
Most of the mathematical notation in use today was not invented until the 16th century.[22] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[23] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.
Mathematical language can also be hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
The infinity symbol ∞ in several typefaces.Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[24] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[25]
Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[26]
Mathematics as science
Carl Friedrich Gauss, himself known as the "prince of mathematicians",[27] referred to mathematics as "the Queen of the Sciences".Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[28] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[29] However, in the 1930s important work in mathematical logic convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[30] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.[31] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.[citation needed] In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians[who?] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others[who?] feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[citation needed]
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[32][33] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics
An abacus, a simple calculating tool used since ancient times.Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
Natural numbers Integers Rational numbers Real numbers Complex numbers
Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.
Combinatorics Number theory Group theory Graph theory Order theory
Space
The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.
Geometry Trigonometry Differential geometry Topology Fractal geometry Measure Theory
Change
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis
Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[34] Some disagreement about the foundations of mathematics continues to present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.
Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory.[citation needed] Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.
Mathematical logic Set theory Category theory
Theoretical computer science
Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems.[35] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.
Theory of computation Cryptography
Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas.
Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[36]
Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using ideas of functional analysis and techniques of approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Other areas of computational mathematics include computer algebra and symbolic computation.
Mathematical physics
Fluid dynamics
Numerical analysis
Optimization
Probability theory
Statistics
Financial mathematics
Game theory
Mathematical biology
Mathematical chemistry
Mathematical economics
Control theory
See also
Mathematics portal
Book:Mathematics
Books are collections of articles that can be downloaded or ordered in print.
Main article: Lists of mathematics topics
Definitions of mathematics
Dyscalculia
Iatromathematicians
Logics
Mathematical anxiety
Mathematical game
Mathematical model
Mathematical problem
Mathematical structure
Mathematics and art
Mathematics competitions
Mathematics education
Mathematics portal
Pattern
Philosophy of mathematics
Pseudomathematics
Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]
There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.[7]
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]
Contents
1 Etymology
2 History
3 Inspiration, pure and applied mathematics, and aesthetics
4 Notation, language, and rigor
5 Mathematics as science
6 Fields of mathematics
6.1 Quantity
6.2 Structure
6.3 Space
6.4 Change
6.5 Foundations and philosophy
6.6 Theoretical computer science
6.7 Applied mathematics
7 See also
8 Notes
9 References
10 External links
Etymology
The word "mathematics" comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
History
Main article: History of mathematics
Pythagoras (c.570-c.495 BC) has commonly been given credit for discovering the Pythagorean theorem. Well-known figures in Greek mathematics also include Euclid, Archimedes, and Thales.The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[11] was probably that of numbers: the realization that a collection of two apples and a collection two oranges (for example) have something in common, namely quantity of their members.
In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.[12] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.
Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed] Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]
Mayan numeralsThe earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[13] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.[14]
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[15]
Inspiration, pure and applied mathematics, and aesthetics
Main article: Mathematical beauty
Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[16] Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".[17] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[18] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[19] Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.[20][21] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
Leonhard Euler, who created and popularized much of the mathematical notation used todayMain article: Mathematical notation
Most of the mathematical notation in use today was not invented until the 16th century.[22] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[23] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.
Mathematical language can also be hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
The infinity symbol ∞ in several typefaces.Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[24] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[25]
Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[26]
Mathematics as science
Carl Friedrich Gauss, himself known as the "prince of mathematicians",[27] referred to mathematics as "the Queen of the Sciences".Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[28] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[29] However, in the 1930s important work in mathematical logic convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[30] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.[31] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.[citation needed] In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians[who?] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others[who?] feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[citation needed]
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[32][33] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics
An abacus, a simple calculating tool used since ancient times.Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
Natural numbers Integers Rational numbers Real numbers Complex numbers
Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.
Combinatorics Number theory Group theory Graph theory Order theory
Space
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HEAT TRANSFER
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Heat transfer, also known as heat flow, heat exchange, or transfer of thermal energy is the movement of heat from one place to another. When an object is at a different temperature from its surroundings, heat transfer occurs so that the body and the surroundings reach the same temperature (thermal equilibrium). Heat transfer always occurs from a higher-temperature region to a cooler-temperature one as described by the second law of thermodynamics or the Clausius statement.
Where there is a temperature difference between objects in proximity, heat transfer between them cannot be stopped although its rate can be controlled.
Contents
1 Overview
2 Heat transfer mechanisms
2.1 Conduction
2.2 Convection
2.3 Radiation
2.4 Mass Transfer
2.5 Convection vs. conduction
3 Heat transfer in phase changes
3.1 Boiling
3.2 Condensation
4 Modelling approaches
4.1 Heat equation
4.2 Computational methods
4.3 Lumped system analysis
5 Applications and techniques
5.1 Insulation and radiant barriers
5.1.1 Critical insulation thickness
5.2 Heat exchangers
5.3 Heat dissipation
5.3.1 Heat sinks
5.3.2 Buildings
5.4 Thermal energy storage
5.5 Evaporative cooling
5.6 Radiative cooling
5.7 Laser cooling
5.8 Magnetic cooling
5.9 Other
6 See also
7 References
8 Further reading
9 External links
[edit] Overview
See also: Heat and Thermodynamics
This article deals with specific methods by which heat transfer occurs.
Note that although the definition of "heat" implicitly means the movement of energy, the term "heat transfer" has acquired a traditional usage in engineering and other contexts (despite its literal redundancy).
Fundamental methods of heat transfer include conduction, convection, and radiation. Although separate physical laws have been discovered to describe the behaviour of each of these methods, real systems may exhibit a complicated combination. Various mathematical methods have been developed to solve or approximate the results of heat transfer in systems.
Heat transfer is a path function (process quantity), as opposed to a point function (state quantity).
Heat transfer is typically studied as part of a general chemical engineering or mechanical engineering curriculum. Typically, thermodynamics is a prerequisite to undertaking a course in heat transfer, as the laws of thermodynamics are essential in understanding the mechanism of heat transfer. Other courses related to heat transfer include energy conversion, thermofluids and mass transfer.
Heat transfer methodologies are used in the following disciplines, among others:
Automotive engineering
Thermal management of electronic devices and systems
HVAC
Insulation
Materials processing
Power plant engineering
There are some notable similarities in equations for heat, momentum, and mass transfer[1]. The molecular transfer equations of Newton's law for fluid momentum, Fourier's law for heat, and Fick's law for mass are very similar. A great deal of effort has been devoted to developing analogies among these three transport processes so as to allow prediction of one from any of the others.
