MATHEMATICS

Rough Lecture Plan

• What is Mathematics ?

• Information about various Exams.

• Careers in Mathematics.

• Philosophy of Mathematics.• Philosophy of Mathematics.

What is Mathematics?

• <R, +, *>

R denotes the set of real numbers,

+ denotes the operation of addition,

* denotes the operation of multiplication.* denotes the operation of multiplication.

What is Mathematics?

• Mathematics is a study of Sets with

Structures. There are three basic structures.

Three Basic Structures of Mathematics

• Algebraic Structures

Three Basic Structures of Mathematics

• Algebraic Structures → • An algebraic

structure consists of

one or more sets

closed under one or

more operations more operations

satisfying some axioms.

<R, +>

Three Basic Structures of Mathematics

• Algebraic Structures

• Topological Structures

Three Basic Structures of Mathematics

• Algebraic Structures

• Topological Structures →

• Topological structures allow the formal definition of concepts such as convergence, connectedness, and continuity in every and continuity in every mathematical space where it is possible to do so.

The branch of mathematics that studies topological spaces in their own right is called topology.

Three Basic Structures of Mathematics

• Algebraic Structures

• Topological Structures

• Ordered Structures

Three Basic Structures of Mathematics

• Algebraic Structures

• Topological Structures

• Ordered Structures →

• In Mathematical space:

the relations are defined

to relate or compare

different elements.

These relations These relations

together with the set

under consideration

gives rise to ordered

structure in

Mathematics.

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Logic

What is Logic?

• Logic is the basis of Mathematical reasoning.

• It makes possible to assign meaning to

statements in Mathematics.

• The rules of logic allows us to distinguish • The rules of logic allows us to distinguish

between valid and invalid statements in

mathematics.

Rules of Logic

How the empty set is subset of

every set?

To prove A is subset of B, We prove

So to prove φ is subset of S, We prove

Consider the statement

It is the Statement of the form

qq

(*)

It is the Statement of the form

where p is always false. Since p implies q is always true when p is false (regardless of truth value of q), the statement (*) is true.

Definition

The definition of definition consists

of certain axioms. It is studied in

the subject Logic. the subject Logic.

Two properties of Definition

1. Definition is always Fundamental.

2. Definition is if and only if type of 2. Definition is if and only if type of statement.

DEFINITIONFundamental

Level

1. Definition is always Fundamental

Consider the following argument

Where is the Mistake?

The proper way to calculate

product Is to use the DEFINITION

of product of complex numbers.

Definition of product of two

complex numbers is

2. Definition is an If and only If

statement.

Triangle is equilateral

All its sides are equal.

Definition of SET

In formal mathematics, you begin with

some "undefined objects", some axioms or

postulates about those objects, possibly some

definitions, and from those, you prove

theorems. In Euclidean geometry, for example,

the terms "point, line, plane" are undefined

objects.

Euclid tried to define a point as "A point is Euclid tried to define a point as "A point is

that which has no part." Of course as soon as he

said that, we can ask, "What is a part?" If he had

given the definition of "part", it would have had

undefined terms, and the process would never

end.

The only thing we officially know about these

things is whatever the postulates tell us. For example,

there is a postulate that tells us that every two points

lie on exactly one line. From this, we can prove the

following very simple theorem: "If the system contains

at least two points, then it contains a line."

Because of this, more modern mathematicians Because of this, more modern mathematicians

realized that it is impossible to define everything--you

will always be forced into an infinite set of definitions--

so the only solution is to begin with a set of undefined

terms, and then to write down a series of properties of

these terms that you call axioms or postulates and

work from those.

Axiomatic Set Theory

The axioms for set theory now most often

studied and used, although put in their final

form by Skolem, are called the Zermelo-Fraenkel

axioms (ZF). Actually, this term usually excludes

the axiom of choice, which was once more

controversial than it is today. When this axiom is

included, the resulting system is called ZFC.included, the resulting system is called ZFC.

An important feature of ZFC is that every object

that it deals with is a set. In particular, every

element of a set is itself a set. Other familiar

mathematical objects, such as numbers, must be

subsequently defined in terms of sets.

The ten axioms of ZFC are listed below. (Strictly speaking, the axioms of ZFC

are just strings of logical symbols. What follows should therefore be viewed

only as an attempt to express the intended meaning of these axioms in

English. Moreover, the axiom of separation, along with the axiom of

replacement, is actually an infinite schema of axioms, one for each formula.)

Each axiom has further information in its own article.

1. Axiom of extensionality: Two sets are the same if and only if they have the

same elements.

2. Axiom of empty set: There is a set with no elements. We will use {} to 2. Axiom of empty set: There is a set with no elements. We will use {} to

denote this empty set.

3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as

its only elements.

4. Axiom of union: For any set x, there is a set y such that the elements

of y are precisely the elements of the elements of x.

5. Axiom of infinity: There exists a set x such that {} is in x and whenever y is

in x, so is the union y U {y}.

