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UNIVERSITY OF MUMBAI

SYLLABUS for the F.Y.B.A/B.Sc.

Programme: B.A./B.Sc.

Subject: Mathematics

Choice Based Credit System (CBCS)

with eect from the

academic year 2016-17

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Semester I

Calculus I

Course Code Unit Topics Credits L/Week

USMT101,UAMT101Unit I Real Number system

Unit II Sequences 3 3

Unit III Limits & Continuity

Algebra I

USMT102Unit I Integers & divisibility

Unit II Functions & Equivalence relation 3 3

Unit III Polynomials

Semester II

Calculus II

Course Code Unit Topics Credits L/Week

USMT201,UAMT201Unit I Series

Unit II Continuous functions & Dierentiation 3 3

Unit III Applications of dierentiation

Linear Algebra

USMT202Unit I System of Linear Equations & Matrices

Unit II Vector spaces 3 3

Unit III Basis & Linear transformations

Teaching Pattern

1. Three lectures per week per course. Each lecture is of 1 hour duration.

2. One tutorial per week per course (the batches to be formed as prescribed

by the University)

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Syllabus for Semester I & II

Note : All topics have to be covered with proof in details (unless mentioned

otherwise) and with examples.

USMT101/UAMT101 CALCULUS I

Unit I: Real Number System (15 Lectures)

Real number system Rand order properties of R, Absolute valuej:jand its

properties.

AM-GM inequality, Cauchy-Schwarz inequality, Intervals and neighbourhoods,

Hausdor property.

Bounded sets, statement of l.u.b. axiom, g.l.b. axiom and its consequences,

Supremum and inmum, Maximum and minimum, Archimedean property and

its applications, density of rationals.

Unit II: Sequences (15 Lectures)

Denition of a sequence and examples, Convergence of sequence, every conver-

gent sequence is bounded, Limit of a convergent sequence and uniqueness of

limit , Divergent sequences.

Convergence of standard sequences like

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1 +na

8a>0;(bn)80**n)8c>0;& (n1n);algebra of convergent sequences, sandwich theorem, monotone sequences, mono-tone convergence theorem and consequences such as convergence of (1 +1n)n):Denition of subsequence, subsequence of a convergent sequence is conver-gent and converges to the same limit, denition of a Cauchy sequence, everyconvergent sequence is a Cauchy sequence and converse.Unit III: Limits & Continuity (15 Lectures)Brief review: Domain and range of a function, injective function, surjective func-tion, bijective function , composite of two functions (when dened), Inverse ofa bijective function.Graphs of some standard functions such as jxj;ex;logx;ax2+bx+c;1x;xn(n3);sinx;cosx;tanx;xsin(1x);x2sin(1x)over suitable intervals of R:3**

Denition of Limit lim

x!af(x)of a function f(x);evaluation of limit of simple

functions using the denition, uniqueness of limit if it exists, algebra of lim-

its , limit of composite function, sandwich theorem, left-hand-limit lim

x!af(x),

right-hand-limit lim

x!a+f(x), non-existence of limits, lim

x!1f(x);lim

x!1f(x)

and lim

x!af(x) =1:

Continuous functions: Continuity of a real valued function on a set in terms of

limits, examples, Continuity of a real valued function at end points of domain,

Sequential continuity, Algebra of continuous functions, Discontinuous functions,

examples of removable and essential discontinuity.

Reference Books :

1. R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

2. K.G. Binmore, Mathematical Analysis, Cambridge University Press, 1982.

3. R.G. Bartle- D.R. Sherbert, Introduction to Real Analysis, John Wiley &

Sons, 1994.

Additional Reference Books

1. T. M. Apostol, Calculus Volume I, Wiley & Sons (Asia) Pte. Ltd.

2. Richard Courant-Fritz John, A Introduction to Calculus and Analysis, Vol-

ume I, Springer.

3. AjitKumar-S. Kumaresan, A Basic Course in Real Analysis, CRC Press, 2014.

4. James Stewart, Calculus, Third Edition, Brooks/cole Publishing Company,

1994.

5. Ghorpade, Sudhir R.- Limaye, Balmohan V., A Course in Calculus and Real

Analysis, Springer International Ltd, 2000.

Tutorials for USMT101, UAMT101:

1) Application based examples of Archimedean property, intervals, neighbour-

hood. 2) Consequences of l.u.b. axiom, inmum and supremum of sets. 3)

Calculating limits of sequences. 4) Cauchy sequences, monotone sequences. 5)

Limit of a function and Sandwich theorem. 6) Continuous and discontinuous

functions.

