A famous Jain Mathematician,
Ācārya Mahāvira (9th century) writes that-“Bahubhirvi pralāpaih kim trailokye
sacarācare, Yatkimcidvastu tatsarva ganitena binā nahi”. The verse shows the
importance of Mathematics as-“What is good of saying much in vain? Whatever
there is in all three worlds, which are possessed of moving and non-moving
being all that indeed cannot exist as apart from Mathematics”.
Here are some main disciplines in which Role of Mathematics is widely accepted.
Here are some main disciplines in which Role of Mathematics is widely accepted.
Mathematics in
Physical Sciences
In Physics, every rule and
principle takes the mathematical form ultimately. Mathematics gives a
final shape to the rules of physics. It presents them in a workable form.
Mathematical calculations occur at every step in physics.
The
units of measurement are employed to substances in physics as frequently as in
mathematics. The Chare’s law of expansion of gases is based upon mathematical
calculations. Graduation of the stem of thermometer and then the conversion of
scales is also a mathematical work.
In mathematical physics, some
basic axioms about mass, momentum, energy, force, temperature, heat etc. are
abstracted, from observations and physical experiments and then the techniques
of abstraction, generalization and logical deduction are used. It is the branch
of mathematical analysis that emphasizes tools and techniques of particular use
to physicists and engineers. It focuses on vector spaces, matrix algebra,
differential equations, integral equations, integral transforms, infinite
series, and complex variables. Its approach can be adapted to applications in
electromagnetism, classical mechanics, and quantum mechanics.
In mathematics, a particle is a
point-like, perfectly rigid, solid object. Particle mechanics deals with the
results of subjecting particles to forces. It includes celestial mechanics —
the study of the motion of celestial objects. The role of Space Dynamics is
very important in mechanics. Here we have to consider the trajectories which
are time-optimal i.e. which take the least time in going from one point to
another and in which the object starts and reaches the destination with zero
velocity. Similarly we may have to consider energy- optimal trajectories. We
have also to consider the internal and external ballistic of rocket and the
path of inter continental ballistic missiles.
Fluid Dynamics :
Understanding the conditions that
result in avalanches, and developing ways to predict when they might occur,
uses an area of mathematics called fluid mechanics. Many mathematicians and
physicists applied the basic laws of Newton to obtain mathematical models for
solid and fluid mechanics. This is one of the most widely applied areas of
mathematics, and is also used in understanding volcanic eruptions, flight,
ocean currents.
Civil and mechanical engineers
still base their models on this work, and numerical analysis is one of their
basic tools. In the 19th century, phenomena involving heat, electricity, and
magnetism were successfully modeled; and in the 20th century, relativistic
mechanics, quantum mechanics, and other theoretical constructs were created to
extend and improve the applicability of earlier ideas. One of the most
widespread numerical analysis techniques for working with such models involves
approximating a complex, continuous surface, structure, or process by a finite
number of simple elements, known as the finite element method (FEM). This
technique was developed by the American engineer Harold Martin and others to
help the Boeing Company analyze stress forces on new jet wing designs in the
1950s. FEM is widely used in stress analysis, heat transfer, fluid flow, and
torsion analysis.
Fluid Dynamics is also very
important in Atmospheric Sciences, in dynamic meteorology and weather
prediction. Another use in the study of diffusion of pollutants in the
atmosphere e.g. to find out what proportion of pollutants emitted from chimneys
or refineries reach hospitals and other buildings. It is also needed for the
study of effect of leakages of poisonous gases.
Computational Fluid
Dynamics :
Computational Fluid Dynamics is a
discipline wherein we use computers to solve the Navier – Stokes equations for
specified initial and boundary condition for subsonic, transonic and hypersonic
flows. Many of our research workers use computers, but usually these are used
at the final stage when drastic simplifications have already been made, partial
differential equation have been reduced to ordinary differential equations and
those equations have even been solved.
Physical Oceanography :
Important fluid dynamics problem
arise in physical oceanography. Problems of waves, tides, cyclones flows in
bays and estuaries, the effect of efflux of pollutants from nuclear and other
plants in sea water, particularly on fish population in the ocean are important
for study. From defense point of view, the problem of under-water explosions,
the flight of torpedoes in water, the sailing of ships and submarines are also
important.
