This was an Inter Collegiate and a State Level Contest named SIGMA '08. Won a special prize for this paper. This research emphasized on how simple concepts of Mathematics helps into constructing complex mathematical models for space programming and their individual importance in real time applications.

1. Application of Mathematics
A short research on the application of a few selected mathematical concepts, what do they
signify in the world of numerical science and a case study of a single project titled “Global
Precipitation Measurement” that encompasses the amalgamation of all the concepts
considered for this research.
Karthik Murali
9/1/08
Numerical Methods

2. Application of Mathematics
INTRODUCTION
We keep the broad definition here, that mathematics includes all the related areas which
touch on quantitative, geometric, and logical themes. This includes Statistics, Computer
Science, Logic, Applied Mathematics, and other fields which are frequently considered
distinct from mathematics, as well as fields which study the study of mathematics i.e.
History of Mathematics, Mathematics Education and so on. We draw the line only at
experimental sciences, philosophy, and computer applications. Personal perspectives vary
widely, of course.
A fairly standard definition is the one in the Columbia
Encyclopedia (5th edition):"Mathematics is deductive study of numbers, geometry, and
various abstract constructs, or structures. The latter often
arise from analytical models in the empirical sciences, but
may emerge from purely mathematical considerations."
Some definitions of mathematics heard from others:

That which mathematicians do.

The study of well-defined things.

The study of statements of the form "P implies Q".

The science of patterns (Keith Devlin)
Contrary to common perception, mathematics does not consist of ‘crunching numbers’ or
‘solving equations’. As we shall see there are branches of mathematics concerned with
setting up equations, or analyzing their solutions, and there are parts of mathematics
devoted to creating methods for doing computations. But there are also parts of
mathematics which have nothing at all to do with numbers or equations.

3. ORIGIN OF MATHEMATICS
The word ‘mathematics’ (Greek: μαθηματικά or mathēmatiká) comes from the Greek
word μάθημα (máthēma) which means learning, study, science, and additionally came to
have the narrower and more technical meaning "mathematical study", even in Classical
times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious,
which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη
(mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.
APPLIED MATHEMATICS
Applied mathematics considers the use of abstract mathematical tools in solving concrete
problems in the sciences, business, and other areas. An important field in applied
mathematics is statistics, which uses probability theory as a tool and allows the
description, analysis, and prediction of phenomena where chance plays a role. Most
experiments, surveys and observational studies require the informed use of statistics.
(Many statisticians, however, do not consider themselves to be mathematicians, but rather
part of an allied group.) Numerical analysis investigates computational methods for
efficiently solving a broad range of mathematical problems that are typically too large for
human numerical capacity; it includes the study of rounding errors or other sources of
error in computation.
Mathematical Physics
Mathematical Fluid Dynamics
Numerical Analysis
Optimization
Probability
Statistics
Financial Mathematics
Game Theory

4. MATHEMATICAL PHYSICS
Mathematical
physics
is
the
scientific
discipline
concerned with ‘the application of mathematics to
problems in physics and the development of mathematical
methods suitable for such applications and for the
formulation of physical theories.’
It can be seen as underpinning both theoretical physics and computational physics.
MATHEMATICAL FLUID DYNAMICS
Fluid mechanics is the study of the physics of continuous materials which take the shape
of their container.
Like any mathematical model of the real world, fluid mechanics makes some
basic assumptions about the materials being studied. These assumptions are turned into
equations that must be satisfied if the assumptions are to hold true. For example, consider
an incompressible fluid in three dimensions. The assumption that mass is conserved
means that for any fixed closed surface (such as a sphere) the rate of mass passing from
outside to inside the surface must be the same as rate of mass passing the other way.
(Alternatively, the mass inside remains constant, as does the mass outside). This can be
turned into an integral equation over the surface.
NUMERICAL ANALYSIS
Numerical analysis is the study of algorithms or the problems of continuous mathematics
as distinguished from discrete mathematics.

