MADE
BY
NAME : INDU MEGHA
CLASS : Xl th
GUIDED
BY : Mr. LAWANIA
USES
OF MATHEMATICS IN …
I Degree
The
idea of fields was first introduced into mathematics in the
early 19th century by French mathematician Évariste Galois
and Norwegian mathematician Niels Henrik Abel in their studies
of the roots of polynomials (see Equations, Theory of). Fields
were used to show that, although there is a quadratic formula
for solving second-degree polynomials, there is no analogous
formula for the general solution of fifth-degree or higher
polynomials. German mathematician Julius Dedekind who, along
with his colleague Leopold Kronecker, developed abstract field
theory and its application to the theory of numbers first
used the word field at the end of the 19th century.
II Degree
In
advanced branches of mathematics, especially those involving
calculus, angles are usually measured in units called radians
(rad). There are 2 p rad, or about 6.28 rad, in 360°.
In
military usage, angles are usually measured in mils, especially
for artillery aiming. A mil is the measure of a central angle
subtended by an arc that is 1/6400 of the circle. One mil
equals 0.05625° and is approximately 0.001 rad.
III Science
Scientific knowledge in Egypt and Mesopotamia
was chiefly of a practical nature, with little rational organization.
Among the first Greek scholars to seek the fundamental causes
of natural phenomena was the philosopher Thales, in the 6th
century bc, who introduced the concept that the Earth was
a flat disc floating on the universal element, water. The
mathematician and philosopher Pythagoras, who followed him,
established a movement in which mathematics became a discipline
fundamental to all scientific investigation. The Pythagorean
scholars postulated a spherical Earth moving in a circular
orbit about a central fire. In Athens, in the 4th century
bc, Ionian natural philosophy and Pythagorean mathematical
science combined to produce the syntheses of the logical philosophies
of Plato and Aristotle. At the Academy of Plato, deductive
reasoning and mathematical representation were emphasized;
at the Lyceum of Aristotle, inductive reasoning and qualitative
description were stressed. The interplay between these two
approaches to science has led to most subsequent advances.
During
the so-called Hellenistic Age following the death of Alexander
the Great, the mathematician, astronomer, and geographer Eratosthenes
made a remarkably accurate measurement of the Earth. Also,
the astronomer Aristarchus of Samos espoused a heliocentric
(Sun-centred) planetary system, although this concept did
not gain acceptance in ancient times. The mathematician and
inventor Archimedes laid the foundations of mechanics and
hydrostatics (part of fluid mechanics); the philosopher and
scientist Theophrastus became the founder of botany; the astronomer
Hipparchus developed trigonometry; and the anatomists and
physicians Herophilus and Erasistratus based anatomy and physiology
on dissection.
Following
the destruction of Carthage and Corinth by the Romans in 146
bc, scientific inquiry lost its impetus until a brief revival
took place in the 2nd century ad under the Roman emperor and
philosopher Marcus Aurelius. At this time the geocentric (Earth-centred)
Ptolemaic System, advanced by the astronomer Ptolemy, and
the medical works of the physician and philosopher Galen became
standard scientific treatises for the ensuing age. A century
later the new experimental science of alchemy arose, springing
from the practice of metallurgy. By 300, however, alchemy
had acquired an overlay of secrecy and symbolism that obscured
the advantages such experimentation might have brought to
science
IV Geometry
Geometry
(Greek geo, “Earth”; metrein, “to measure”), branch of mathematics
that deals with the properties of space. In its most elementary
form geometry is concerned with such metrical problems as
determining the areas and diameters of two-dimensional figures
and the surface areas and volumes of solids. Other fields
of geometry include analytic geometry, descriptive geometry,
topology, the geometry of spaces having four or more dimensions,
fractal geometry, and non-Euclidean geometry.
V Population
Population
study as an academic discipline is known as demography.
Demography
is an interdisciplinary field involving mathematics and statistics,
biology, medicine, sociology, economics, history, geography,
and anthropology. The field of demography has a relatively
brief history. Its beginning is often dated from the publication
in 1798 of An Essay on the Principle of Population by the
British economist Thomas Robert Malthus. In this work Malthus
warned of the constant tendency for human population growth
to outstrip food production and classified the various ways
that such growth would, in consequence, be slowed. He distinguished
between “positive checks” to population growth (such as war,
famine, and disease) and “preventive checks” (celibacy and
contraception).
