Calculus
Part of a series of articles about |
Calculus |
---|
Part of a series on | ||
Mathematics | ||
---|---|---|
|
||
Mathematics Portal | ||
Calculus is the
Originally called infinitesimal calculus or "the calculus of
Infinitesimal calculus was developed independently in the late 17th century by
Etymology
In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances,[5] tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton.[6]
In addition to differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories that seek to model a particular concept in terms of mathematics. Examples of this convention include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
History
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India.
Ancient precursors
Egypt
Calculations of
Greece
Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus (c. 390 – 337 BC) developed the method of exhaustion to prove the formulas for cone and pyramid volumes.
During the
China
The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle.[10][11] In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method[12][13] that would later be called Cavalieri's principle to find the volume of a sphere.
Medieval
Middle East
In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[14]
India
Bhāskara II was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function.[15] In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if then This can be interpreted as the discovery that
Modern
Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus.[17] Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.[18]
Significant work was a treatise, the origin being Kepler's methods,[18] written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus brought together Cavalieri's infinitesimals with the
The
These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.[26] He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with 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 put painstaking effort into his choices of notation.[27]
Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics. Leibniz developed much of the notation used in calculus today.[28]: 51–52 The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.
When Newton and Leibniz first published their results, there was
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and
Foundations
In calculus, foundations refers to the
Several mathematicians, including
In modern mathematics, the foundations of calculus are included in the field of
Limits are not the only rigorous approach to the foundation of calculus. Another way is to use
Significance
While many of the ideas of calculus had been developed earlier in
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.[43]
Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization.[44]: 341–453 Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure.[44]: 685–700 More advanced applications include power series and Fourier series.
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
Principles
Limits and infinitesimals
Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols and were taken to be infinitesimal, and the derivative was their ratio.[34]
The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the
Differential calculus
Differential calculus is the study of the definition, properties, and applications of the
In more explicit terms the "doubling function" may be denoted by g(x) = 2x and the "squaring function" by f(x) = x2. The "derivative" now takes the function f(x), defined by the expression "x2", as an input, that is all the information—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 output another function, the function g(x) = 2x, as will turn out.
In
If the input of the function represents time, then the derivative represents change concerning time. For example, if f is a function that takes 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.[28]: 18–20
If a function is linear (that is if the graph of the function is a straight line), then the function can be written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-intercept, and:
This gives an exact value for the slope of a straight line.[46]: 6 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 concerning 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
Geometrically, the derivative is the slope of the
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.
The slope of the 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 computation similar to the one above shows that the derivative of the squaring function is the doubling function.[46]: 63
Leibniz 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.[46]: 74 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".[46]: 79 Another example of correct notation could be:
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.
Integral calculus
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.
A motivating example is the distance traveled in a given time.[46]: 153 If the speed is constant, only multiplication is needed:
But if the speed changes, a more powerful method of finding the distance is necessary. 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.
When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, traveling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.[44]: 535 This connection between the area under a curve and the distance traveled can be extended to any irregularly shaped region exhibiting a fluctuating velocity over a given period. If f(x) represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the region between f(x) and the x-axis, between x = a and x = b.
To approximate that area, an intuitive method would be to divide up the distance between a and b into several 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.[44]: 512–522
The symbol of integration is , an elongated S chosen to suggest summation.[44]: 529 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.[28]: 44
The indefinite integral, or antiderivative, is written:
Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is a family of functions differing only by a constant.[47]: 326 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:
The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.[48]: 135
Fundamental theorem
The fundamental theorem of calculus states that differentiation and integration are inverse operations.[47]: 290 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
Applications
Calculus is used in every branch of the physical sciences,
Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion, which states that the derivative of an object's momentum concerning time equals the net force upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times it's acceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.[55]
Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus.[56][57]: 52–55 Chemistry also uses calculus in determining reaction rates[58]: 599 and in studying radioactive decay.[58]: 814 In biology, population dynamics starts with reproduction and death rates to model population changes.[59][60]: 631
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.[61] 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 to maximize flow.[62] Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a cancerous tumor grows.[63]
In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.[64]: 387
See also
- Glossary of calculus
- List of calculus topics
- List of derivatives and integrals in alternative calculi
- List of differentiation identities
- Publications in calculus
- Table of integrals
References
- OCLC 527896.
- OCLC 643872.
- ISBN 1-56025-706-7.
- ISBN 0-07-242432-X.
- ^ See, for example:
- "History – Were metered taxis busy roaming Imperial Rome?". Skeptics Stack Exchange. 17 June 2020. Archived from the original on 25 May 2012. Retrieved 13 February 2022.