6. Axiom of separation (or subset axiom): Given any set and

any proposition P(x), there is a subset of the original set containing precisely

those elements x for which P(x) holds.

7. Axiom of replacement: Given any set and any mapping, formally defined as

a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set

containing precisely the images of the original set's elements.

8. Axiom of power set: Every set has a power set. That is, for any set x there

exists a set y, such that the elements of y are precisely the subsets of x.exists a set y, such that the elements of y are precisely the subsets of x.

9. Axiom of regularity (or axiom of foundation): Every non-empty

set x contains some element y such that x and y are disjoint sets.

10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint

nonempty sets, there is a set y (a choice set for x) containing exactly one

element from each member of x.

The axioms of choice and regularity are still controversial today among a

minority of mathematicians.

Set theory foundations for

mathematicsIt is often asserted that axiomatic set theory is an adequate

foundation for current mathematical practice, in the sense

that in principle all proofs produced by the mathematical

community could be written formally in set theory terms. It is

also generally believed that no serious advantage would come

from doing that, in almost all cases: the axiomatic foundations from doing that, in almost all cases: the axiomatic foundations

normally used are sufficiently closely aligned to the underlying

set theory, that full axiomatic translation yields only a little extra,

compared to argument in the usual, traditional informal style.

One area where a gap can appear between practice and easy

formalisation is in category theory, where for example a concept

like 'the category of all categories' requires more careful set-

theoretic handling.

Objections to set theorySince its inception, there have been some mathematicians who

have objected to using set theory as a foundation for

mathematics, claiming that it is just a game which includes

elements of fantasy. Notably,

Henri Poincare said

"set theory is a disease from which mathematics will one day

recover", and

Errett Bishop dismissed set theory as God's mathematics, which Errett Bishop dismissed set theory as God's mathematics, which

we should leave for God to do.

The most frequent objection to set theory is based on the

constructivist view that, loosely, mathematics has something to

do with computation. See mathematical constructivism.

On the other hand this is not really an objection to axiomatic set

theory, as a formal theory. It is a comment on the naive set

theory that is being formalised, and its admission of non-

computational elements.

Set is the most fundamental

concept of Mathematics

that is

Every definition in Mathematics is

ultimately a Set.ultimately a Set.

e.g. definition of function,

sequence, group is a set. (How?)

Concept of a function:

A function f from set A to set B is

the relation from set A to set B

such that every element of set A is

related to unique element of B.

In other words it is the subset of In other words it is the subset of

A B in which every element of B

appears exactly once as first

coordinate. It is a subset of A B

i.e. it is a set.

Sequence

It is a function whose domain is a

set of natural numbers. Since a

function is a set, sequence is a set.function is a set, sequence is a set.

Group

Group = { <G, *>/ * is a binary operation

on set G satisfying p1, p2, p3 }

where

p : * is associativep1 : * is associative

p2 : G has identity with respect to *

p3 : every element of G has an inverse

element with respect to * .

Conclusions

1. Set is the most fundamental

concept of Mathematics.

2. Every Definition in Mathematics

is ultimately a Set.

3. In set theory, the word "set" is

formally undefined.

4. All we know about sets are what

the axioms of set theory tell us. the axioms of set theory tell us.

Various Entrance Exams/Interviews at

Graduate and Post-Graduate level

1. IIT JAM Entrance

Examination

2. TIFR Entrance Examination

7. NBHM Ph.D. Scholarship

Examination

8. GATE Entrance

Examination3. IISC Bangalore (I-Math)

Entrance Examination

4. NBHM MA/M.Sc.

Scholarship Examination

5. JEST Entrance Examination

6. INAT Entrance Examination

Examination

9. IISER Ph.D. Interview

10. IMSC Ph.D. Interview

11. ISI Examination

12. SET Exam

13. NET Exam

Basic Books For these Exams

Analysis

i) Real Analysis

Author: N. L. Carothers

Publisher: Cambridge University Press

ii) Real Analysisii) Real Analysis

Author: H.L. Royden

Publisher: Prentice Hall of India

iii) Analysis on Manifolds

Author: J. R. Munkres

Publisher: Addison Wesley Publishing Company.

iv) Functions of One Complex Variable

Author: John B. Conway

Publisher: Narosa Publishing House

Algebra

• i) An Introduction to AlgebraAuthor- Donald LewisPublisher- Harper and Row, New York.

• ii) A First Course in Abstract Algebra• ii) A First Course in Abstract AlgebraAuthor- John B. FraleighPublisher- Narosa publishing house.

• iii) Linear AlgrbraAuthor- Seymour LipschutzPublisher- Schaum Series

Topology

• i) Topology Author- James R. MunkresPublisher- Prentice Hall of India

• ii) An Introduction to Topology & Modern Analysis • ii) An Introduction to Topology & Modern Analysis Author- G.F. SimmonsPublisher- Tata McGraw-Hill.