[edit] Heat transfer mechanisms
There are a number of distinct, or fundamental, modes of heat transfer:
Conduction: Transfer of energy between objects in physical contact
Convection: Transfer of energy between an object and its environment, due to fluid motion
Radiation: Transfer of energy from or to a body by the emission or absorption of electromagnetic radiation
Mass transfer: Movement of physical objects represents a movement of their internal energy
[edit] Conduction
Fire test used to test the heat transfer through firestops and penetrants used in construction listing and approval use and compliance.Main article: Heat conduction
On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring atoms. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in contact. Fluids (and especially gases) are less conductive.
Steady state conduction is a form of conduction which happens when the temperature difference driving the conduction is constant so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further. In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out.
Transient conduction occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories, and/or numerical analysis by computer.
[edit] Convection
Main article: Convective heat transfer
Convection is the transfer of heat from one place to another by the movement of fluids. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid.[2] Convection is usually the dominant form of heat transfer in liquids and gases. Although often discussed as a third method of heat transfer, convection actually describes the combined effects of conduction and fluid flow.
Free or natural convection occurs when the fluid motion is caused by buoyancy forces that result from the density variations due to variations of temperature in the fluid. Forced convection is when the fluid is forced to flow over the surface by external source such as fans, stirrers, and pumps, creating an artificially induced convection current.[3]
Convection is described by Newton's law of cooling, which states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings.
[edit] Radiation
A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through radiation.Main article: Thermal radiation
Thermal radiation is the transfer of heat energy through empty space by electromagnetic waves. All objects with a temperature above absolute zero radiate energy. No medium is necessary for radiation to occur, for it is transferred by electromagnetic waves; radiation takes place even in and through a perfect vacuum. For instance, the energy from the Sun travels through the vacuum of space before warming the earth. Radiation is the only form of heat transfer that can occur in the absence of any form of medium (i.e., through a vacuum).
Thermal radiation is a direct result of the movements of atoms and molecules in a material. Since these atoms and molecules are composed of charged particles (protons and electrons), their movements result in the emission of electromagnetic radiation, which carries energy away from the surface. At the same time, the surface is constantly bombarded by radiation from the surroundings, resulting in the transfer of energy to the surface. Since the amount of emitted radiation increases with increasing temperature, a net transfer of energy from higher temperatures to lower temperatures results.
Unlike conductive and convective forms of heat transfer, thermal radiation can be concentrated in a tiny spot by using reflecting mirrors. Concentrating solar power takes advantage of this fact. For example, the sunlight reflected from mirrors heats the PS10 solar power tower, and during the day it can heat water to 285°C (558.15°K) or 545°F.
[edit] Mass Transfer
Please help improve this article by expanding it. Further information might be found on the talk page. (August 2010)
Main article: Mass transfer
It is possible to move heat by physical transfer of a hot or cold object from one place to another. This can be as simple as placing hot water in a bottle and heating your bed or the movement of an iceberg and changing ocean currents.
[edit] Convection vs. conduction
Consider a body of fluid where heat is produced or enters at its lower end and leaves at its upper end. Ignoring radiation, conduction and convection can be considered to "compete" for dominance. If heat conduction is too great, fluid moving down by convection will be heated by conduction so fast that its downward movement will be stopped due to its buoyancy, while fluid moving up by convection will be cooled by conduction so fast that its driving buoyancy will diminish. If, on the other hand, heat conduction is very low, a large temperature gradient will be formed and convection will be very strong.
The Rayleigh number (Ra) is a measure determining the result of this competition.
where
g is acceleration due to gravity
ρ is the density with Δρ being the density difference between the lower and upper ends
μ is the dynamic viscosity
α is the Thermal diffusivity
β is the volume thermal expansivity (sometimes denoted α elsewhere)
T is the temperature and ν is the kinematic viscosity.
The Rayleigh number can be understood as the ratio between the rate of heat transfer by convection to the rate of heat transfer by conduction, or, equivalently, the ratio between the corresponding timescales (i.e. conduction timescale divided by convection timescale), up to a numerical factor. This can be seen as follows, where all calculations are up to numerical factors depending on the geometry of the system.
The buoyancy force driving the convection is roughly gΔρL3 so the corresponding pressure is roughly gΔρL. In steady state, this is canceled by the shear stress due to viscosity and therefore roughly equals μV / L = μ / Tconv, where V is the typical fluid velocity due to convection and Tconv the order of its timescale. The conduction timescale, on the other hand, is of the order of Tcond = L2 / α.
Convection occurs when Rayleigh number is above 1000-2000. For example, the Earth's mantle, exhibiting non-stable convection, has Rayleigh number of the order of 103, and Tconv as calculated above is around 108 years.
[edit] Heat transfer in phase changes
Transfer of heat through a phase transition in the medium such as water-to-ice or water-to-steam or steam-to-water or ice-to-water involves significant energy and is exploited in many ways: steam engine, refrigerator etc. (see latent heat of fusion). For example, the Mason equation is an approximate analytical expression for the growth of a water droplet based on the effects of heat transport on evaporation and condensation.
[edit] Boiling
See also: boiling and critical heat flux
Heat transfer in boiling fluids is complex but of considerable technical importance. It is characterised by an s-shaped curve relating heat flux to surface temperature difference (see say Kay & Nedderman 'Fluid Mechanics & Transfer Processes', CUP, 1985, p. 529).
At low driving temperatures, no boiling occurs and the heat transfer rate is controlled by the usual single-phase mechanisms. As the surface temperature is increased, local boiling occurs and vapour bubbles nucleate, grow into the surrounding cooler fluid, and collapse. This is sub-cooled nucleate boiling and is a very efficient heat transfer mechanism. At high bubble generation rates the bubbles begin to interfere and the heat flux no longer increases rapidly with surface temperature (this is the departure from nucleate boiling DNB). At higher temperatures still, a maximum in the heat flux is reached (the critical heat flux). The regime of falling heat transfer which follows is not easy to study but is believed to be characterised by alternate periods of nucleate and film boiling. Nucleate boiling slowing the heat transfer due to gas phase {bubbles} creation on the heater surface, as mentioned, gas phase thermal conductivity is much lower than liquid phase thermal conductivity, so the outcome is a kind of "gas thermal barrier".
At higher temperatures still, the hydrodynamically quieter regime of film boiling is reached. Heat fluxes across the stable vapour layers are low, but rise slowly with temperature. Any contact between fluid and the surface which may be seen probably leads to the extremely rapid nucleation of a fresh vapour layer ('spontaneous nucleation').