- Cousineau, Phil (2010). Wordcatcher: An Odyssey into the World of Weird and Wonderful Words. Simon and Schuster. p. 58. from the original on 1 March 2023. Retrieved 15 February 2022.
- ^ "calculus". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
- ISBN 978-0-19-506135-2. Archivedfrom the original on 1 March 2023. Retrieved 20 February 2022.
- OCLC 934433864.
- ^ See, for example:
- Powers, J. (2020). ""Did Archimedes do calculus?"" (PDF). Mathematical Association of America. Archived (PDF) from the original on 9 October 2022.
- Jullien, Vincent (2015). "Archimedes and Indivisibles". Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies. Vol. 49. Cham: Springer International Publishing. pp. 451–457. ISBN 978-3-319-00130-2.
- Plummer, Brad (9 August 2006). "Modern X-ray technology reveals Archimedes' math theory under forged painting". Stanford University. Archived from the original on 20 January 2022. Retrieved 28 February 2022.
- Archimedes (2004). The Works of Archimedes, Volume 1: The Two Books On the Sphere and the Cylinder. Translated by Netz, Reviel. Cambridge University Press. ISBN 978-0-521-66160-7.
- Gray, Shirley; Waldman, Cye H. (20 October 2018). "Archimedes Redux: Center of Mass Applications from The Method". The College Mathematics Journal. 49 (5): 346–352. S2CID 125411353.
- ISBN 978-0-7923-3463-7. Archived from the original on 1 March 2023. Retrieved 15 November 2015.,pp. 279ff Archived 1 March 2023 at the Wayback Machine
- ^ OCLC 32272485.
- ISBN 978-0-321-38700-4.
- ISBN 978-0-7637-5995-7. Archived from the original on 1 March 2023. Retrieved 15 November 2015. Extract of page 27 Archived 1 March 2023 at the Wayback Machine
- ^ JSTOR 2691411.
- ^ Shukla, Kripa Shankar (1984). "Use of Calculus in Hindu Mathematics". Indian Journal of History of Science. 19: 95–104.
- ISBN 0-471-18082-3.
- ^ "Johannes Kepler: His Life, His Laws and Times". NASA. 24 September 2016. Archived from the original on 24 June 2021. Retrieved 10 June 2021.
- ^ a b Chisholm, Hugh, ed. (1911). . Encyclopædia Britannica. Vol. 14 (11th ed.). Cambridge University Press. p. 537.
- ISBN 0-8176-4565-9.
- S2CID 165043307.
The most interesting to us are Lectures X–XII, in which Barrow comes close to providing a geometrical demonstration of the fundamental theorem of the calculus... He did not realize, however, the full significance of his results, and his rejection of algebra means that his work must remain a piece of mid-17th century geometrical analysis of mainly historic interest.
- S2CID 21473035.
- ISBN 978-1-931914-59-8. Archivedfrom the original on 1 March 2023. Retrieved 31 August 2017.
- ISBN 978-0-387-73468-2. Archivedfrom the original on 1 March 2023. Retrieved 31 August 2017.
- ISBN 978-0-444-50871-3.
[Newton] immediately realised that quadrature problems (the inverse problems) could be tackled via infinite series: as we would say nowadays, by expanding the integrand in power series and integrating term-wise.
- ^ OCLC 60416766.
- ISBN 978-1-605-20533-5. Archivedfrom the original on 1 March 2023. Retrieved 5 June 2022.
- ISBN 978-0-691-17337-5.
Leibniz understood symbols, their conceptual powers as well as their limitations. He would spend years experimenting with some—adjusting, rejecting, and corresponding with everyone he knew, consulting with as many of the leading mathematicians of the time who were sympathetic to his fastidiousness.
- ^ OCLC 227002144.
- JSTOR 27956626.
- ISBN 978-0-191-63396-6.
- S2CID 214472568.
- ISBN 978-0-7734-5226-8.
- ^ Unlu, Elif (April 1995). "Maria Gaetana Agnesi". Agnes Scott College. Archived from the original on 3 December 1998. Retrieved 7 December 2010.
- ^ a b c d e f Bell, John L. (6 September 2013). "Continuity and Infinitesimals". Stanford Encyclopedia of Philosophy. Archived from the original on 16 March 2022. Retrieved 20 February 2022.
- George Allen & Unwin Ltd. p. 857.
The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish calculus without infinitesimals, and thus, at last, made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.
- ISBN 978-0-387-90527-3.
- OCLC 682200048.
- OCLC 682200048.
- OCLC 682200048.
- .
- OCLC 682200048.
- OCLC 726764443.
- ISBN 981-02-2201-7.
- ^ OCLC 1022848630. Archivedfrom the original on 23 September 2022. Retrieved 26 July 2022.
