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Calculus biography, Calculus discography
For other uses, see Calculus (disambiguation).Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.Historically, it was sometimes referred to as "the calculus of infinitesimals", but that usage is seldom seen today.Most basically, calculus is the study of change, in the same way that geometry is the study of space.Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient.Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus.In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.More generally, calculus (plural calculi) can refer to any method or system of calculation guided by the symbolic manipulation of expressions.Sir 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 history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods.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.BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.From the school of Greek mathematics, Eudoxus (c.BC) developed this idea further, inventing heuristics which resemble integral calculus.Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method that is readily generalizable to finding the formula for the sum of any integral powers, which was fundamental to the development of integral calculus.Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.In the modern period, independent discoveries in calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method of exhaustion.James Gregory proved a special case of the second fundamental theorem of calculus in AD 1668.Gottfried Wilhelm Leibniz was originally accused of plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus.Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus.Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts.The basic insight that both Newton and Leibniz had was the fundamental theorem of calculus.When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit.Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society.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.It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane.Lebesgue further generalized the notion of the integral.Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development.Significance
While some of the ideas of calculus were developed earlier, in 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 the basic principles of calculus.This work had a strong impact on the development of physics.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.Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.The usual one 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.Calculus is usually developed by manipulating very small quantities.Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean property.From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals.Limits describe the value of a function at a certain input in terms of its values at nearby input.From this viewpoint, calculus is a collection of techniques for manipulating certain limits.Differential calculus is the study of the definition, properties, and applications of the derivative or slope of a graph.The process of finding the derivative is called differentiation.In technical language, the derivative is a linear operator, which inputs a function and outputs a second function, so that at every point the value of the output is the slope of the input.The concept of the derivative is fundamentally more advanced than the concepts encountered in algebra.In algebra, students learn about functions which input a number and output another number.For example, if the derivative inputs the squaring function, then it outputs the doubling function, because the doubling function gives the slope of the squaring function at any given point.If the input of a function is time, then the derivative of that function is the rate at which the function changes.If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies, and we can use calculus to find an exact value at a given point.Note that y and f(x) represent the same thing: the output of the function.Working out one particular case, we find the slope of the squaring function at the point where the input is 3 and the output is 9 (i.The procedure can be visualized as in the following figure.This slope is determined by considering the limiting value of the slopes of secant lines.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.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.The fundamental theorem of calculus states that differentiation and integration are inverse operations.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 is also a prototype solution of a differential equation.Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.Physics makes particular use of calculus; all concepts in classical mechanics are interrelated through calculus.In the subfields of electricity and magnetism calculus can be used to find the total flux of electromagnetic fields.Acceleration involves differential calculus because acceleration can be expressed as the derivative of velocity.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.Calculus can be used in conjunction with other mathematical disciplines.In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maximums and minimums), slope, concavity and inflection points.In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.Calculus can be used to find approximate solutions to equations, in methods such as Newton's method, fixed point iteration, and linear approximation.There is no exact evidence on how it was done; some, including Morris Kline (Mathematical thought from ancient to modern times Vol."Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), pp.Tusi's Muadalat", Journal of the American Oriental Society 110 (2), pp.Science and technology in free India.Transactions of the Royal Asiatic Society of Great Britain and Ireland.Calculus: Early Transcendentals, 5th ed.Introduction to calculus and analysis 1.Survey, Mathematical Association of America No.Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998.Florian Cajori, "The History of Notations of the Calculus."Annals of Mathematics, 2nd Ser.Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch.The Tools of Calculus", Princeton Univ.Calculus and Pizza: A Math Cookbook for the Hungry Mind.Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY.Calculus and Analytic geometry 9th, Addison Wesley."Second Fundamental Theorem of Calculus.""Calculus" Light and Matter, Fullerton."Notes on first year calculus" University of Minnesota."Difference Equations to Differential Equations: An introduction to calculus"."Calculus" Massachusetts Institute of Technology.Topics on Calculus at PlanetMath.Calculus Made Easy (1914) by Silvanus P.ThompsonFull text in PDF