[edit] Condensation
Condensation occurs when a vapor is cooled and changes its phase to a liquid. Condensation heat transfer, like boiling, is of great significance in industry. During condensation, the latent heat of vaporization must be released. The amount of the heat is the same as that absorbed during vaporization at the same fluid pressure.
There are several modes of condensation:
Homogeneous condensation (as during a formation of fog).
Condensation in direct contact with subcooled liquid.
Condensation on direct contact with a cooling wall of a heat exchanger-this is the most common mode used in industry:
Filmwise condensation (when a liquid film is formed on the subcooled surface, usually occurs when the liquid wets the surface).
Dropwise condensation (when liquid drops are formed on the subcooled surface, usually occurs when the liquid does not wet the surface). Dropwise condensation is difficult to sustain reliably; therefore, industrial equipment is normally designed to operate in filmwise condensation mode.
[edit] Modelling approaches
A number of approaches to modelling complex heat transfer phenomena have been developed.
[edit] Heat equation
Main article: Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. In some cases, exact solutions of the equation are available; in other cases the equation must be solved numerically (see computational methods, below).
[edit] Computational methods
Please help improve this article by expanding it. Further information might be found on the talk page. (August 2010)
See also: Computational fluid dynamics
For example, simplified climate models may use Newtonian cooling instead of a full (and computationally expensive) radiation code to maintain atmospheric temperatures.
[edit] Lumped system analysis
Main article: Lumped capacitance model
See also: Biot number
Lumped system analysis is a common approximation in transient conduction which may be used whenever heat conduction within an object is much faster than heat conduction across the boundary of the object.
This is a method of approximation that reduces one aspect of the transient conduction system (that within the object) to an equivalent steady state system (that is, it is assumed that the temperature within the object is completely uniform, although its value may be changing in time).
In this method, a term known as the Biot number is calculated, which is defined as the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the object's boundary. For small Biot numbers, the approximation of spatially uniform temperature within the object can begin to be used, since it can be presumed that heat transferred into the object has time to uniformaly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.
[edit] Applications and techniques
Heat transfer has broad application to the functioning of numerous devices and systems. Objectives of studying and applying heat transfer principles may be to preserve (i.e., insulate), increase, or decrease temperature in a wide variety of circumstances.
[edit] Insulation and radiant barriers
Main articles: Thermal insulation and radiant barrier
Heat exposure as part of a fire test for firestop products.Thermal insulators are materials specifically designed to reduce the flow of heat by limiting conduction, convection, or both. Radiant barriers are materials which reflect radiation and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice versa. Metal, for instance, is an excellent reflector and poor insulator.
The effectiveness of an insulator is indicated by its R- (resistance) value. The R-value of a material is the inverse of the conduction coefficient (k) multiplied by the thickness (d) of the insulator. The units of resistance value are in SI units: (K·m²/W)
Rigid fiberglass, a common insulation material, has an R-value of 4 per inch, while poured concrete, a poor insulator, has an R-value of 0.08 per inch.[4]
The tog is a measure of thermal resistance, commonly used in the textile industry, and often seen quoted on, for example, duvets and carpet underlay.
The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa (at any specific wavelength, reflectivity = 1 - emissivity). An ideal radiant barrier would have a reflectivity of 1 and would therefore reflect 100% of incoming radiation. Vacuum bottles (Dewars) are 'silvered' to approach this. In space vacuum, satellites use multi-layer insulation which consists of many layers of aluminized (shiny) mylar to greatly reduce radiation heat transfer and control satellite temperature.
[edit] Critical insulation thickness
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Low thermal conductivity (k) materials reduce heat fluxes. The smaller the k value, the larger the corresponding thermal resistance (R) value.
The units of thermal conductivity(k) are W·m−1·K−1 (watts per meter per kelvin), therefore increasing width of insulation (x meters) decreases the k term and as discussed increases resistance.
This follows logic as increased resistance would be created with increased conduction path (x).
However, adding this layer of insulation also has the potential of increasing the surface area and hence thermal convection area (A).
An obvious example is a cylindrical pipe:
As insulation gets thicker, outer radius increases and therefore surface area increases.
The point where the added resistance of increasing insulation width becomes overshadowed by the effects of surface area is called the critical insulation thickness. In simple cylindrical pipes:[5]
For a graph of this phenomenon in a cylidrical pipe example see: External Link: Critical Insulation Thickness diagram as at 26/03/09
[edit] Heat exchangers
Main article: Heat exchanger
A heat exchanger is a tool built for efficient heat transfer from one fluid to another, whether the fluids are separated by a solid wall so that they never mix, or the fluids are directly contacted. Heat exchangers are widely used in refrigeration, air conditioning, space heating, power generation, and chemical processing. One common example of a heat exchanger is the radiator in a car, in which the hot radiator fluid is cooled by the flow of air over the radiator surface.
Common types of heat exchanger flows include parallel flow, counter flow, and cross flow. In parallel flow, both fluids move in the same direction while transferring heat; in counter flow, the fluids move in opposite directions and in cross flow the fluids move at right angles to each other. The common constructions for heat exchanger include shell and tube, double pipe, extruded finned pipe, spiral fin pipe, u-tube, and stacked plate.
When engineers calculate the theoretical heat transfer in a heat exchanger, they must contend with the fact that the driving temperature difference between the two fluids varies with position. To account for this in simple systems, the log mean temperature difference (LMTD) is often used as an 'average' temperature. In more complex systems, direct knowledge of the LMTD is not available and the number of transfer units (NTU) method can be used instead.
[edit] Heat dissipation
[edit] Heat sinks
Main article: Heat sink
A heat sink is a term for a component that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems and the radiator (also a heat exchanger) in a car. Heat sinks also help to cool electronic and optoelectronic devices, such as CPUs, higher-power lasers, and light emitting diodes (LEDs). A heat sink uses its extended surfaces to increase the surface area in contact with the cooling fluid, the air for example.
[edit] Buildings
In cold climates, houses with their heating systems form dissipative systems. In spite of efforts to insulate such houses to reduce heat losses to their exteriors, considerable heat is lost which can make their interiors uncomfortably cool or cold. For the comfort of the inhabitants, the interiors must be maintained out of thermal equilibrium with the external surroundings. In effect, these domestic residences are oases of warmth in a sea of cold and the thermal gradient between the inside and outside is often quite steep. This can lead to problems such as condensation and uncomfortable draughts (drafts) which, if left unaddressed, can cause cosmetic or structural damage to the property. Such issues can be prevented by use of insulation techniques for reducing heat loss.