• iii) Topology Author- Seymour LipschutzPublisher- Schaum Series

1. IIT JAM Entrance Examination

SYLLABUS

• Sequences, Series and Differential Calculus : Sequences of real numbers. Convergent sequences and series, absolute and conditional convergence. Mean value theorem. Taylor 's theorem. Maxima and minima of functions of a single variable. Functions of two and three variables. Partial derivatives, maxima and minima.

• Integral Calculus : Integration, Fundamental theorem of calculus. Double and Triple, integrals, Surface areas and volumes.

• Differential Equations : Ordinary differential equations of the first order of the form y'=f(x,y). Linear differential equations of second order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.

• Vector Calculus : Gradient, divergence, curl and Laplacian. Green's, Stokes' and Gauss' theorems • Vector Calculus : Gradient, divergence, curl and Laplacian. Green's, Stokes' and Gauss' theorems and their applications.

• Algebra : Groups, subgroups and normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups, rings, ideals, quotient rings and fields.

• Linear Algebra : Systems of linear equations. Matrices, rank, determinant, inverse. Eigenvalues and eigenvectors. Finite Dimensional Vector Spaces over Real and Complex Numbers, Basis, Dimension, Linear Transformations.

• Real Analysis : Open and closed sets, limit points, completeness of R, Uniform Continuity, Uniform convergence, Power series.

How to Prepare?

• Basic Books on Algebra and Analysis

(except : Functions of One Complex Variable)

1) Calculus Volume II1) Calculus Volume II

Author- Tom M. Apostol

Publisher- John Wiley & Sons.

2) Differential Equations

Author- Gupta, Malik, Mittal

Publisher- Pragati Publications, Meerut.

2. TIFR Entrance Examination

(Tata Institute of Fundamental

Research, Mumbai)

• Website:

http://univ.tifr.res.in/gs2010/index.html

Question Paper Pattern

• There are questions on Algebra, Analysis,

Topology and simple questions on

Combinatorics and on other topics.

• Questions are of only True/False type!• Questions are of only True/False type!

• There is negative marking.

How to Prepare?

• Basic Books on Algebra and Analysis and Topology

• Combinatorics

Author- V. K. Balakrishnan

Publisher- Schaum Series.

3. IISC Bangalore (I-Math) Entrance

Examination

(Indian Institute of Science, Bangalore)

• At IISC Bangalore:

1. Ph. D. Programme (After M.Sc.)1. Ph. D. Programme (After M.Sc.)

2. Integrated Ph.D. Programme (After B.Sc.)

How to get into IISC?

• By Qualifying in one of the following Tests

CSIR-UGC NET for JRF; or UGC-NET for JRF; or

DBT JRF or ICMR JRF; or JEST; or NBHM; or

IISc Entrance Test; or GATE.IISc Entrance Test; or GATE.

How to prepare for IISC Entrance?

• Basic Books on Algebra and Analysis and

Topology

• Topics from Applied Mathematics chosen from

the syllabus

4. NBHM MA/M.Sc. Scholarship

Examination

(National Board for Higher

Mathematics, Department of Atomic Mathematics, Department of Atomic

Energy, Mumbai)

• Website:

http://www.nbhm.dae.gov.in/msc.html

What is NBHM MA/M.Sc. Scholarship?

• NBHM (National Board of Higher

Mathematics) gives scholarship to the

selected students doing MA/M.Sc. In

Mathematics. Around Ten students are Mathematics. Around Ten students are

selected from all over India on the basis of

their performance on written test and

interview. At present this scholarship is Rs.

6000 per month which amounts to Rs.

1,44,000 for two years of post graduation.

Who can apply for this Scholarship?

• Student appearing/appeared in T.Y.B.Sc, M.Sc.

I year or in M.Sc. II year.

How To Prepare?

• There are three sections in the question

paper:

1. Algebra

2. Analysis2. Analysis

3. Topology/Geometry (optional)

Student can choose Topology or Geometry

from third section.

Books

• 1. Algebra

• i) Introduction to AlgebraAuthor- Donald LewisPublisher- Harper and Row, New York.(This is a MUST MUST have book).You just need to go through all the statements and results in this You just need to go through all the statements and results in this book.By doing so you will have Complete, Concise, Clear collection of all the results in Group Theory, Ring Theory, Field Theory, Linear Algebra (Vector Spaces). This collection will help you to solve any question on these topics. Great Book!!!

2. Analysis

• i) Real AnalysisAuthor: H.L. RoydenPublisher: Prentice Hall of India

Prepare only first Nine chapters.

ii) Complex Variables and ApplicationsAuthors- Churchil and BrownPublisher- McGraw HillPrepare only the first Six Chapters of this book. This book is written in simple language and is very useful.

3. Geometry (optional)

• i) Book: Solid GeometryAuthor: M.L. KhannaPublisher: Jai Prakash Nath & Co.,

Prepare only first Eight chapters of this book.

ii) Book: Elements of Coordinate GeometryAuthor: S.l. LoneyPublisher: S. Chand

Prepare only first Fourteen Chapters.