Math is extremely important in
physical chemistry especially advanced topics such as quantum or statistical
mechanics. Quantum relies heavily on group theory and linear algebra and
requires knowledge of mathematical/physical topics such as Hilbert spaces and
Hamiltonian operators. Statistical mechanics relies heavily on probability
theory.
Other fields of chemistry also use
a significant amount of math. For example, most modern IR and NMR spectroscopy
machines use the Fourier transform to obtain spectra. Even biochemistry has
important topics which rely heavily on math, such as binding theory and
kinetics.
Even Pharmaceutical companies
require teams of mathematicians to work on clinical data about the
effectiveness or dangers of new drugs. Pure scientific research in chemistry
and biology also needs mathematicians, particularly those with higher degrees
in computer science, to help develop models of complicated processes.
All chemical combinations and
their equations are governed by certain mathematical laws. Formation of
chemical compounds is governed by mathematical calculations. For
instance, water is a compound, its formation is possible when exactly two atoms
of hydrogen combines with one atom of oxygen. Without this strict
observance of the mathematical fact, the preparation is improbable.
In
the manufacture of any chemical, there is some mathematical ratio in which
different elements have to be mixed. For estimation of elements in organic
compounds, the use of percentage and ratio has to be made.
Molecular
weights of organic compounds are calculated mathematically.
Mathematics in
Biological Sciences
Biomathematics is a rich fertile
field with open, challenging and fascination problems in the areas of
mathematical genetics, mathematical ecology, mathematical neuron- physiology,
development of computer software for special biological and medical problems,
mathematical theory of epidemics, use of mathematical programming and
reliability theory in biosciences and mathematical problems in biomechanics, bioengineering
and bioelectronics.
Mathematical and computational
methods have been able to complement experimental structural biology by adding
the motion to molecular structure. These techniques have been able to bring
molecules to life in a most realistic manner, reproducing experimental data of
a wide range of structural, energetic and kinetic properties. Mathematical
models have played, and will continue to play, an important role in cellular
biology. A major goal of cell biology is to understand the cascade of events
that controls the response of cells to external legends (hormones, transport
proteins, antigens, etc.). Mathematical modeling has also made an enormous
impact on neuroscience. . Three-dimensional topology and two-dimensional
differential geometry are two additional areas of mathematics when it interacts
with biology. Its application is also very important to cellular and molecular
biology in the area of structural biology. This area is at the interface of
three disciplines: biology, mathematics and physics.
In Population Dynamics, we study
deterministic and stochastic models for growth of population of micro-organisms
and animals, subject to given laws of birth, death, immigration and emigration.
The models are in terms of differential equations, difference equations,
differential difference equations and integral equations.
In Internal physiological Fluid
Dynamics, we study flows of blood and other fluids in the complicated network
of cardiovascular and other systems. We also study the flow of oxygen through
lung airways and arteries to individual cells of the human or animal body and
the flow of synovial fluid in human joints. In External Physiological Fluid
Dynamics we study the swimming of micro organisms and fish in water and the
flight of birds in air.
In Mathematical Ecology, we study
the prey predator models and models where species in geographical space are
considered. Epidemic models for controlling epidemics in plants and animals are
considered and the various mathematical models pest control is critically
examined.
In Mathematical Genetics, we study
the inheritance of genetic characteristics from generation to generation and
the method for genetically improving plant and animal species. Decoding of the
genetic code and research in genetic engineering involve considerable
mathematical modeling.
Mathematical theory of the Spread
of Epidemics determines the number of susceptible, infected and immune persons
at any time by solving systems of differential equations. The control of
epidemics subject to cost constraints involves the use of control theory and
dynamic programming. We have also to take account of the incubation period, the
number of carriers and stochastic phenomena. The probability generating
function for the stochastic case satisfies partial differential equations which
cannot be solved in the absence of sufficient boundary and initial conditions.
In Drug kinetics, we study the
spread of drugs in the various compartments of the human body. In mathematical
models for cancer and other diseases, we develop mathematical models for the
study of the comparative effects of various treatments. Solid Biomechanics
deals with the stress and strain in muscles and bones, with fractures and
injuries in skulls etc. and is very complex because of non symmetrical shapes
and the composite structures of these substances. This involves solution of
partial differential equations.
In Pollution Control Models, we
study how to obtain maximum reduction in pollution levels in air, water or
noise with a given expenditure or how to obtain a given reduction in pollution
with minimum cost. Interesting non- conventional mathematical programming
problems arise here.