Thermal transmittance is the rate of transfer of heat through a structure divided by the difference in temperature across the structure. It is expressed in watts per square metre per kelvin, or W/m²K. Well-insulated parts of a building have a low thermal transmittance whereas poorly-insulated parts of a building have a high thermal transmittance.
A thermostat is a device capable of starting the heating system when the house's interior falls below a set temperature, and of stopping that same system when another (higher) set temperature has been achieved. Thus the thermostat controls the flow of energy into the house, that energy eventually being dissipated to the exterior.
[edit] Thermal energy storage
Main article: Thermal energy storage
Thermal energy storage refers to a number of technologies that store energy in a thermal reservoir for later use. They can be employed to balance energy demand between day time and night time. The thermal reservoir may be maintained at a temperature above (hotter) or below (colder) than that of the ambient environment. Applications include later use in space heating, domestic or process hot water, or to generate electricity. Most practical active solar heating systems have storage for a few hours to a day's worth of heat collected.
[edit] Evaporative cooling
Main article: Evaporative cooler
Evaporative cooling is a physical phenomenon in which evaporation of a liquid, typically into surrounding air, cools an object or a liquid in contact with it. Latent heat describes the amount of heat that is needed to evaporate the liquid; this heat comes from the liquid itself and the surrounding gas and surfaces. The greater the difference between the two temperatures, the greater the evaporative cooling effect. When the temperatures are the same, no net evaporation of water in air occurs, thus there is no cooling effect. A simple example of natural evaporative cooling is perspiration, or sweat, which the body secretes in order to cool itself. An evaporative cooler is a device that cools air through the simple evaporation of water.
[edit] Radiative cooling
Main article: Radiative cooling
Radiative cooling is the process by which a body loses heat by radiation and is an important effect in the Earth's atmosphere. In the case of the earth-atmosphere system it refers to the process by which long-wave (infra red) radiation is emitted to balance the absorption of short-wave (visible) energy from the sun. Convective transport of heat and evaporative transport of latent heat both remove heat from the surface and redistributes it in the atmosphere, making it available for radiative transport at higher altitudes.
[edit] Laser cooling
Main article: Laser cooling
Laser cooling refers to a number of techniques in which atomic and molecular samples are cooled through the interaction with one or more laser light fields. The most common method of laser cooling is Doppler cooling. In Doppler cooling, the frequency of light is tuned slightly below an electronic transition in the atom. Thus the atoms would absorb more photons if they moved towards the light source, due to the Doppler effect. If an excited atom then emits a photon spontaneously, it will be accelerated. The result of the absorption and emission process is to reduce the speed of the atom. Eventually the mean velocity, and therefore the kinetic energy of the atoms will be reduced. Since the temperature of an ensemble of atoms is a measure of the random internal kinetic energy, this is equivalent to cooling the atoms.
Sympathetic cooling is a process in which particles of one type cool particles of another type. Typically, atomic ions that can be directly laser cooled are used to cool nearby ions or atoms. This technique allows cooling of ions and atoms that can't be cooled directly by laser cooling.
[edit] Magnetic cooling
Main article: Magnetic evaporative cooling
Magnetic evaporative cooling is a technique for lowering the temperature of a group of atoms. The process confines atoms using a magnetic field. Collisions mean that over time, individual atoms will become much more energetic than the others, and will escape, removing energy from the system and reducing the temperature of the remaining group. This process is similar to the familiar process by which standing water becomes water vapor.
[edit] Other
A heat pipe is a passive device that is constructed in such a way that it acts as though it has extremely high thermal conductivity. Heat pipes use latent heat and capillary action to move heat, and can carry many times as much heat as a similar-sized copper rod. Originally invented for use in satellites, they are starting to have applications in personal computers.
A thermocouple is a junction between two different metals that produces a voltage related to a temperature difference. Thermocouples are a widely used type of temperature sensor for measurement and control and can also be used to convert heat into electric power.
A thermopile is an electronic device that converts thermal energy into electrical energy. It is composed of thermocouples. Thermopiles do not measure the absolute temperature, but generate an output voltage proportional to a temperature difference. Thermopiles are widely used, e.g., they are the key component of the infrared thermometers that are widely used to measure body temperature via the ear.
A thermal diode or thermal rectifier is a device that preferentially passes heat in one direction; a "one-way valve" for heat.
Heat transfer, also known as heat flow, heat exchange, or transfer of thermal energy is the movement of heat from one place to another. When an object is at a different temperature from its surroundings, heat transfer occurs so that the body and the surroundings reach the same temperature (thermal equilibrium). Heat transfer always occurs from a higher-temperature region to a cooler-temperature one as described by the second law of thermodynamics or the Clausius statement.
Where there is a temperature difference between objects in proximity, heat transfer between them cannot be stopped although its rate can be controlled.
Contents
1 Overview
2 Heat transfer mechanisms
2.1 Conduction
2.2 Convection
2.3 Radiation
2.4 Mass Transfer
2.5 Convection vs. conduction
3 Heat transfer in phase changes
3.1 Boiling
3.2 Condensation
4 Modelling approaches
4.1 Heat equation
4.2 Computational methods
4.3 Lumped system analysis
5 Applications and techniques
5.1 Insulation and radiant barriers
5.1.1 Critical insulation thickness
5.2 Heat exchangers
5.3 Heat dissipation
5.3.1 Heat sinks
5.3.2 Buildings
5.4 Thermal energy storage
5.5 Evaporative cooling
5.6 Radiative cooling
5.7 Laser cooling
5.8 Magnetic cooling
5.9 Other
6 See also
7 References
8 Further reading
9 External links
[edit] Overview
See also: Heat and Thermodynamics
This article deals with specific methods by which heat transfer occurs.
Note that although the definition of "heat" implicitly means the movement of energy, the term "heat transfer" has acquired a traditional usage in engineering and other contexts (despite its literal redundancy).
Fundamental methods of heat transfer include conduction, convection, and radiation. Although separate physical laws have been discovered to describe the behaviour of each of these methods, real systems may exhibit a complicated combination. Various mathematical methods have been developed to solve or approximate the results of heat transfer in systems.
Heat transfer is a path function (process quantity), as opposed to a point function (state quantity).
Heat transfer is typically studied as part of a general chemical engineering or mechanical engineering curriculum. Typically, thermodynamics is a prerequisite to undertaking a course in heat transfer, as the laws of thermodynamics are essential in understanding the mechanism of heat transfer. Other courses related to heat transfer include energy conversion, thermofluids and mass transfer.