3. Topology (optional)

• i) Book: An Introduction to Topology & Modern AnalysisAuthor: G.F. SimmonsPublisher:Tata McGraw-Hill

Prepare only chapters Three to SixPrepare only chapters Three to Six

ii) Book: General TopologyAuthor: S. LipschutzPublisher: Schaum Series

•

5. JEST Entrance Examination

Joint Entrance Screening Test

• Website: http://jest09.veccal.ernet.in/

What is JEST?

• A number of reputed institutes in the country have come together to conduct a Joint Entrance Screening Test (JEST) for enrolment of students in Ph.D. / Integrated M.Sc.-Ph.D. programs in physics (including various inter-disciplinary research areas in physics) and Theoretical Computer Science.

JEST is conducted for selecting candidates to be interviewed for admission to Ph. D. Programmes in Physics/ Theoretical Computer admission to Ph. D. Programmes in Physics/ Theoretical Computer Science in these institutions in the country.

Applications are invited from motivated students, with consistently good academic record, for appearing in JEST leading to enrolment in a Ph. D. Programme in Physics (including various inter-disciplinary research areas in Physics)/ Theoretical Computer Science, at any of the 21 premier participating institutions.

JEST score forms an important component in the selection of candidates for the Ph.D. programmes in these institutions.

Institutes which allow admission on

qualifying JEST1. ARIES, Nainital

2. IIA, Bangalore

3. IISC, Bangalore

4. IISER MOHALI, Mohalii

5. IISER PUNE, Pune

11. SNBNCBS, Kolkata

12. BARC, Mumbai

13. UGC-DAE CSR, Indore

14. HRI, Allahabad

15. IGCAR, Kalpakkam5. IISER PUNE, Pune

6. IUCAA, Pune

7. JNCASR, Bangalore

8. NCRA, Pune

9. PRL, Ahmedabad

10. RRI, Bangalore

15. IGCAR, Kalpakkam

16. IOP, Bhubaneswar

17. IMSC, Chennai.

18. IPR, Gandhinagar

19. RRCAT, Indore

20. SINP, Kolkata

21. VECC, Kolkata

Eligibility

• Students who expect to complete their final examination by August 2009 are also

eligible to apply.

• Minimum Qualifications & Selection Procedures

• Ph.D. Programme (PHYSICS)

• M.Sc. in Physics or M.Sc. / M.E. / M.Tech. in related disciplines. Candidates with

M.Sc. in Mathematics and B.Tech (in Engg Physics/Applied Physics/Post B.Sc. Hons M.Sc. in Mathematics and B.Tech (in Engg Physics/Applied Physics/Post B.Sc. Hons

only) will also be considered at IIA. Graduates with a B.E. or B.Tech. will also be

considered at IISc, IMSc, IUCAA, JNCASR, NCRA, RRI, IISER Mohali, IISER Pune and

SNBNCBS. At IPR candidates should have a Masters degree in Physics, Engineering

Physics, Applied Physics, etc. with at least 55% aggregate. Talented final year B. Sc.

Graduates will also be considered for pre-selection at IUCAA.

• Ph.D. Programme (Theoretical Computer Science)

• M.Sc./ M.E. / M.Tech. / M.C.A. in Computer Science and related disciplines, and

should be interested in the mathematical aspects of computer science. Talented

B.Sc. / B.E. / B.Tech. graduates will also be considered for integrated M.Sc.- Ph.D.

programme.

1. At HRI, candidates with a Bachelor's degree will also be considered for the integrated M.Sc. -

Ph.D. programme in Physics.

2. At IMSc, graduates with B.Sc. / B.E. / B. Tech. degree will also be considered for admission to

M.Sc. (by Research) programme in Physics and Theoretical Computer Science, as part of an

integrated Ph.D. programme.

3. At SNBNCBS, graduates with B.Sc. (Physics) / B.E. / B.Tech. degree will be considered for the

integrated Post-B.Sc.-Ph.D. programme in subject areas mentioned above.integrated Post-B.Sc.-Ph.D. programme in subject areas mentioned above.

4. At IIA, graduates with B.Sc. (Physics/Mathematics) / B.E. / B.Tech. degree will be considered

for the integrated M.Sc.-Ph.D. programme in subject areas mentioned above.

5. Integrated M.Tech - Ph.D. Programme at IIA

6. At IIA, graduates with M.Sc. (Physics / Applied Physics) / B.E. / B.Tech. degree will be

considered for the integrated M.Tech-Ph.D. programme in subject areas mentioned above.

Details of the programme can be found on the IIA webpages.

Mere qualifying in JEST does not entitle one to get a Research Fellowship. Using the JEST

results, each institute will call a limited number of candidates for its further selection

procedure depending on its requirements. All candidates selected after interview will

receive Research Fellowship from the respective institutes.