Mathematics in
Engineering and Technology
The use of mathematics in
engineering is very well known. It is considered to be the foundation of
engineering. Engineering deals with surveying, levelling, designing,
estimating, construction etc., In all these processes, application of
mathematics is very important. By the application of geometric principles
to design and constructions, the durability of things constructed can be
increased. With its help, results can often be verified in engineering.
Mathematics has played an
important role in the development of mechanical, civil, aeronautical and
chemical engineering through its contributions to mechanics of rigid bodies,
hydro-dynamics, aero-dynamics, heat transfer, lubrication, turbulence,
elasticity, etc.. It has become of great interest to electrical engineers
through its applications to information theory, cybernetics, analysis and
synthesis of networks, automatic control systems, design of digital computers
etc. The new mathematical sciences of magneto-hydrodynamics and plasma dynamics
are used for making flow meters, magneto-hydrodynamic generates and for
experiments in controlled nuclear fusion.
It is well known that most of the
technological processes in industry are described effectively by using
mathematical frame work. This frame work is then subsequently used to analyze and
comprehend advantages and disadvantages in adopting efficient and novel
methodologies in these processes, resulting into the introduction of
Mathematical Technology.
The defense sector is an important
employer of mathematicians; it needs people who can design, build and operate
planes and ships, and work on other advanced technologies. It also needs
clear-thinking and analytical strategists.
Agriculture as a science is going
to depend extensively on mathematics. It needs a direct application of
mathematics, such as, measurement of land or area, average investment and
expenditure, average return or income, production per unit area, cost of
labour, time and work, seed rate etc., Progress of the farm can be judged by
drawing graphs of different items of production.
Mathematics and
Economics
The level of mathematical literacy
required for personal and social activities is continually increasing.
Mastery of the fundamental processes is necessary for clear thinking. The
social sciences are also beginning to draw heavily upon mathematics.
Mathematical language and methods
are used frequently in describing economic phenomena. According to Marshall –
“The direct application of mathematical reasoning to the discovery of economic
truths has recently rendered great services in the hand of master
mathematicians.” Statistical methods are used in economic forecast different
issues of economics can be represented statistically such as ‘Trade Cycles’,
Volume trade, trend of exports and imports, population trends, industrial
trends, thrift, expenditure of public money etc.,
In economic theory and
econometrics, a great deal of mathematical work is being done all over the
world. In econometrics, tools of matrices, probability and statistics are used.
A great deal of mathematical thinking goes in the task of national economic
planning, and a number of mathematical models for planning have been
developed.
The models may be stochastic or
deterministic, linear or non-linear, static or dynamic, continuous or discrete,
microscopic or macroscopic and all types of algebraic, differential, difference
and integral equations arise for the solution of these models. At a later stage
more sophisticated models for international economies, for predicting the
results of various economic policies and for optimizing the results are
developed.
Another important subject for
economics is Game theory. The whole economic situation is regarded as a game
between consumers, distributors, and producers, each group trying to optimize
its profits. The subject tries to develop optimal strategies for each group and
the equilibrium values of games.
Mathematics and
Psychology
The great educationist Herbart has
said, “It is not only possible, but necessary that mathematics be applied to
psychology”.
Now, experimental psychology has
become highly mathematical due to its concern with such factors as intelligence
quotient, standard deviation, mean, median, mode, correlation coefficients and
probable errors. Statistical analysis is the only reliable method of
attacking social and psychological phenomena. Until mathematicians entered into
the field of psychology, it was nothing but a flight of imagination.
Suppose, we want to know whether
ability is general or particular [General means, that if a student be brilliant
in mathematics, he will be equally brilliant in language or history.
Particular ability means, that if a student is brilliant in mathematics, it is
not essential that he will be equally brilliant in langue.] To know the
general ability or particular ability, we can use the co-efficient of
correlation.
Mathematics and
Actuarial Science, Insurance and Finance
Actuaries use mathematics and
statistics to make financial sense of the future. For example, if an
organization is embarking on a large project, an actuary may analyze the
project, assess the financial risks involved, model the future financial
outcomes and advise the organization on the decisions to be made. Much of their
work is on pensions, ensuring funds stay solvent long into the future, when
current workers have retired. They also work in insurance, setting premiums to
match liabilities.
Mathematics is also used in many
other areas of finance, from banking and trading on the stock market, to
producing economic forecasts and making government policy.