Heat transfer methodologies are used in the following disciplines, among others:
Automotive engineering
Thermal management of electronic devices and systems
HVAC
Insulation
Materials processing
Power plant engineering
There are some notable similarities in equations for heat, momentum, and mass transfer[1]. The molecular transfer equations of Newton's law for fluid momentum, Fourier's law for heat, and Fick's law for mass are very similar. A great deal of effort has been devoted to developing analogies among these three transport processes so as to allow prediction of one from any of the others.
[edit] Heat transfer mechanisms
There are a number of distinct, or fundamental, modes of heat transfer:
Conduction: Transfer of energy between objects in physical contact
Convection: Transfer of energy between an object and its environment, due to fluid motion
Radiation: Transfer of energy from or to a body by the emission or absorption of electromagnetic radiation
Mass transfer: Movement of physical objects represents a movement of their internal energy
[edit] Conduction
Fire test used to test the heat transfer through firestops and penetrants used in construction listing and approval use and compliance.Main article: Heat conduction
On a microscopic scale, conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring atoms. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in contact. Fluids (and especially gases) are less conductive.
Steady state conduction is a form of conduction which happens when the temperature difference driving the conduction is constant so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further. In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out.
Transient conduction occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories, and/or numerical analysis by computer.
[edit] Convection
Main article: Convective heat transfer
Convection is the transfer of heat from one place to another by the movement of fluids. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid.[2] Convection is usually the dominant form of heat transfer in liquids and gases. Although often discussed as a third method of heat transfer, convection actually describes the combined effects of conduction and fluid flow.
Free or natural convection occurs when the fluid motion is caused by buoyancy forces that result from the density variations due to variations of temperature in the fluid. Forced convection is when the fluid is forced to flow over the surface by external source such as fans, stirrers, and pumps, creating an artificially induced convection current.[3]
Convection is described by Newton's law of cooling, which states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings.
[edit] Radiation
A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through radiation.Main article: Thermal radiation
Thermal radiation is the transfer of heat energy through empty space by electromagnetic waves. All objects with a temperature above absolute zero radiate energy. No medium is necessary for radiation to occur, for it is transferred by electromagnetic waves; radiation takes place even in and through a perfect vacuum. For instance, the energy from the Sun travels through the vacuum of space before warming the earth. Radiation is the only form of heat transfer that can occur in the absence of any form of medium (i.e., through a vacuum).
Thermal radiation is a direct result of the movements of atoms and molecules in a material. Since these atoms and molecules are composed of charged particles (protons and electrons), their movements result in the emission of electromagnetic radiation, which carries energy away from the surface. At the same time, the surface is constantly bombarded by radiation from the surroundings, resulting in the transfer of energy to the surface. Since the amount of emitted radiation increases with increasing temperature, a net transfer of energy from higher temperatures to lower temperatures results.
Unlike conductive and convective forms of heat transfer, thermal radiation can be concentrated in a tiny spot by using reflecting mirrors. Concentrating solar power takes advantage of this fact. For example, the sunlight reflected from mirrors heats the PS10 solar power tower, and during the day it can heat water to 285°C (558.15°K) or 545°F.
[edit] Mass Transfer
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Main article: Mass transfer
It is possible to move heat by physical transfer of a hot or cold object from one place to another. This can be as simple as placing hot water in a bottle and heating your bed or the movement of an iceberg and changing ocean currents.
[edit] Convection vs. conduction
Consider a body of fluid where heat is produced or enters at its lower end and leaves at its upper end. Ignoring radiation, conduction and convection can be considered to "compete" for dominance. If heat conduction is too great, fluid moving down by convection will be heated by conduction so fast that its downward movement will be stopped due to its buoyancy, while fluid moving up by convection will be cooled by conduction so fast that its driving buoyancy will diminish. If, on the other hand, heat conduction is very low, a large temperature gradient will be formed and convection will be very strong.
The Rayleigh number (Ra) is a measure determining the result of this competition.
where
g is acceleration due to gravity
ρ is the density with Δρ being the density difference between the lower and upper ends
μ is the dynamic viscosity
α is the Thermal diffusivity
β is the volume thermal expansivity (sometimes denoted α elsewhere)
T is the temperature and ν is the kinematic viscosity.
The Rayleigh number can be understood as the ratio between the rate of heat transfer by convection to the rate of heat transfer by conduction, or, equivalently, the ratio between the corresponding timescales (i.e. conduction timescale divided by convection timescale), up to a numerical factor. This can be seen as follows, where all calculations are up to numerical factors depending on the geometry of the system.
The buoyancy force driving the convection is roughly gΔρL3 so the corresponding pressure is roughly gΔρL. In steady state, this is canceled by the shear stress due to viscosity and therefore roughly equals μV / L = μ / Tconv, where V is the typical fluid velocity due to convection and Tconv the order of its timescale. The conduction timescale, on the other hand, is of the order of Tcond = L2 / α.
Convection occurs when Rayleigh number is above 1000-2000. For example, the Earth's mantle, exhibiting non-stable convection, has Rayleigh number of the order of 103, and Tconv as calculated above is around 108 years.
[edit] Heat transfer in phase changes
Transfer of heat through a phase transition in the medium such as water-to-ice or water-to-steam or steam-to-water or ice-to-water involves significant energy and is exploited in many ways: steam engine, refrigerator etc. (see latent heat of fusion). For example, the Mason equation is an approximate analytical expression for the growth of a water droplet based on the effects of heat transport on evaporation and condensation.
[edit] Boiling
See also: boiling and critical heat flux
Heat transfer in boiling fluids is complex but of considerable technical importance. It is characterised by an s-shaped curve relating heat flux to surface temperature difference (see say Kay & Nedderman 'Fluid Mechanics & Transfer Processes', CUP, 1985, p. 529).
At low driving temperatures, no boiling occurs and the heat transfer rate is controlled by the usual single-phase mechanisms. As the surface temperature is increased, local boiling occurs and vapour bubbles nucleate, grow into the surrounding cooler fluid, and collapse. This is sub-cooled nucleate boiling and is a very efficient heat transfer mechanism. At high bubble generation rates the bubbles begin to interfere and the heat flux no longer increases rapidly with surface temperature (this is the departure from nucleate boiling DNB). At higher temperatures still, a maximum in the heat flux is reached (the critical heat flux). The regime of falling heat transfer which follows is not easy to study but is believed to be characterised by alternate periods of nucleate and film boiling. Nucleate boiling slowing the heat transfer due to gas phase {bubbles} creation on the heater surface, as mentioned, gas phase thermal conductivity is much lower than liquid phase thermal conductivity, so the outcome is a kind of "gas thermal barrier".