- OCLC 1003309980.
- ^ OCLC 135567.
- ^ OCLC 794034942.
- OCLC 961352944.
- ^ See, for example:
- Mahoney, Michael S. (1990). "Barrow's mathematics: Between ancients and moderns". In Feingold, M. (ed.). Before Newton. Cambridge University Press. pp. 179–249. ISBN 978-0-521-06385-2.
- Feingold, M. (June 1993). "Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation". S2CID 144019197.
- Probst, Siegmund (2015). "Leibniz as Reader and Second Inventor: The Cases of Barrow and Mengoli". In Goethe, Norma B.; Beeley, Philip; Rabouin, David (eds.). G.W. Leibniz, Interrelations Between Mathematics and Philosophy. Archimedes: New Studies in the History and Philosophy of Science and Technology. Vol. 41. Springer. pp. 111–134. ISBN 978-9-401-79663-7.
- Mahoney, Michael S. (1990). "Barrow's mathematics: Between ancients and moderns". In Feingold, M. (ed.). Before Newton. Cambridge University Press. pp. 179–249.
- OCLC 1127050110. Archivedfrom the original on 26 July 2022. Retrieved 26 July 2022.
- OCLC 892067655.
- ^ Kayaspor, Ali (28 August 2022). "The Beautiful Applications of Calculus in Real Life". Medium. Archived from the original on 26 September 2022. Retrieved 26 September 2022.
- S2CID 233384462.
- OCLC 860391091.
- OCLC 921230825.
- S2CID 502776.
- OCLC 704518582.
- ^ OCLC 501943698.
- OCLC 53165394.
- OCLC 426065941.
- JSTOR 2320679.
- S2CID 8259705.
- SIAM News. 37 (1). Archived(PDF) from the original on 9 October 2022.
- OCLC 1064041906.
Further reading
- Adams, Robert A. (1999). Calculus: A complete course. Addison-Wesley. ISBN 978-0-201-39607-2.
- Albers, Donald J.; Anderson, Richard D.; Loftsgaarden, Don O., eds. (1986). Undergraduate Programs in the Mathematics and Computer Sciences: The 1985–1986 Survey. Mathematical Association of America.
- Anton, Howard; Bivens, Irl; Davis, Stephen (2002). Calculus. John Wiley and Sons Pte. Ltd. ISBN 978-81-265-1259-1.
- ISBN 978-0-471-00005-1.
- ISBN 978-0-471-00007-5.
- ISBN 978-0-521-62401-5. Uses synthetic differential geometryand nilpotent infinitesimals.
- Boelkins, M. (2012). Active Calculus: a free, open text (PDF). Archived from the original on 30 May 2013. Retrieved 1 February 2013.
- ISBN 0-486-60509-4.
- JSTOR 1967725.
- ISBN 978-3-540-65058-4.
- OCLC 932781617.
- Keisler, H.J. (2000). Elementary Calculus: An Approach Using Infinitesimals. Retrieved 29 August 2010 from http://www.math.wisc.edu/~keisler/calc.html Archived 1 May 2011 at the Wayback Machine
- ISBN 0-8218-2830-4.
- Lebedev, Leonid P.; Cloud, Michael J. (2004). "The Tools of Calculus". Approximating Perfection: a Mathematician's Journey into the World of Mechanics. Princeton University Press. Bibcode:2004apmj.book.....L.
- ISBN 978-0-547-16702-2.
- McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers. University Science Books. ISBN 978-1-891389-24-5.
- ISBN 978-0-471-26987-8.
- Salas, Saturnino L.; ISBN 978-0-471-69804-3.
- ISBN 978-0-914098-89-8.
- ISBN 0-88385-058-3.
- ISBN 978-0-538-49790-9.
- ISBN 978-0-201-53174-9.
- ISBN 978-0-321-48987-6.
- ISBN 978-0-312-18548-0.
External links
- "Calculus", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric W. "Calculus". MathWorld.
- Topics on Calculus at PlanetMath.
- Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF
- Calculus on In Our Time at the BBC
- Calculus.org: The Calculus page at University of California, Davis – contains resources and links to other sites
- Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
- The Role of Calculus in College Mathematics Archived 26 July 2021 at the Wayback Machine from ERICDigests.org
- OpenCourseWare Calculus from the Massachusetts Institute of Technology
- Infinitesimal Calculus – an article on its historical development, in Encyclopedia of Mathematics, ed. Michiel Hazewinkel.
- Daniel Kleitman, MIT. "Calculus for Beginners and Artists".
- Calculus training materials at imomath.com
- (in English and Arabic) The Excursion of Calculus, 1772