How to Prepare?

• Basic Books on Analysis and Topology

1. Fundamentals of Physics

Author- Halliday, Rescnick, Walker

Publisher- John Wiley & Sons.Publisher- John Wiley & Sons.

2. Fundamental Laws of Mechanichs

Author- I.E. Irodov

Publisher- MIR Publishers Moscow/CBS Publisher India

3. Fundamental Laws of Electromagnetism

Author- I.E. Irodov

Publisher- MIR Publishers Moscow/CBS Publisher India

4. Problems in General Physics

Author- I.E. Irodov

Publisher- MIR Publishers Moscow/CBS Publisher India

6. INAT Entrance Examination

IUCAA-NCRA Admission Test

What is INAT Test?

• IUCCA-NCRA Admisson Test is being conducted to select candidate for research scholarship to join Ph.D. programme either at

Inter University for Astronomy and Inter University for Astronomy and Astrophysics (IUCAA), Pune

OR

National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research (NCRA-TIFR), Pune.

Eligibility and Preparation

• Same as that of JEST test.

7. NBHM Ph.D. Scholarship

Examination

• Website:

http://www.nbhm.dae.gov.in/phd.html

What is NBHM Ph.D. Scholarship?

• NBHM (National Board of Higher

Mathematics) gives scholarship to the

selected students for doing Ph.D. in

Mathematics. Around Ten students are Mathematics. Around Ten students are

selected from all over India on the basis of

their performance on written test and

interview. At present this scholarship is Rs.

22,000 per month. The scholarship is given

maximum up to 5 years.

Who can apply for this Scholarship?

• Student appearing/appeared in T.Y.B.Sc., M.Sc.

I year M.Sc. II year or those who have

completed M.Sc.

Pattern of the Exam

• There are five sections, containing ten questions each

1. Algebra

2. Analysis

3. Topology

4. Applied Mathematics4. Applied Mathematics

5. Miscellaneous.

• Answer as many questions as possible.

• The assessment of the paper will be based on the best FOUR sections.

• Question Types: Objective and Short answer.

How to prepare?

• Basic Books on Algebra and Analysis and

Topology

• Selected topics from Applied Mathematics

8. GATE Entrance Examination

• Graduate Aptitude Test in Engineering

What is GATE?

• Clearing GATE (Graduate Aptitude Test in Engineering) examination in Mathematics opens door to do Ph.D. in many National Institutes such as IIT, IISC, IISER etc.

• GATE qualified candidates (in Mathematics) can • GATE qualified candidates (in Mathematics) can also apply for M.Tech. in selected subjects in selected institutes including IIT’s.

• All GATE qualified students who have taken admissions for Ph.D./M.Tech get scholarship from MHRD (Ministry of Human Resource and Development).

Eligibility

• M.Sc. II year appeared or passed.

Syllabus

• Linear Algebra: Finite dimensional vector spaces; Linear transformations and their matrix representations, rank; systems of linear equations, eigen values and eigen vectors, minimal polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian, Skew-Hermitian and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators.

• Complex Analysis: Analytic functions, conformal mappings, bilinear transformations; complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle; Taylor and Laurent’s series; residue theorem and applications for evaluating real integrals.

• Real Analysis: Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass approximation theorem, compactness; Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem.

• Ordinary Differential Equations: First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients; method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality.

Syllabus

• Algebra: Normal subgroups and homomorphism theorems, automorphisms; Group actions, Sylow’stheorems and their applications; Euclidean domains, Principle ideal domains and unique factorization domains. Prime ideals and maximal ideals in commutative rings; Fields, finite fields.

• Functional Analysis: Banach spaces, Hahn-Banach extension theorem, open mapping and closed graph theorems, principle of uniform boundedness; Hilbert spaces, orthonormal bases, Rieszrepresentation theorem, bounded linear operators.

• Numerical Analysis: Numerical solution of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration; interpolation: error of polynomial interpolation, Lagrange, Newton interpolations; numerical differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss Legendre quadrature, method of undetermined parameters; Trapezoidal and Simpson rules, Gauss Legendre quadrature, method of undetermined parameters; least square polynomial approximation; numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition); iterative methods (Jacobi and Gauss-Seidel); matrix eigenvalue problems: power method, numerical solution of ordinary differential equations: initial value problems: Taylor series methods, Euler’s method, Runge-Kutta methods.

• Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in two variables; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations.

• Mechanics: Virtual work, Lagrange’s equations for holonomic systems, Hamiltonian equations.

• Topology: Basic concepts of topology, product topology, connectedness, compactness, countabilityand separation axioms, Urysohn’s Lemma.

Syllabus

• Probability and Statistics: Probability space, conditional probability, Bayestheorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, conditional expectation, moments; Weak and strong law of large numbers, central limit theorem; Sampling distributions, UMVU estimators, maximum likelihood estimators, Testing of hypotheses, standard parametric tests based on normal, X2 , t, F – distributions; Linear regression; Interval estimation.