At higher temperatures still, the hydrodynamically quieter regime of film boiling is reached. Heat fluxes across the stable vapour layers are low, but rise slowly with temperature. Any contact between fluid and the surface which may be seen probably leads to the extremely rapid nucleation of a fresh vapour layer ('spontaneous nucleation').
[edit] Condensation
Condensation occurs when a vapor is cooled and changes its phase to a liquid. Condensation heat transfer, like boiling, is of great significance in industry. During condensation, the latent heat of vaporization must be released. The amount of the heat is the same as that absorbed during vaporization at the same fluid pressure.
There are several modes of condensation:
Homogeneous condensation (as during a formation of fog).
Condensation in direct contact with subcooled liquid.
Condensation on direct contact with a cooling wall of a heat exchanger-this is the most common mode used in industry:
Filmwise condensation (when a liquid film is formed on the subcooled surface, usually occurs when the liquid wets the surface).
Dropwise condensation (when liquid drops are formed on the subcooled surface, usually occurs when the liquid does not wet the surface). Dropwise condensation is difficult to sustain reliably; therefore, industrial equipment is normally designed to operate in filmwise condensation mode.
[edit] Modelling approaches
A number of approaches to modelling complex heat transfer phenomena have been developed.
[edit] Heat equation
Main article: Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. In some cases, exact solutions of the equation are available; in other cases the equation must be solved numerically (see computational methods, below).
[edit] Computational methods
Please help improve this article by expanding it. Further information might be found on the talk page. (August 2010)
See also: Computational fluid dynamics
For example, simplified climate models may use Newtonian cooling instead of a full (and computationally expensive) radiation code to maintain atmospheric temperatures.
[edit] Lumped system analysis
Main article: Lumped capacitance model
See also: Biot number
Lumped system analysis is a common approximation in transient conduction which may be used whenever heat conduction within an object is much faster than heat conduction across the boundary of the object.
This is a method of approximation that reduces one aspect of the transient conduction system (that within the object) to an equivalent steady state system (that is, it is assumed that the temperature within the object is completely uniform, although its value may be changing in time).
In this method, a term known as the Biot number is calculated, which is defined as the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the object's boundary. For small Biot numbers, the approximation of spatially uniform temperature within the object can begin to be used, since it can be presumed that heat transferred into the object has time to uniformaly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.
[edit] Applications and techniques
Heat transfer has broad application to the functioning of numerous devices and systems. Objectives of studying and applying heat transfer principles may be to preserve (i.e., insulate), increase, or decrease temperature in a wide variety of circumstances.
[edit] Insulation and radiant barriers
Main articles: Thermal insulation and radiant barrier
Heat exposure as part of a fire test for firestop products.Thermal insulators are materials specifically designed to reduce the flow of heat by limiting conduction, convection, or both. Radiant barriers are materials which reflect radiation and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice versa. Metal, for instance, is an excellent reflector and poor insulator.
The effectiveness of an insulator is indicated by its R- (resistance) value. The R-value of a material is the inverse of the conduction coefficient (k) multiplied by the thickness (d) of the insulator. The units of resistance value are in SI units: (K·m²/W)
Rigid fiberglass, a common insulation material, has an R-value of 4 per inch, while poured concrete, a poor insulator, has an R-value of 0.08 per inch.[4]
The tog is a measure of thermal resistance, commonly used in the textile industry, and often seen quoted on, for example, duvets and carpet underlay.
The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa (at any specific wavelength, reflectivity = 1 - emissivity). An ideal radiant barrier would have a reflectivity of 1 and would therefore reflect 100% of incoming radiation. Vacuum bottles (Dewars) are 'silvered' to approach this. In space vacuum, satellites use multi-layer insulation which consists of many layers of aluminized (shiny) mylar to greatly reduce radiation heat transfer and control satellite temperature.
[edit] Critical insulation thickness
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Low thermal conductivity (k) materials reduce heat fluxes. The smaller the k value, the larger the corresponding thermal resistance (R) value.
The units of thermal conductivity(k) are W·m−1·K−1 (watts per meter per kelvin), therefore increasing width of insulation (x meters) decreases the k term and as discussed increases resistance.
This follows logic as increased resistance would be created with increased conduction path (x).
However, adding this layer of insulation also has the potential of increasing the surface area and hence thermal convection area (A).
An obvious example is a cylindrical pipe:
As insulation gets thicker, outer radius increases and therefore surface area increases.
The point where the added resistance of increasing insulation width becomes overshadowed by the effects of surface area is called the critical insulation thickness. In simple cylindrical pipes:[5]
For a graph of this phenomenon in a cylidrical pipe example see: External Link: Critical Insulation Thickness diagram as at 26/03/09
[edit] Heat exchangers
Main article: Heat exchanger
A heat exchanger is a tool built for efficient heat transfer from one fluid to another, whether the fluids are separated by a solid wall so that they never mix, or the fluids are directly contacted. Heat exchangers are widely used in refrigeration, air conditioning, space heating, power generation, and chemical processing. One common example of a heat exchanger is the radiator in a car, in which the hot radiator fluid is cooled by the flow of air over the radiator surface.
Common types of heat exchanger flows include parallel flow, counter flow, and cross flow. In parallel flow, both fluids move in the same direction while transferring heat; in counter flow, the fluids move in opposite directions and in cross flow the fluids move at right angles to each other. The common constructions for heat exchanger include shell and tube, double pipe, extruded finned pipe, spiral fin pipe, u-tube, and stacked plate.
When engineers calculate the theoretical heat transfer in a heat exchanger, they must contend with the fact that the driving temperature difference between the two fluids varies with position. To account for this in simple systems, the log mean temperature difference (LMTD) is often used as an 'average' temperature. In more complex systems, direct knowledge of the LMTD is not available and the number of transfer units (NTU) method can be used instead.
[edit] Heat dissipation
[edit] Heat sinks
Main article: Heat sink
A heat sink is a term for a component that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems and the radiator (also a heat exchanger) in a car. Heat sinks also help to cool electronic and optoelectronic devices, such as CPUs, higher-power lasers, and light emitting diodes (LEDs). A heat sink uses its extended surfaces to increase the surface area in contact with the cooling fluid, the air for example.
[edit] Buildings
In cold climates, houses with their heating systems form dissipative systems. In spite of efforts to insulate such houses to reduce heat losses to their exteriors, considerable heat is lost which can make their interiors uncomfortably cool or cold. For the comfort of the inhabitants, the interiors must be maintained out of thermal equilibrium with the external surroundings. In effect, these domestic residences are oases of warmth in a sea of cold and the thermal gradient between the inside and outside is often quite steep. This can lead to problems such as condensation and uncomfortable draughts (drafts) which, if left unaddressed, can cause cosmetic or structural damage to the property. Such issues can be prevented by use of insulation techniques for reducing heat loss.