5. OPTIMIZATIONS
In
mathematics,
the
term
optimization,
or
mathematical programming, refers to the study of
problems in which one seeks to minimize or
maximize a real function by systematically choosing
the values of real or integer variables from within an
allowed set.
Many real-world and theoretical problems may be
modeled in a general framework. The branch of
applied mathematics and numerical analysis that is concerned with the development of
deterministic algorithms that are capable of guaranteeing convergence in finite time to the
actual optimal solution of a non-convex problem is called global optimization.
PROBABILITY
Probability is the likelihood or chance that
something is the case or will happen. Probability
theory is used extensively in areas such as
statistics, mathematics, science and philosophy to
draw conclusions about the likelihood of potential
events and the underlying mechanics of complex
systems.
Two major applications of probability theory in everyday life are in risk assessment and
in trade commodity market. Governments typically apply probabilistic methods in
environmental regulation where it is called ‘pathway analysis’, often measuring well
being using methods that are stochastic in nature, and choosing projects to undertake
based on statistical analyses of their probable effect on the population as a whole.

6. It is not correct to say that statistics are involved in the modeling itself, as typically the
assessments of risk are one-time and thus require more fundamental probability models,
e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such
choices and perception of the effect of such choices, which makes probability, measures a
political matter.
STATISTICS
Statistics is a mathematical science pertaining to the collection, analysis, interpretation or
explanation, and presentation of data. It is applicable to a wide variety of academic
disciplines, from the natural and social sciences to the humanities. Statistics is also used
for making informed decisions in government and business.
Statistical methods can be used to summarize or describe a collection of data; this is
called descriptive statistics. In addition, patterns in the data may be modeled in a way that
accounts for randomness and uncertainty in the observations, and then used to draw
inferences about the process or population being studied; this is called inferential
statistics. Both descriptive and inferential statistics comprise applied statistics. There is
also a discipline called mathematical statistics, which is concerned with the theoretical
basis of the subject.
The word statistics is also the
plural of statistic (singular),
which refers to the result of
applying a statistical algorithm
to a set of data, as in economic
statistics, crime statistics, etc.

7. MATHEMATICAL FINANCE
Mathematical finance is the branch of applied mathematics concerned with the financial
markets. The subject has a close relationship with the discipline of financial economics,
which is concerned with much of the underlying theory. Generally, mathematical finance
will derive, and extend, the mathematical or numerical models suggested by financial
economics. Thus, for example, while a financial economist might study the structural
reasons why a company may have a certain share price, a financial mathematician may
take the share price as a given, and attempt to use stochastic calculus to obtain the fair
value of derivatives of the stock. In terms of practice, mathematical finance also overlaps
heavily with the fields of financial engineering and computational finance.
Many universities around the world now offer degree and research programs in
mathematical finance.
GAME THEORY
Game theory is a branch of applied mathematics that is often used in the context of
economics. It studies strategic interactions between agents. In strategic games, agents
choose strategies that will maximize their return, given the strategies the other agents
choose. The essential feature is that it provides a formal modeling approach to social
situations in which decision makers interact with other agents. Game theory extends the
simpler optimization approach developed in neoclassical economics.
These were some of the applications of Mathematics in the real world. Mathematics is an
essential part of our everyday lives. It is very much vivid that without mathematics, many
of the world’s complex problems would not have turned so easy.

8. CASE STUDY: - Global Precipitation Measurement [GPM] Goddard NASA
Some of the concepts of applied mathematics that
are discussed above are put to use or we could say,
are implemented in the GPM Project taken up by
NASA. Mathematical Physics, Mathematical Fluid
Dynamics, Numerical
Analysis, Optimization,
Probability & Statistics are some of the few that we
have seen so far. Mathematics is still an unexplored
area. There is still so much to be revealed. Global
Precipitation is a project initialized for the betterment of mankind. Indirectly,
Mathematics too, becomes a factor that is very clearly helping the Earth in many ways.
CONCLUSION:Mathematics is a powerful subject with the context and constraints of its own. Its
applications are being developed on such a large scale that it will nearly take light years
to study all of them. Mathematics is seriously a boon to mankind as it is indirectly and
unknowingly applied in most of the complex problems; the world faces. Mathematics has
no boundaries and will always hold its importance till eternity.