The Online Calculus course for transfer, notes, video lectures, active forum at San Francisco State University by Professor Arek Goetz
Calculus.Online Integrator (WebMathematica) from Wolfram Research
The Role of Calculus in College Mathematics from ERICDigests.Encyclopaedia of Mathematics, Michiel Hazewinkel ed.This page was last modified on 23 May 2008, at 21:25.All text is available under the terms of the GNU Free Documentation License.Works for PCs, Macs and Linux.Cover of Calculus, by Professor Gilbert Strang.Image courtesy of Gilbert Strang.OCW is pleased to make this textbook available online.It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications.The Velocity at an Instant, pp.The Derivative of a Function, pp.The Slope and the Tangent Line,
pp.Applications of the Derivative,
pp.Maximum and Minimum Problems,
pp.Ellipses, Parabolas, and Hyperbolas,
pp.Newton's Method and Chaos,
pp.The Mean Value Theorem and l'Hopital's Rule, pp.Derivatives by the Charin Rule,
pp.Inverses of Trigonometric Functions,
pp.The Idea of an Integral, pp.Indefinite Integrals and Substitutions,
pp.Powers Instead of Exponentials,
pp.Length of a Plane Curve, pp.Area of a Surface of Revolution, pp.Polar Equations and Graphs, pp.Convergence Tests: All Series, pp.Cross Products and Determinants, pp.Matrices and Linear Equations, pp.Linear Algebra in Three Dimensions, pp.Plane Motion: Projectiles and Cycloids, pp.Tangent Vector and Normal Vector, pp.Polar Coordinates and Planetary Motion, pp.Surface and Level Curves, pp.Directional Derivatives and Gradients, pp.Maxima, Minima, and Saddle Points, pp.Changing to Better Coordinates, pp.Stokes' Theorem and the Curl of F, pp.GRAPHICS FOR THE CALCULUS CLASSROOM
Douglas N.These are excerpts from a collection of graphical demonstrations I
developed for first year calculus.Those interested in higher math may
also want to visit my page of graphics for complex
analysis.Also the animation is a
bit smoother, and the frames shuttle (first to last and then backward
to first, etc.An older version of this page
using the MPEG animation format is available, but no longer actively
maintained, and so not recommended.Differentials and differences
This animation
expands upon the classic calculus diagram above.The point is
that if the green segment is small, the yellow segment is
very small.The first image reviews the basic principle.The
other images treat a specific volume, that of the wedge of water formed
when a cylindrical class of equal height and diameter is tipped until
the water line runs through the center of the base.The pictures are
frozen frames from AVS, and can only convey a rough idea of the
interactive classroom presentation (which typically lasts about 30
minutes).Three different ways to slice the
same volume
And now for the quiz: Compute the percentage of the glass filled by
water using each of the the three slicings depicted in the last slide
and verify that they all lead to the same answer.Archimedes calculated the value of
to an accuracy of one accuracy of one part in a thousand.His
technique was based on inscribing and circumscribing polygons in a
circle, and is very much akin to the method of lower and upper sums
used to define the Riemann integral.His approach is presented in the
following sequence of slides.How the ball bounces
As a way to help students appreciate functions, their applications,
and their graphs, I involve them in a small project to describe the
functions determined by the height of a bouncing ball.The students view the
animation (in slow motion, with manual frame advance, etc.As a homework assignment they are
asked to determine the function algebraically.This is a piecewise
quadratic and helps the students to realize that piecewise defined
functions do exist outside of calculus books.Secants and tangents
This is a pretty straightforward animation depicting the geometric
convergence of secant lines to the tangent line.The slope of the secant
(which converges to the derivative) is also displayed.Students can easily demonstrate this themselves
using a graphics calculator equipped with a zoom button.In this
animation, we provide some extra distance queueing by showing the grid
and striping the tangent line.The proof then follows from the "squeeze theorem."Here are some instructions for creating
it in class.During the presentation I make frequent recourse to plotting
software to verify the various inequalities.Given good classroom graphics facilities
such an exploration is easy, but it is almost hopeless without them.This plot of such a function was
produced with a few lines of Matlab code
following Weierstrass's classical construction.On a slower machine it is preferable to
use fewer points, and decrease the point spacing as you zoom in.Students are often puzzled by the appearance of the number e,
which is given above (to 35 decimal places).By manipulating the frame advance, you can
adjust a so that the tangent has slope close to 1.The
second
animation is similar to the first, but drawn on a larger
scale, and from it one can read off the first few decimal places of e.The intersection of two cylinders
Here's a demonstration by my
colleague David Sibley illustrating the computation of the volume of the
region formed by two intersecting cylinders.Note:
On the morning of Wednesday, April 18, the Temple University network will be unavailable for several hours due to
an upgrade.Warning:
The COW system will be unavailable for a few hours on the morning of Tuesday, December 19 for a system upgrade.No systems will be available during this maintenance window.If you wish to log in for a recorded
session, click on the Login button.
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