• Linear programming: Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex

• Linear programming: Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, big-M and two phase methods; infeasible and unbounded LPP’s, alternate optima; Dual problem and duality theorems, dual simplex method and its application in post optimality analysis; Balanced and unbalanced transportation problems, u -u method for solving transportation problems; Hungarian method for solving assignment problems.

• Calculus of Variation and Integral Equations: Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions.

Question Paper Pattern

Patterns of Question papers Negative Marks for wrong Answer

Q.1 to Q.20 : Will carry one mark

each (sub-total 20 marks).

1/3 mark will be deducted for each wrong

answer.

Q.21 to Q.50 : Will carry two marks

each (sub-total 60 marks)

2/3 mark will be deducted for each wrong

answer.

Question pairs (Q.57, Q.58) and (Q.59, There will be negative marks only for Question pairs (Q.57, Q.58) and (Q.59,

Q.60) will be linked answer questions.

The answer to the second question of the

last two pairs will depend on the answer

to the first question of the pair.

If the first question in the linked pair is

wrongly answered or is un-attempted,

then the answer to the second question

in the pair will not be evaluated. Each

question will carry two marks

There will be negative marks only for

wrong answer to the first question of the

linked answer question pair i.e. for Q.57

and Q.59, 2/3 mark will be deducted for

each wrong answer. There is no negative

marking for Q.58 and Q.60.

How to prepare?

• Basic Books on Algebra and Analysis and Topology

1) Differential Equations

Author- Gupta, Malik, MittalAuthor- Gupta, Malik, Mittal

Publisher- Pragati Publications, Meerut.

2) For functional analysis:

Book: An introduction to Topology and Modern Analysis

Author- G.F. Simmons

Publisher- Tata McGraw Hill

Books

3) For Numerical Analysis:

Book: Numerical Methods for Mathematics, Science and Engineering.

Author- John H. Mathews.Author- John H. Mathews.

Publisher- Prentice Hall of India

4) For Partial Differential Equations:

Book: Partial Differential Equations

Author- W. E. Williams

Publisher- Clarendon Press Oxford

Books

5) For Linear Programming Theory:

I used internet resources for preparing

theory of various methods of LPP.

6) For Linear Programming Problems:6) For Linear Programming Problems:

Book: Operations Research

Author- V.K. Kapoor

Publisher-

Books

7) For Mechanics:

Book: Classical Mechanics

Author- N.C. Rana, P.C. Joag

Publisher- Tata McGraw HillPublisher- Tata McGraw Hill

8) For Calculus of Variation and Integral Equations

Book:

Author-

Publisher-

Books

9) For Probability and Statistics

Book:

Author-

Publisher-Publisher-

9. IISER Ph.D. Interview

(Indian Institute of Science Education

and research)

About IISER

• The Government of India, through the Ministry of Human Resource Development (MHRD), has established five Indian Institutes of Science Education and Research (IISER). These institutes are located in Bhopal, Mohali, Pune, Kolkata and Thiruvanantapuram. The IISERs represent a unique Thiruvanantapuram. The IISERs represent a unique initiative in India where teaching and education are totally integrated with state-of-the-art research nurturing both curiosity and creativity in an intellectually vibrant atmosphere of research. Each IISER is an autonomous institution awarding its own Masters and Doctoral degrees.

•

IISER Ph.D. Interview- Eligibility

• To appear for interview candidate should have

cleared any one of these exams:

1. UGC CSIR NET with JRF

2. GATE (Minimum percentile 95)2. GATE (Minimum percentile 95)

3. NBHM Ph.D. Scholarship Examination

For more information, visit the IISER’s

websites:

1. IISER Pune:

Website: www.iiserpune.ac.in

2. IISER Bhopal:

Website: www.iiserbhopal.ac.in

3. IISER Mohali:3. IISER Mohali:

Website: www.iisermohali.ac.in

4. IISER Kolkata:

Website: www.iiserkol.ac.in

5. IISER Thiruvanantapuram:

Website: www.iisertvm.ac.in

10. IMSC Ph.D. Interview

Institute of Mathematical Sciences,

Chennai

• Website: www.imsc.res.in

IMSCInstitute of Mathematical SciencesChennai

At IMSC Chennai:

1. Ph. D. Programme (After M.Sc.)

2. Integrated Ph.D. Programme (After B.Sc.)

Eligibility

• Applicants for the PhD programme should

have completed a Masters degree in

Mathematics or Statistics by the time they

actually join the programme.actually join the programme.

• Applicants for the integrated PhD programme

should have completed a Bachelors degree in

Mathematics or Statistics by the time they

actually join the programme.

How to get into IMSC?

• Admission to both of these programmes (Ph.Dand integrated Ph.D.) is based on the candidate's performance in an interview. Candidates are selected for the interview by screening applications. This screening is screening applications. This screening is typically based on the applicant's performance in the PhD Scholarship screening test of the NBHM.