USMT102 ALGEBRA I

Prerequisites:

Set Theory: Set, subset, union and intersection of two sets, empty set, univer-

sal set, complement of a set, De Morgan's laws, Cartesian product of two sets,

Relations, Permutations nPrand Combinations nCr:

Complex numbers: Addition and multiplication of complex numbers, modulus,

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amplitude and conjugate of a complex number.

Unit I: Integers & divisibility (15 Lectures)

Statements of well-ordering property of non-negative integers, Principle of nite

induction (rst and second) as a consequence of well-ordering property, Bino-

mial theorem for non-negative exponents, Pascal Triangle.

Divisibility in integers, division algorithm, greatest common divisor (g.c.d.) and

least common multiple (l.c.m.) of two integers, basic properties of g.c.d. such

as existence and uniqueness of g.c.d. of integers a&band that the g.c.d. can

be expressed as ma+nbfor somem;n2Z;Euclidean algorithm, Primes, Eu-

clid's lemma, Fundamental theorem of arithmetic, The set of primes is innite.

Congruences, denition and elementary properties, Eulers 'function, State-

ments of Eulers theorem, Fermats theorem and Wilson theorem, Applications.

Unit II: Functions and Equivalence relations (15 Lectures)

Denition of function; domain, co-domain and range of a function; compos-

ite functions, examples, Direct image f(A)and inverse image f1(B)for a

functionf;Injective, surjective, bijective functions; Composite of injective, sur-

jective, bijective functions when dened; invertible functions, bijective functions

are invertible and conversely; examples of functions including constant, identity,

projection, inclusion; Binary operation as a function, properties, examples.

Equivalence relation, Equivalence classes, properties such as two equivalences

classes are either identical or disjoint, Denition of partition, every partition

gives an equivalence relation and vice versa.

Congruence is an equivalence relation on Z;Residue classes and partition of Z;

Addition modulo n;Multiplication modulo n;examples.

Unit III: Polynomials (15 Lectures)

Denition of a polynomial, polynomials over the eld FwhereF=Q;RorC;

Algebra of polynomials, degree of polynomial, basic properties,

Division algorithm in F[X](without proof), and g.c.d. of two polynomials

and its basic properties (without proof), Euclidean algorithm (without proof),

applications, Roots of a polynomial, relation between roots and coecients,

multiplicity of a root, Remainder theorem, Factor theorem,

A polynomial of degree over nhas at most nroots, Complex roots of a polyno-

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mial in R[X]occur in conjugate pairs, Statement of Fundamental Theorem of

Algebra, A polynomial of degree in C[X]has exactly ncomplex roots counted

with multiplicity, A non constant polynomial in R[X]can be expressed as a

product of linear and quadratic factors in R[X];necessary condition for a ra-

tional numberp

qto be a root of a polynomial with integer coecients, simple

consequences such asppis a irrational number where pis a prime number,

roots of unity, sum of all the roots of unity.

Reference Books

1. David M. Burton, Elementary Number Theory, Seventh Edition, McGraw

Hill Education (India) Private Ltd.

2. Norman L. Biggs, Discrete Mathematics, Revised Edition, Clarendon Press,

Oxford 1989.

Additional Reference Books

1. I. Niven and S. Zuckerman, Introduction to the theory of numbers, Third

Edition, Wiley Eastern, New Delhi, 1972.

2. G. Birko and S. Maclane, A Survey of Modern Algebra, Third Edition,

MacMillan, New York, 1965.

3. N. S. Gopalkrishnan, University Algebra, Ne Age International Ltd, Reprint

2013.

4. I .N. Herstein, Topics in Algebra, John Wiley, 2006.

5. P. B. Bhattacharya S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra,

New Age International, 1994.

6. Kenneth Rosen, Discrete Mathematics and its applications, Mc-Graw Hill

International Edition, Mathematics Series.

Tutorials:

1. Mathematical induction (The problems done in F.Y.J.C. may be avoided).

2. Division Algorithm and Euclidean algorithm in Z, primes and the Funda-

mental Theorem of Arithmetic. 3. Functions (direct image and inverse image),

Injective, surjective, bijective functions, nding inverses of bijective functions.

4. Congruences and Eulers-function, Fermat's little theorem, Euler's theorem

and Wilson's theorem. 5. Equivalence relation. 6. Factor Theorem, rela-

tion between roots and coecients of polynomials, factorization and reciprocal

polynomials.

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SEMESTER II

USMT 201 CALCULUS II

Unit I: Series (15 Lectures)

Series1X

n=1anof real numbers, simple examples of series, Sequence of partial

sums of a series, convergence of a series, convergent series, divergent series,

Necessary condition:1X

n=1anconverges)an!0;but converse is not true, al-

gebra of convergent series, Cauchy criterion, divergence of harmonic series,

convergence of1X

n=11

np(p>1);Comparison test, limit comparison test, alter-

nating series, Leibnitz's theorem (alternating series test) and convergence of

1X

n=1(1)n

n, absolute convergence, conditional convergence, absolute conver-

gence implies convergence but not conversely, Ratio test (without proof), root

test (without proof), and examples.