The
development of demography has been closely tied to the gradually
increasing availability of data on births and deaths from
parish and civil registers, and on population size and composition
from the censuses that became common in the 19th century.
The growth of the behavioural sciences in the 20th century
and advances in the fields of statistics and computer science
further stimulated demographic research. Subfields of mathematical,
economic, and social demography have grown rapidly in recent
decades.
VI Heat and Temperature
Different
sensations are experienced when hot and cold bodies are touched,
leading to the qualitative and subjective concept of temperature.
The transfer of energy to a body generally leads to an increase
in temperature when no melting or boiling occurs, and in the
case of two bodies at different temperatures brought into
contact, energy flows from one to the other until their temperatures
become the same and thermal equilibrium is reached. Energy
that flows from one body to another as a consequence of temperature
differences is called heat. To arrive at a scientific measure
of temperature, scientists used the observation that the addition
or subtraction of heat produced a change in at least one well-defined
property of a body. For example, heating a column of liquid
maintained at constant pressure increased the length of the
column, while heating a gas confined in a container raised
its pressure. Temperature, therefore, can invariably be measured
by one other physical property, as in the length of the mercury
column in an ordinary thermometer, provided the other relevant
properties remain unchanged. The mathematical relationship
between the relevant physical properties of a body or system
and its temperature is known as the equation of state. Thus,
for a so-called ideal gas, a simple relationship exists between
the pressure, p, volume V, number of moles n, and the absolute
temperature T, given by pV = nRT, where R is the same constant
for all ideal gases. Boyle’s law, named after the British
physicist and chemist Robert Boyle, and Gay-Lussac’s, or Charles’s,
law, named after the French physicists and chemists Joseph
Louis Gay-Lussac and Jacques Alexandre César Charles, are
both contained in this equation of state (seeGases).
Until
well into the 19th century, heat was considered to be a massless
fluid called caloric, contained in matter and capable of being
squeezed out of or into it. Although the so-called caloric
theory answered most early questions on thermometry and calorimetry,
it failed to provide a sound explanation of many early 19th-century
observations. The first true connection between heat and other
forms of energy was observed in 1798 by the Anglo-American
physicist and statesman Benjamin Thompson, Count von Rumford,
who noted that the heat produced in the boring of cannon was
roughly proportional to the amount of work done. (In mechanics,
work is the product of a force on a body and the distance
through which the body moves in the direction of the force
during its application.)
VII MODERN WEATHER FORECASTING
It
has long been recognized that the only reliable method of
producing useful weather forecasts for more than a day ahead
is numerical weather prediction, or NWP. The basis of NWP
is the set of mathematical equations that govern the behaviour
of the atmosphere. These are combined in a complex mathematical
model, and this is applied to observations of the real atmosphere.
The first attempt at NWP was carried out by Lewis Fry Richardson
in 1922. He was unsuccessful because of insufficient data
and computing power, but showed it to be possible. The first
experimental forecast to be completed was at Princeton University
in 1950, involving a simplified set of equations over a model
atmosphere with only one level. That 24-hour forecast took
one day to compute. Subsequent improvements in the mathematical
formulation of the equations and vast increases in computer
power have established NWP as the foundation of weather forecasting
worldwide.
The
laws of physics and the mathematical equations governing the
motion of fluids have been well known for more than a century.
They incorporate the principles of conservation of momentum,
mass, energy, and water, and include laws of motion applied
to a fluid on a rotating sphere as well as the laws of thermodynamics,
radiation, and gases. The Earth's size, rotation rate, geography,
and topography are known, as are daily and seasonal variations
of incoming solar radiation. Other factors include surface
reflectivity (albedo), melting, evaporation, cloud, rain,
friction, and sea temperatures. Many of these factors vary
through the period of a forecast and must be updated accordingly.
The
complex set of equations cannot be solved directly over the
whole atmosphere. They are adapted to operate on the atmosphere
at individual points, each representing an area of the Earth's
surface. The model is applied to a large array of points,
laid out as a grid in the model atmosphere. Each point includes
several levels up through the atmosphere, and can be regarded
as a “stack” of “parcels” of air, each of which represents
a particular level over the area of a grid square.
