Everything about Calculus totally explained
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. Some examples of other well-known calculi are
propositional calculus,
predicate calculus,
relational calculus, and
lambda calculus.
History
Development
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 doesn't 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. 1800 BC), in which an Egyptian successfully calculated the
volume of a
pyramidal
frustum. 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. The
method of exhaustion was later used in
China by
Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by
Zu Chongzhi in the 5th century AD, who used it to find the volume of a
sphere. Around AD 1000, the
Islamic mathematician Ibn al-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's readily generalizable to finding the formula for the sum of any
integral powers, which was fundamental to the development of integral calculus. In the 12th century, the
Persian mathematician Sharaf al-Din al-Tusi discovered the
derivative of
cubic polynomials, an important result in differential calculus. In the 14th century,
Madhava of Sangamagrama, along with other mathematician-astronomers of the
Kerala school of astronomy and mathematics, described special cases of
Taylor series, which are treated in the text
Yuktibhasa.
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. In Europe, the second half of the 17th century was a time of major innovation. Calculus provided a new opportunity in
mathematical physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably
John Wallis and
Isaac Barrow.
James Gregory proved a special case of the
second fundamental theorem of calculus in AD 1668.
Leibniz and
Newton pulled these ideas together into a coherent whole and they're 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 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. 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. 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.
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 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. 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.
Principles
Limits and infinitesimals
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". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals don't satisfy the
Archimedean property. From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint 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 using ordinary numbers. From this viewpoint, 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 easy to put on rigorous foundations, and for this reason they're the standard approach to calculus.
Derivatives
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 doubling function inputs 3, then it outputs 6, while if the squaring function inputs 3, it outputs 9. But the derivative inputs a function and outputs another function. 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.
To understand the derivative, students must learn mathematical notation. In mathematical notation, one common symbol for the derivative of a function is an apostrophe-like mark called
prime. Thus the derivative of
f is
f′ (spoken "f prime"). The last sentence of the preceding paragraph, in mathematical notation, would be written
»
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.
Applications
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. 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. In the subfields of
electricity and
magnetism calculus can be used to find the total
flux of electromagnetic fields. A more historical example of the use of calculus in physics is
Newton's second law of motion, it expressly uses the term "rate of change" which refers to the derivative:
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. Even the common expression of Newton's second law as Force = Mass × 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. For example, it can be used with
linear algebra to find the "best fit" linear approximation for a set of points in a domain.
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. For instance, spacecraft use a variation of the
Euler method to approximate curved courses within zero gravity environments.
Further Information
Get more info on 'Calculus'.
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