Unit II: Limits & Continuity of functions (15 Lectures)

Denition of Limit lim

x!af(x)of a function f(x);evaluation of limit of simple

functions using the denition, uniqueness of limit if it exists, algebra of lim-

its , limit of composite function, sandwich theorem, left-hand-limit lim

x!af(x),

right-hand-limit lim

x!a+f(x), non-existence of limits, lim

x!1f(x);lim

x!1f(x)

and lim

x!af(x) =1:

Continuous functions: Continuity of a real valued function on a set in terms of

limits, examples, Continuity of a real valued function at end points of domain,

Sequential continuity, Algebra of continuous functions, Discontinuous functions,

examples of removable and essential discontinuity. Intermediate value theorem

and its applications, Bolzano- Weierstrass theorem (statement only): A continu-

ous function on a closed and bounded interval is bounded and attains its bounds.

Dierentiation of real valued function of one variable: Denition of dieren-

tiation at a point of an open interval, examples of dierentiable and non dif-

ferentiable functions, dierentiable functions are continuous but not conversely,

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algebra of dierentiable functions.

Chain rule, Higher order derivatives, Leibnitz rule, Derivative of inverse func-

tions, Implicit dierentiation (only examples).

Unit III: Applications of dierentiation (15 Lectures)

Denition of local maximum and local minimum, necessary condition, stationary

points, second derivative test, examples, Graphing of functions using rst and

second derivatives, concave, convex functions, points of in
ection.

Rolle's theorem, Lagrange's and Cauchy's mean value theorems, applications

and examples, Monotone increasing and decreasing function, examples,

L-hospital rule without proof, examples of indeterminate forms, Taylor's the-

orem with Lagrange's form of remainder with proof, Taylor polynomial and

applications.

Reference Books :

1. R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

2. James Stewart, Calculus, Third Edition, Brooks/cole Publishing Company,1994.

3. T. M. Apostol, Calculus Vol I, Wiley & Sons (Asia) Pte. Ltd.

Additional Reference Books:

1. Richard Courant-Fritz John, A Introduction to Calculus and Analysis, Vol-

ume I, Springer.

2. Ajit Kumar- S. Kumaresan, A Basic Course in Real Analysis, CRC Press,

2014.

4. Ghorpade, Sudhir R.- Limaye, Balmohan V., A Course in Calculus and Real

Analysis, Springer International Ltd, 2000.

5. K.G. Binmore, Mathematical Analysis, Cambridge University Press, 1982.

6. G. B. Thomas, Calculus, 12th Edition, 2009.

Tutorials: 1. Calculating limit of series, Convergence tests. 2. Properties of

continuous functions. 3. Dierentiability, Higher order derivatives, Leibnitz

theorem. 4. Mean value theorems and its applications. 5. Extreme values,

increasing and decreasing functions. 6. Applications of Taylors theorem and

Taylors polynomials.

USMT 202/ UAMT 201 LINEAR ALGEBRA

Prerequisites : Review of vectors in R2;R3and as points, Addition and scalar

multiplication of vectors in terms of co-ordinates, dot-product structure, Scalar

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triple product, Length (norm) of a vector.

Unit I: System of Linear equations and Matrices (15 Lectures)

Parametric equation of lines and planes, system of homogeneous and non-

homogeneous linear equations, the solution of system of mhomogeneous linear

equations in nunknowns by elimination and their geometrical interpretation for

(n;m) = (1;2);(1;3);(2;2);(2;3);(3;3);denition of ntuples of real num-

bers, sum of two ntuples and scalar multiple of an ntuple.

Matrices with real entries; addition, scalar multiplication and multiplication of

matrices; transpose of a matrix, types of matrices: zero matrix, identity matrix,

scalar matrices, diagonal matrices, upper triangular matrices, lower triangu-

lar matrices, symmetric matrices, skew-symmetric matrices, Invertible matrices;

identities such as (AB)t=BtAt;(AB)1=B1A1:

System of linear equations in matrix form, elementary row operations, row

echelon matrix, Gaussian elimination method, to deduce that the system of m

homogeneous linear equations in nunknowns has a non-trivial solution if m

Denition of a real vector space, examples such as Rn;R[X]; Mmn(R);space

of all real valued functions on a non empty set.