Archaeologists use a variety of
mathematical and statistical techniques to present the data from archaeological
surveys and try to distinguish patterns in their results that shed light on
past human behavior. Statistical measures are used during excavation to monitor
which pits are most successful and decide on further excavation. Finds are
analyzed using statistical and numerical methods to spot patterns in the way
the archaeological record changes over time, and geographically within a site
and across the country. Archaeologists also use statistics to test the
reliability of their interpretations.
Mathematics and Logic
D’Alembert says, “Geometry is a practical logic, because in it, rules of
reasoning are applied in the most simple and sensible manner”. Pascal says
– “Logic has borrowed the rules of geometry, the method of avoiding error
is sought by everyone. The logicians profess to lead the way, the
geometers alone reach it, and aside form their science there is no true
demonstration”. C.J.Keyser – “Symbolic logic is mathematics, mathematics is
symbolic logic”.
The symbols and methods used in
the investigation of the foundation of mathematics can be transferred to the
study of logic. They help in the development and formulation of logical
laws.
Mathematics in Music
Leibritz, the great mathematician had said, - “Music is a hidden
exercise in arithmetic of a mind unconscious of dealing with numbers”.
Pythogoras has said – “Where harmony is, there are numbers”.
Calculations are the root of all
sorts of advancement in different disciplines. The rhythm that we find in all
music notes is the result of innumerable permutations and combinations of
SAPTSWAR. Music theorists often use mathematics to understand musical structure
and communicate new ways of hearing music. This has led to musical applications
of set theory, abstract algebra, and number theory. Music scholars have also
used mathematics to understand musical scales, and some composers have
incorporated the Golden ratio and Fibonacci numbers into their work.
Most of today's music is produced
using synthesizers and digital processors to correct pitch or add effects to the
sound. These tools are created by audio software engineers who work out ways of
manipulating the digital sound, by using a mathematical technique called
Fourier analysis. This is part of the area of digital signal processing (DSP)
which has many other applications including speech recognition, image
enhancement and data compression.
Mathematics in Arts
Mathematics in Arts
"Mathematics and art are just two different languages that can be
used to express the same ideas." It is considered that the universe is
written in the language of mathematics, and its characters are triangles,
circles, and other geometric figures. The old Goethic Architecture is based on
geometry. Even the Egyptian Pyramids, the greatest feat of human architecture
and engineering, were based on mathematics. Artists who strive and seek to
study nature must therefore first fully understand mathematics. On the other
hand, mathematicians have sought to interpret and analyze art though the lens
of geometry and rationality. This branch of mathematics studies the nature of
geometric objects by allowing them to distort and change. An area that benefits
most from the visual approach is topology.
Moreover the study of origami and
mathematics can be classified as topology, although some feel that it is more
closely aligned with combinatorics, or, more specifically, graph theory.
Huzita's axioms are one important contribution to this field of study. Beauty of a piece of art depends on the
manner in which it expresses truth. Mathematics is knowledge of truth and
realities. It is in itself a piece of fine art. Mathematics exists
in music. Everybody cannot appreciate fully a piece of architecture, a
painting or musical notes. Only a mathematical mind can appreciate these
arts with some sense of confidence. Mathematics provides a basis and
background for aesthetic appreciation. Appreciation of rhythm,
proportion, balance and symmetry postulates a mathematical mind.
Mathematics in Philosophy
The function of mathematics in the
development of philosophical thought has been very aptly put by the great
educationist Herbart, in his words. The real finisher of our education is
Philosophy, but it is the office of mathematics to ward off the dangers of
philosophy.”
Mathematics occupies a central
place between natural philosophy and mental philosophy. It was in their
search of distinction between fact and fiction that plato and other thinkers
came under the influence of mathematics.
Usually, philosophy is defined as
the science that investigates the ultimate reality of things, whereas in
mathematics, the philosophers find orderly and systematic achievements of
unambiguous truths. By warding of the dangers, mathematics puts the
philosophers on the right path of acquiring true knowledge. By eliminating
irrationality, mathematical methods produced the realistic school of thought in
philosophy.
Mathematics in Social Networks
Graph theory, text analysis,
multidimensional scaling and cluster analysis, and a variety of special models
are some mathematical techniques used in analyzing data on a variety of social
networks.
Mathematics in Political Science
In Mathematical Political Science, we analyze past election results to
see changes in voting patterns and the influence of various factors on voting
behavior, on switching of votes among political parties and mathematical models
for Conflict Resolution. Here we make use of Game Theory.