The
British Meteorological Office's Global Model is one of the
most powerful current NWP models. Its grid comprises 288 points
on each of 217 circles of latitude, with a stack of 19 levels
at each. Thus the set of equations must be solved for well
over a million “parcels” of air to advance the model a step
in time. Every forecast starts with a “first guess” of the
initial state of the atmosphere. This is based on a short-period
forecast from a previous model run, adjusted by thousands
of observations from around the world. Advancing the model
in time can proceed only in short steps of ten minutes or
so, because changes at each “parcel” influence its neighbours.
The “time step” is repeated until the required forecast period
is covered. A 24-hour forecast involves more than a trillion
calculations, and currently takes about 5 minutes. Major NWP
systems are continually being refined as understanding of
the atmosphere improves, computing power increases, and mathematical
techniques advance.
The
grid spacing or horizontal resolution of the British model
averages about 100 km (62 mi). This is important because it
dictates the minimum size of atmospheric disturbance the model
can be expected to forecast. Even the highest resolution model
cannot be expected to predict a shower or thunderstorm with
complete accuracy, but it should give a good indication of
areas in which they might develop. Vertical model resolution
is also important, because there are often important variations
in wind and humidity over depths less than 1 km (.62 mi),
especially near the Earth's surface and high in the atmosphere.
For this reason model levels are unevenly spaced, being clustered
at the top and bottom of the troposphere.
For
greater detail over a smaller area of interest, it is possible
to nest a higher resolution model within a Global Model. This
avoids the extra computing needed with thousands of extra
points over the whole globe. The British Model has a system
rather like a Russian doll: the Global Model contains another
with 50-km (30-mi) spacing which spans Europe and the North
Atlantic, and that has within it a model with 15-km (10-mi)
resolution over the British Isles.
There
remains an important role for the forecasters. They must allow
for weaknesses in the model, take account of later information,
and use experience to add detail and value
VIII GRAPHS
Use
of graphs is in the representation, and sometimes the solution,
of mathematical equations. The graph in figure 2 illustrates
this type of use. Suppose it is known that Yolanda is four
years older than Xavier. Using y for Yolanda’s age and x for
Xavier’s age, this relationship can be written as y = x +
4. There are many possible solutions for this equation: one
possible pair of values for x and y is x = 1 and y = 5, which
can be written briefly as (1,5). In figure 2, distances from
left to right represent values of x, and the horizontal axis
is called the x-axis. Distances upward from the x-axis represent
values of y, and the vertical axis is called the y-axis. The
set of all the pairs (x,y) for which y = x + 4 is represented
by the blue straight line plotted in figure 2. In addition
to knowing that Yolanda is four years older than Xavier, it
is also known that Yolanda is three times Xavier’s age: y
= 3x. The problem is to find values for x and y that make
the equations y = x + 4 and y = 3x both true at the same time.
In figure 2, these two equations are plotted together; the
solution of these simultaneous equations is the point at which
the two graphs intersect, (2,6), which shows that Xavier is
two years old and Yolanda is six years old.
Graphs
can also be used to exhibit inequalities. The curve in figure
3 is a plot of the curve y = x2 - 1 (a parabola). The shaded
area, not including the curve itself, is the graph of the
inequality y > x2 - 1.
These
simple mathematical uses of graphs can be extended to cover
very complex equations. Many physical laws have been discovered
by analysis of the raw data produced by experiment when they
were plotted on graphs. The field of mathematics called calculus
is relevant to the study of graphs. It is concerned with such
things as calculating the gradient, or slope, of a line plotted
on a graph from the equation defining that line. The gradient
of a curve can reveal important information. The steeper the
curve, the more rapidly the quantity shown is varying. Thus,
in a graph of numbers of cases of infectious disease against
time, the steeper the curve, the more rapidly the disease
is spreading, and the greater the cause for alarm.
Graphs
need not always have two axes at right angles. For special
purposes, the axes may be inclined at an angle to one another,
or may even be curved. There may not even be two axes. See
Coordinate System (mathematics).
IX Civil Engineering
Civil
Engineering, branch of engineering embracing expert practices
which create, maintain, and operate the social, commercial,
and industrial infrastructure that sustains a modern society.
This includes all building construction, roads, railways,
canals, airports, harbours, docks, water supply, drainage,
flood and erosion control, bridges, tunnels, pipelines, dams,
irrigation systems, electricity generation, and industrial
facilities.
Engineering
uses mathematics, physics, mechanics, and the properties of
materials to produce cost-effective solutions to building
problems. It does this by combining precedent, the best available
expertise, labour and materials with attention to time, cost,
hazards, and social responsibility.