Subspace: denition, examples: lines, planes passing through origin as sub-

spaces of R2;R3respectively; upper triangular matrices, diagonal matrices, sym-

metric matrices, skew-symmetric matrices as subspaces of Mn(R) (n= 2;3);

Pn(X) =fa0+a1X++anXnjai2R80ingas a subspace R[X];

the space of all solutions of the system of mhomogeneous linear equations in

nunknowns as a subspace of Rn:

Properties of a subspace such as necessary and sucient condition for a non

empty subset to be a subspace of a vector space, arbitrary intersection of sub-

spaces of a vector space is a subspace, union of two subspaces is a subspace if

and only if one is a subset of the other.

Finite linear combinations of vectors in a vector space; the linear span L(S)

of a non-empty subset Sof a vector space, Sis a generating set for L(S);L(S)

is a vector subspace of V;linearly independent/linearly dependent subsets of a

vector space, a subset fv1;v2;;vkgof a vector space is linearly dependent if

and only if9i2f1;2;kgsuch thatviis a linear combination of the other

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vectorsv0

js:

Unit III: Basis and Linear Transformations (15 Lectures)

Basis of a vector space, dimension of a vector space, maximal linearly indepen-

dent subset of a vector space is a basis of a vector space, minimal generating

set of a vector space is a basis of a vector space, any two basis of a vector space

have the same number of elements, any set of nlinearly independent vectors

in anndimensional vector space is a basis, any collection of n+ 1 linearly

independent vectors in an ndimensional vector space is linearly dependent; if

W1; W 2are two subspaces of a vector space VthenW1+W2is a subspace

of the vector space Vof dimension dim (W1) +dim(W1)dim(W1\W2));ex-

tending any basis of a subspace Wof a vector space Vto a basis of the vector

spaceV:

Linear transformations; kernel kernel (T)of a linear transformation, matrix as-

sociated with a linear transformation, properties such as: for a linear transfor-

mationTkernel (T)is a subspace of the domain space of Tand the image

image (T)is a subspace of the co-domain space of T:IfV;W are real vector

spaces withfv1;v2;;vnga basis ofVandfw1;w2;;wngany vectors

inWthen there exists a unique linear transformation T:V!Wsuch that

T(vj) =wj81jn;Rank nullity theorem ( statement only) and examples.

Reference Books :

1. Serge Lang, Introduction to Linear Algebra, Second Edition, Springer.

2. S. Kumaresan, Linear Algebra, A Geometric Approach, Prentice Hall of India

Pvt. Ltd, 2000.

Additional Reference Books:

1. M. Artin: Algebra, Prentice Hall of India Private Limited, 1991.

2. K. Homan and R. Kunze: Linear Algebra, Tata McGraw-Hill, New Delhi,

1971.

3. Gilbert Strang: Linear Algebra and its applications, International Student

Edition.

3. L. Smith: Linear Algebra, Springer Verlag.

4. A. Ramachandra Rao and P. Bhima Sankaran: Linear Algebra, Tata McGraw-

Hill, New Delhi.

5. T. Bancho and J. Warmers: Linear Algebra through Geometry, Springer

Verlag, New York, 1984.

6. Sheldon Axler: Linear Algebra done right, Springer Verlag, New York.

7. Klaus Janich: Linear Algebra.

8. Otto Bretcher: Linear Algebra with Applications, Pearson Education.

9. Gareth Williams: Linear Algebra with Applications.

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Tutorials:

1) Solving homogeneous system of m equations in n unknowns by elimination

for(m;n) = (1;2);(1;3);(2;2);2;3);(3;3);row echelon form.

2) Solving system Ax=bby Gauss elimination, Solutions of system of linear

Equations.

3) Verifying whether given (V;+;)is a vector space with respect to addition

+and scalar multiplication

4) Linear span of a non-empty subset of a vector space, determining whether

a given subset of a vector space is a subspace. Showing the set of convergent

real sequences is a subspace of the space of real sequences etc.

5. Finding basis of a vector space such as P3(X); M 3(R)etc. Verifying whether

a set is a basis of a vector space. Extending basis of a subspace to a basis of a

nite dimensional vector space.

6. Verifying whether a map T:X!Yis a linear transformation, nding kernel

of a linear transformation and matrix associated with a linear transformation,

verifying the Rank Nullity theorem.

Scheme of Examination

There will be a Semester end external Theory examination of 100 marks for

all the courses of Semester I & II.

1. Duration: The examinations shall be of 3 Hours duration.

2. Question Paper Pattern: There shall be FOUR questions. The rst three

questions shall be of 25 marks on each unit, and the fourth question shall be of

25 marks based on Unit I, II, & III .

3. All the questions shall be compulsory with internal choices within the ques-

tions. Including the choices, the marks for each question shall be 38-40.

4. Questions may be subdivided into sub questions as a, b, c, d & e, etc & the

allocation of marks depends on the weightage of the topic.

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