Thermal transmittance is the rate of transfer of heat through a structure divided by the difference in temperature across the structure. It is expressed in watts per square metre per kelvin, or W/m²K. Well-insulated parts of a building have a low thermal transmittance whereas poorly-insulated parts of a building have a high thermal transmittance.
A thermostat is a device capable of starting the heating system when the house's interior falls below a set temperature, and of stopping that same system when another (higher) set temperature has been achieved. Thus the thermostat controls the flow of energy into the house, that energy eventually being dissipated to the exterior.
[edit] Thermal energy storage
Main article: Thermal energy storage
Thermal energy storage refers to a number of technologies that store energy in a thermal reservoir for later use. They can be employed to balance energy demand between day time and night time. The thermal reservoir may be maintained at a temperature above (hotter) or below (colder) than that of the ambient environment. Applications include later use in space heating, domestic or process hot water, or to generate electricity. Most practical active solar heating systems have storage for a few hours to a day's worth of heat collected.
[edit] Evaporative cooling
Main article: Evaporative cooler
Evaporative cooling is a physical phenomenon in which evaporation of a liquid, typically into surrounding air, cools an object or a liquid in contact with it. Latent heat describes the amount of heat that is needed to evaporate the liquid; this heat comes from the liquid itself and the surrounding gas and surfaces. The greater the difference between the two temperatures, the greater the evaporative cooling effect. When the temperatures are the same, no net evaporation of water in air occurs, thus there is no cooling effect. A simple example of natural evaporative cooling is perspiration, or sweat, which the body secretes in order to cool itself. An evaporative cooler is a device that cools air through the simple evaporation of water.
[edit] Radiative cooling
Main article: Radiative cooling
Radiative cooling is the process by which a body loses heat by radiation and is an important effect in the Earth's atmosphere. In the case of the earth-atmosphere system it refers to the process by which long-wave (infra red) radiation is emitted to balance the absorption of short-wave (visible) energy from the sun. Convective transport of heat and evaporative transport of latent heat both remove heat from the surface and redistributes it in the atmosphere, making it available for radiative transport at higher altitudes.
[edit] Laser cooling
Main article: Laser cooling
Laser cooling refers to a number of techniques in which atomic and molecular samples are cooled through the interaction with one or more laser light fields. The most common method of laser cooling is Doppler cooling. In Doppler cooling, the frequency of light is tuned slightly below an electronic transition in the atom. Thus the atoms would absorb more photons if they moved towards the light source, due to the Doppler effect. If an excited atom then emits a photon spontaneously, it will be accelerated. The result of the absorption and emission process is to reduce the speed of the atom. Eventually the mean velocity, and therefore the kinetic energy of the atoms will be reduced. Since the temperature of an ensemble of atoms is a measure of the random internal kinetic energy, this is equivalent to cooling the atoms.
Sympathetic cooling is a process in which particles of one type cool particles of another type. Typically, atomic ions that can be directly laser cooled are used to cool nearby ions or atoms. This technique allows cooling of ions and atoms that can't be cooled directly by laser cooling.
[edit] Magnetic cooling
Main article: Magnetic evaporative cooling
Magnetic evaporative cooling is a technique for lowering the temperature of a group of atoms. The process confines atoms using a magnetic field. Collisions mean that over time, individual atoms will become much more energetic than the others, and will escape, removing energy from the system and reducing the temperature of the remaining group. This process is similar to the familiar process by which standing water becomes water vapor.
[edit] Other
A heat pipe is a passive device that is constructed in such a way that it acts as though it has extremely high thermal conductivity. Heat pipes use latent heat and capillary action to move heat, and can carry many times as much heat as a similar-sized copper rod. Originally invented for use in satellites, they are starting to have applications in personal computers.
A thermocouple is a junction between two different metals that produces a voltage related to a temperature difference. Thermocouples are a widely used type of temperature sensor for measurement and control and can also be used to convert heat into electric power.
A thermopile is an electronic device that converts thermal energy into electrical energy. It is composed of thermocouples. Thermopiles do not measure the absolute temperature, but generate an output voltage proportional to a temperature difference. Thermopiles are widely used, e.g., they are the key component of the infrared thermometers that are widely used to measure body temperature via the ear.
A thermal diode or thermal rectifier is a device that preferentially passes heat in one direction; a "one-way valve" for heat.
RUDOLF CLAUSIUS
Rudolf Clausius
From Wikipedia, the free encyclopedia
(Redirected from Clausius)
Jump to: navigation, search
"Clausius" redirects here. For the lunar crater, see Clausius (crater).
Rudolf Clausius
Born 2 January 1822(1822-01-02)
Died 24 August 1888 (aged 66)
Nationality Germany
Fields Physics
Known for Thermodynamics and
originator of the concept of entropy
Rudolf Julius Emanuel Clausius (Born Rudolf Gottlieb,[1] 2 January 1822 – 24 August 1888), was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics.[2] By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis. His most important paper, On the mechanical theory of heat, published in 1850, first stated the basic ideas of the second law of thermodynamics. In 1865 he introduced the concept of entropy.
Contents
1 Life
2 Work
2.1 Entropy
3 Tributes
4 Quotes
5 See also
6 References
7 External links
[edit] Life
Clausius was born in Köslin (now Koszalin) in the Province of Pomerania. He started his education at the school of his father. After a few years, he went to the Gymnasium in Stettin (now Szczecin). Clausius graduated from the University of Berlin in 1844 where he studied mathematics and physics with, among others, Heinrich Magnus, Johann Dirichlet and Jakob Steiner. He also studied history with Leopold von Ranke. During 1847, he got his doctorate from the University of Halle on optical effects in the Earth's atmosphere. He then became professor of physics at the Royal Artillery and Engineering School in Berlin and Privatdozent at the Berlin University. In 1855 he became professor at the ETH Zürich, the Swiss Federal Institute of Technology in Zürich, where he stayed until 1867. During that year, he moved to Würzburg and two years later, in 1869 to Bonn.
In 1870 Clausius organized an ambulance corps in the Franco-Prussian War. He was wounded in battle, leaving him with a lasting disability. He was awarded the Iron Cross for his services.
His wife, Adelheid Rimpham, died in childbirth in 1875, leaving him to raise their six children. He continued to teach, but had less time for research thereafter.