• There is NO separate entrance test of IMSC.

IMSC Scholarships

• A monthly stipend of Rs. 22,000 for the first

two years and Rs. 24,000 in subsequent years

(subject to satisfactory performance),

together with an annual contingency grant of together with an annual contingency grant of

Rs. 20,000, housing and medical facilities will

be provided.

11. ISI Examination

INDIAN STATISTICAL INSTITUTE

Kolkata• Website: www.isical.ac.in

About ISI

• The Indian Statistical Institute (I.S.I.), founded by Professor Prasanta Chandra Mahalanobis, grew out of the Statistical Laboratory set up by him in the Presidency College in Kolkata. In 1932 the Institute was registered as a non-profit making learned society for the advancement of statistics in India. Within a few years the Institute's achievements in research that included innovative projects on sample surveys of agricultural crops and socio-economic after-effects of the Bengal famine (1943-44) as well as pathbreaking research publications of Professor R.C. Bose on experimental designs in the Annals of Eugenics (1939), brought recognition in India and abroad. The Institute is now considered as one of the foremost centres in the and abroad. The Institute is now considered as one of the foremost centres in the world for training and research in statistics and related sciences. Under the leadership of Professor P. C. Mahalanobis, the Institute also initiated and promoted the interaction of statistics with natural and social sciences to unfold the role of statistics as a key technology which explicated the twin aspects of statistics - its general applicability and its dependence on other disciplines for its own development. In keeping with this long tradition, the Institute has been engaged in developing statistical theory and methods and their practical applications in various branches of science and technology.

ISI Exam for JRF/Scholarship

• There will be two tests RM-1, and RM-2 of 2 hours duration each in the forenoon and in the afternoon.

• 1) Topics for MI (Forenoon examination) : Real Analysis, Lebesgue Integration, Complex Analysis, Ordinary Differential Equations and General Topology.Differential Equations and General Topology.

• 2) Topics for MII (Afternoon examination) : Algebra, Linear Algebra, Functional Analysis, Elementary Number Theory and Combinatorics.

• Candidates will be judged based on their performance in both the tests.

How to prepare?

• Basic Books on Algebra and Analysis and Topology

1) Differential Equations

Author- Gupta, Malik, MittalAuthor- Gupta, Malik, Mittal

Publisher- Pragati Publications, Meerut.

2) For functional analysis:

Book: An introduction to Topology and Modern Analysis

Author- G.F. Simmons

Publisher- Tata McGraw Hill

Books

3) For Elementary Number Theory:

Book: An Introduction to Number Theory

Author- I. Niven, H. Zuckerman

Publisher- Wiley Eastern LimitedPublisher- Wiley Eastern Limited

4) For Combinatorics:

Book: Combinatorics

Author- V. K. Balakrishnan

Publisher- Schaum Series

12. SET Exam

13. CSIR NET/JRF Exam

12. SET Exam

13. CSIR NET/JRF Exam

Toughest Exam/Interview

Toughest Exam/Interview

• TIFR Interview (after

clearing written test)

Toughest Exam/Interview

• TIFR Interview (after

clearing written test)

Easiest Exam

Toughest Exam/Interview

• TIFR Interview (after

clearing written test)

Easiest Exam

SET Exam

The Joy of Learning Mathematics

A great discovery solves a great problem but

there is a grain of discovery in the solution of

any problem. Your problems may be modest ;

but if it challenges your curiosity and brings into

play your inventive faculties, and if you solve it play your inventive faculties, and if you solve it

by your own means , you may experience the

tension and enjoy the triumph of victory. Such

experiences at a susceptible age may create a

taste for mental work and leave their imprint on

mind and character for a lifetime.

Thus a student whose college curriculum

includes some Mathematics has a singular

opportunity. This opportunity is lost, of course, if

he regards mathematics as a subject in which he

has to earn so and so much marks and which he

should forget after the final examination as

quickly as possible. The opportunity may be lost quickly as possible. The opportunity may be lost

even if the student has some natural talent for

Mathematics because he, as everybody else,

must discover his talents or tastes; he cannot

know that he likes cake if he has never tasted

cake.

You may mange to find out, however, a

Mathematics problem may be as much fun as a

crossword puzzle, or that vigorous mental work

may be an exercise as desirable as a fast game of

football. Having tasted the pleasure in football. Having tasted the pleasure in

Mathematics you will not forget it easily and

then there is a good chance that Mathematics

will become something for you: a hobby, or a

tool of your profession, or your profession or a

great ambition.

Teaching Mathematics

A teacher of Mathematics has great

Opportunity.

If He:

Fills his allotted time with drilling his

students in routine operations-students in routine operations-

Then He:

Kills their interest, hampers their

intellectual development, and misuses the

opportunity.