Mathematics in Linguistics
The concepts of structure and
transformation are as important for linguistic as they are for mathematics.
Development of machine languages and comparison with natural and artificial
language require a high degree of mathematical ability. Information theory,
mathematical biology, mathematical psychology etc. are all needed in the study
of Linguistics. Mathematics has had a great influence on research in
literature. In deciding whether a given poem or essay could have been written
by a particular poet or author, we can compare all the characteristics of the
given composition with the characteristics of the poet or other works of the
author with the help of a computer.
Mathematics in Management
Mathematics in management is a
great challenge to imaginative minds. It is not meant for the routine thinkers.
Different Mathematical models are being used to discuss management problems of
hospitals, public health, pollution, educational planning and administration
and similar other problems of social decisions. In order to apply mathematics
to management, one must know the mathematical techniques and the conditions
under which these techniques are applicable. In addition, one must also
understand the situations under which these can be applied. In all the problems
of management, the basic problem is the maximization or minimization of some
objective function, subject to the constraints in available resources in
manpower and materials. Thus OR techniques is the most powerful mathematical
tool in the field of Management.
Mathematics in Computers
An important area of applications
of mathematics is in the development of formal mathematical theories related to
the development of computer science. Now most applications of Mathematics to
science and technology today are via computers. The foundation of computer
science is based only on mathematics. It includes, logic, relations, functions,
basic set theory, countability and counting arguments, proof techniques,
mathematical induction, graph theory, combinatorics, discrete probability,
recursion, recurrence relations, and number theory, computer-oriented numerical
analysis, Operation Research techniques, modern management techniques like
Simulation, Monte Carlo program, Evaluation Research Technique, Critical Path
Method, Development of new computer languages, study of Artificial
Intelligence, Development of automata theory etc.
All mathematical processes of use
in applications are being rapidly converted into computer package algorithms.
There are computer packages for solution of linear and non linear equations,
inversions of matrices, solution of ordinary and partial differential
equations, for linear, non linear and dynamic programming techniques, for
combinatorial problems and for graph enumeration and even for symbolic
differentiation and integration.
Cryptography is the practice and
study of hiding information. In modern times cryptography is considered a
branch of both mathematics and computer science and is affiliated closely with
information theory, computer security and engineering. Cryptography is used in
applications present in technologically advanced societies; examples include
the security of ATM cards, computer passwords and electronic commerce, which
all depend on cryptography. It is the mathematics behind cryptography that has
enabled the e-commerce revolution and information age. Pattern Recognition
is concerned with training computers to recognize pattern in noisy and complex
situations. e.g. in recognizing signatures on bank cheques, in remote sensing
etc.
In Robotics Vision, computers
built in the robots are trained to recognize objects coming in their way
through the pattern recognition programs built into them. In manufacturing
Robotics, the artificial arms and legs and other organs have to be given the
same degree of flexibility of rotation and motion as human arms, legs and
organs have. This requires special developments in mechanics.
Computerized Tomography uses the
important breakthrough in reconstruction of images of brain and objects from
the knowledge of the proportions of photons observed along different lines sent
through the object. These proportions can be expressed as line integrals of a
function.
Fractals Geometry enable us to
design models of irregular objects like clouds, coast lines, lightening
turbulence etc. and this uses a combination of probability theory, mathematics
and computers. This shows that mathematics can enable us to handle apparently
irregular patterns as much as it can enable us to study regular patterns.
In Computer Graphics we find the
virtual landscapes and things within them are three-dimensional mathematical
objects, and these objects behave and interact according to the equations for
the rules of physics that apply within the game. These rules might cover
gravity, speed and force, and even stop your character falling through a solid
floor but allow them to sink in quicksand. This type of mathematics is used in
computer graphics for movies, and mathematics plays an important part in many
areas of IT, including programming, designing hardware and project management.
Mathematics in Geography
Geography is nothing but a scientific and mathematical description of our earth
in its universe. The dimension and magnitude of earth, its situation and
position in the universe the formation of days and nights, lunar and solar
eclipses, latitude and longitude, maximum and minimum rainfall, etc are some of
the numerous learning areas of geography which need the application of
mathematics. The surveying instruments in geography have to be
mathematically accurate. There are changes in the fertility of the soil,
changes in the distribution of forests, changes in ecology etc., which have to
be mathematically determined, in order to exercise desirable control over them.