In 1886 he remarried Sophie Stack, and then had another child.
Two years later, on 24 August 1888, he died in Bonn, Germany.
[edit] Work
Clausius' PhD thesis concerning the refraction of light proposed that we see a blue sky during the day, and various shades of red at sunrise and sunset (among other phenomena) due to reflection and refraction of light. Later, Lord Rayleigh would show that it was in fact due to the scattering of light, but regardless, Clausius used a far more mathematical approach than some have used.
His most famous paper, "Über die bewegende Kraft der Wärme" ("On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom") [3] was published in 1850, and dealt with the mechanical theory of heat. In this paper, he showed that there was a contradiction between Carnot's principle and the concept of conservation of energy. Clausius restated the two laws of thermodynamics to overcome this contradiction (the third law was developed by Walther Nernst, during the years 1906–1912). This paper made him famous among scientists.
During 1857, Clausius contributed to the field of kinetic theory after refining August Krönig's very simple gas-kinetic model to include translational, rotational and vibrational molecular motions. In this same work he introduced the concept of 'Mean free path' of a particle. [4] [5] [6]
Clausius deduced the Clausius-Clapeyron relation from thermodynamics. This relation, which is a way of characterizing the phase transition between two states of matter such as solid and liquid, had originally been developed in 1834 by Émile Clapeyron.
[edit] Entropy
Main article: history of entropy
In 1865, Clausius gave the first mathematical version of the concept of entropy, and also gave it its name. He used the now abandoned unit 'Clausius' (symbol: Cl) for entropy. Clausius chose the word "entropy" because the meaning, from Greek, en+tropein, is "content transformative" or "transformation content" ("Verwandlungsinhalt"). [7] [8]
1 Cl = 1 cal/°C = 4.1868 joules per kelvin (J/K)
[edit] Tributes
He was elected a Fellow of the Royal Society of London in 1868 and received its Copley Medal in 1879.
He was elected a member of the Royal Swedish Academy of Sciences in 1878.
He received the Huygens Medal in 1870.
He received the Poncelet Prize in 1883.
He received an honorary doctorate from the University of Würzburg in 1882.
The crater Clausius on the Moon is named in his honor.
[edit] Quotes
The following are two famous quotes made by Clausius in 1865:
The energy of the universe is constant.
The entropy of the universe tends to a maximum.
From Wikipedia, the free encyclopedia
(Redirected from Clausius)
Jump to: navigation, search
"Clausius" redirects here. For the lunar crater, see Clausius (crater).
Rudolf Clausius
Born 2 January 1822(1822-01-02)
Died 24 August 1888 (aged 66)
Nationality Germany
Fields Physics
Known for Thermodynamics and
originator of the concept of entropy
Rudolf Julius Emanuel Clausius (Born Rudolf Gottlieb,[1] 2 January 1822 – 24 August 1888), was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics.[2] By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis. His most important paper, On the mechanical theory of heat, published in 1850, first stated the basic ideas of the second law of thermodynamics. In 1865 he introduced the concept of entropy.
Contents
1 Life
2 Work
2.1 Entropy
3 Tributes
4 Quotes
5 See also
6 References
7 External links
[edit] Life
Clausius was born in Köslin (now Koszalin) in the Province of Pomerania. He started his education at the school of his father. After a few years, he went to the Gymnasium in Stettin (now Szczecin). Clausius graduated from the University of Berlin in 1844 where he studied mathematics and physics with, among others, Heinrich Magnus, Johann Dirichlet and Jakob Steiner. He also studied history with Leopold von Ranke. During 1847, he got his doctorate from the University of Halle on optical effects in the Earth's atmosphere. He then became professor of physics at the Royal Artillery and Engineering School in Berlin and Privatdozent at the Berlin University. In 1855 he became professor at the ETH Zürich, the Swiss Federal Institute of Technology in Zürich, where he stayed until 1867. During that year, he moved to Würzburg and two years later, in 1869 to Bonn.
In 1870 Clausius organized an ambulance corps in the Franco-Prussian War. He was wounded in battle, leaving him with a lasting disability. He was awarded the Iron Cross for his services.
His wife, Adelheid Rimpham, died in childbirth in 1875, leaving him to raise their six children. He continued to teach, but had less time for research thereafter.
In 1886 he remarried Sophie Stack, and then had another child.
Two years later, on 24 August 1888, he died in Bonn, Germany.
[edit] Work
Clausius' PhD thesis concerning the refraction of light proposed that we see a blue sky during the day, and various shades of red at sunrise and sunset (among other phenomena) due to reflection and refraction of light. Later, Lord Rayleigh would show that it was in fact due to the scattering of light, but regardless, Clausius used a far more mathematical approach than some have used.
His most famous paper, "Über die bewegende Kraft der Wärme" ("On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom") [3] was published in 1850, and dealt with the mechanical theory of heat. In this paper, he showed that there was a contradiction between Carnot's principle and the concept of conservation of energy. Clausius restated the two laws of thermodynamics to overcome this contradiction (the third law was developed by Walther Nernst, during the years 1906–1912). This paper made him famous among scientists.
During 1857, Clausius contributed to the field of kinetic theory after refining August Krönig's very simple gas-kinetic model to include translational, rotational and vibrational molecular motions. In this same work he introduced the concept of 'Mean free path' of a particle. [4] [5] [6]
Clausius deduced the Clausius-Clapeyron relation from thermodynamics. This relation, which is a way of characterizing the phase transition between two states of matter such as solid and liquid, had originally been developed in 1834 by Émile Clapeyron.
[edit] Entropy
Main article: history of entropy
In 1865, Clausius gave the first mathematical version of the concept of entropy, and also gave it its name. He used the now abandoned unit 'Clausius' (symbol: Cl) for entropy. Clausius chose the word "entropy" because the meaning, from Greek, en+tropein, is "content transformative" or "transformation content" ("Verwandlungsinhalt"). [7] [8]
1 Cl = 1 cal/°C = 4.1868 joules per kelvin (J/K)
[edit] Tributes
He was elected a Fellow of the Royal Society of London in 1868 and received its Copley Medal in 1879.
He was elected a member of the Royal Swedish Academy of Sciences in 1878.
He received the Huygens Medal in 1870.
He received the Poncelet Prize in 1883.
He received an honorary doctorate from the University of Würzburg in 1882.
The crater Clausius on the Moon is named in his honor.
[edit] Quotes
The following are two famous quotes made by Clausius in 1865:
The energy of the universe is constant.
The entropy of the universe tends to a maximum.
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