But –

If He:

Challenges the Curiosity of students by

setting them problems proportionate to their

knowledge, and helps them to solve their

problems with stimulating questions problems with stimulating questions

Then He:

Stimulates the interest in Mathematics

and gives some means of Independent thinking.

Are You interested in Mathematics?

It is Not necessary

that you are interested in Mathematics

But it is Necessary

that you are interested in Something.

You Must “Discover” that “Something.”

But:

Before You Say

that you are Not interested in Mathematics

You, as everybody else, You, as everybody else,

must discover Your Talents or Tastes;

You cannot know that you like cake if you

have never tasted cake.

Stop Saying that

Mathematics

is Difficult.is Difficult.

If Mathematics is Difficult ,

What is Easy ?

Is Playing Cricket Easy?

Is Acting Easy?

Is Singing Easy?

Careers in Mathematics:

MATEMATICS TEACHINGRESEARCH

Teaching

The task of Teaching Mathematics is not

quite easy.

It demands time, practice , devotion and It demands time, practice , devotion and

sound principles.

I hate 'teaching'....I love lecturing,

and have lectured a

great deal to extremely able great deal to extremely able

classes.

-G. H. Hardy

Who is good teacher of Mathematics?

Beauty of Mathematics

“Mathematics, rightly viewed, possesses not

only truth, but supreme beauty — a beauty cold

and austere, like that of sculpture, without

appeal to any part of our weaker nature,

without the gorgeous trappings of painting or

music, yet sublimely pure, and capable of a

stern perfection such as only the greatest art stern perfection such as only the greatest art

can show. The true spirit of delight, the

exaltation, the sense of being more than Man,

which is the touchstone of the highest

excellence, is to be found in mathematics as

surely as poetry.”

- Bertrand Russell

The good teacher is one who can

convey this Beauty to the students.

What is an Effective teaching in

Mathematics?

It results in

1. an increase in mathematics

communication among students;

2. an improvement in student ability to 2. an improvement in student ability to

read mathematics;

3. an increase in the understanding of

mathematics by the students;

.

4. a greater use of mathematics vocabulary by

students;

5. an increased student interest in the class

because of the

variety in ‘teachers’ throughout the course;variety in ‘teachers’ throughout the course;

6. a positive student attitude toward the class

and mathematics; and

7. a more ‘student-centred’ classroom

Research

Who should do the research?Who should do the research?

If

One feels that his or her natural talent

is not suitable for some topic or even for

the subject Mathematics,

Then

He or She should not proceed for the He or She should not proceed for the

research

Because the most probably he or she

would not contribute anything meaningful

to the subject.

Keep other options open.

A Man of Mathematics need not be in the field of

Mathematics.

A student having Mathematical background

performs well in various competitive exams.

The Mathematics people have Analytical Minds

and they do Well in other fields such as software

development.

Go Ahead and look for other options. Explore the

possibilities and select the one where you can perform

well.

Software

Management (M.B.A)

Banking MATEMATICS Software

Development

Banking

Exams

Other competitive exams

M.P.S.C., U.P.S.C.

Philosophy of MathematicsOn the one hand, philosophy of mathematics is

concerned with problems that are closely related to

central problems of metaphysics At first blush,

mathematics appears to study abstract entities. This

makes one wonder what the nature of mathematical makes one wonder what the nature of mathematical

entities consists in and how we can have knowledge of

mathematical entities. If these problems are regarded

as intractable, then one might try to see if

mathematical objects can somehow belong to the

concrete world after all.

On the other hand, it has turned out that to

some extent it is possible to bring mathematical

methods to bear on philosophical questions

concerning mathematics. The setting in which

this has been done is that of mathematical

logic when it is broadly conceived as comprising logic when it is broadly conceived as comprising

proof theory, model theory, set theory, and

computability theory as subfields. Thus the

twentieth century has witnessed the

mathematical investigation of the consequences

of what are at bottom philosophical theories

concerning the nature of mathematics.

When professional mathematicians are

concerned with the foundations of their subject,

they are said to be engaged in foundational

research. When professional philosophers

investigate philosophical questions concerning

mathematics, they are said to contribute to the

philosophy of mathematics. Of course the

distinction between the philosophy of distinction between the philosophy of

mathematics and the foundations of

mathematics is vague, and the more interaction

there is between philosophers and

mathematical logicians working on questions

pertaining to the nature of mathematics, the

better.

The Beauty of Mathematics is the

BEAUTY OF ITS CREATOR who

made it perfect as He is the only

one who is PERFECT. LET’S PRAISE

HIM.

LET’S ADORE HIM.LET’S ADORE HIM.

AND LET’S LISTEN AND DO WHAT

HE TELLS US TO DO.

-Unknown

Locus of Mathematical Reality

“ I believe that Mathematical Reality lies

outside us,that our function is to

discover or observe it,discover or observe it,

and that

the theorems which we prove,

and which we describe grandiloquently as our

'creations‘ are simply our notes of our

observations. ”

-G. H. Hardy