S Y BSc Mathematics syllabus2018 31 Syllabus Mumbai University

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(UNIVERSITY OF MUMBAI)
Syllabus for: S.Y.B.Sc./S.Y.B.A.
Program: B.Sc./B/A.
Course: Mathematics
Choice based Credit System (CBCS)
with eect from the
academic year 2018-19

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SEMESTER III
CALCULUS III
Course Code UNIT TOPICS Credits L/Week
USMT 301, UAMT 301I Functions of several variables
2 3 II Dierentiation
III Applications
ALGEBRA III
USMT 302 ,UAMT 302I Linear Transformations and Matrices
2 3 II Determinants
III Inner Product Spaces
DISCRETE MATHEMATICS
USMT 303I Permutations and Recurrence Relation
2 3 II Preliminary Counting
III Advanced Counting
PRACTICALS
USMTP03Practicals based on3 5USMT301, USMT 302 and USMT 303
UAMTP03Practicals based on2 4UAMT301, UAMT 302
SEMESTER IV
CALCULUS IV
Course Code UNIT TOPICS Credits L/Week
USMT 401, UAMT 401I Riemann Integration
2 3II Indenite Integrals and Improper Integrals
III Beta and Gamma Functions
And Applications
ALGEBRA IV
USMT 402 ,UAMT 402I Groups and Subgroups
2 3 II Cyclic Groups and Cyclic subgroups
III Lagrange's Theorem and Group
Homomorphism
ORDINARY DIFFERENTIAL EQUATIONS
USMT 403I First order First degree
2 3 Dierential equations
II Second order Linear
Dierential equations
III Linear System of Ordinary
Dierential Equations
PRACTICALS
USMTP04Practicals based on3 5USMT401, USMT 402 and USMT 403
UAMTP04Practicals based on2 4UAMT401, UAMT 402

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Teaching Pattern for Semester III
1. Three lectures per week per course. Each lecture is of 48 minutes duration.
2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and
one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed
by the University. Each practical session is of 48 minutes duration.)
Teaching Pattern for Semester IV
1. Three lectures per week per course. Each lecture is of 48 minutes duration.
2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and
one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed
by the University. Each practical session is of 48 minutes duration.)
S.Y.B.Sc. / S.Y.B.A. Mathematics
SEMESTER III
USMT 301, UAMT 301: CALCULUS III
Note: All topics have to be covered with proof in details (unless mentioned otherwise) and
examples.
Unit I: Functions of several variables (15 Lectures)
1. The Euclidean inner product on Rnand Euclidean norm function on Rn, distance between
two points, open ball in Rn;denition of an open subset of Rn;neighbourhood of a point
inRn;sequences in Rn, convergence of sequences- these concepts should be specically
discussed for n= 3 andn= 3:
2. Functions from Rn!R(scalar elds) and from Rn!Rm(vector elds), limits,
continuity of functions, basic results on limits and continuity of sum, dierence, scalar
multiples of vector elds, continuity and components of a vector elds.
3. Directional derivatives and partial derivatives of scalar elds.
4. Mean value theorem for derivatives of scalar elds.
Reference for Unit I:
Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10 of Calculus, Vol. 2 (Second Edition) by
Apostol.
Unit II: Dierentiation (15 Lectures)
1. Dierentiability of a scalar eld at a point of Rn(in terms of linear transformation) and
on an open subset of Rn;the total derivative, uniqueness of total derivative of a dier-
entiable function at a point, simple examples of nding total derivative of functions such
asf(x;y) =x2+y2;f(x;y;z ) =x+y+z;, dierentiability at a point of a function f
implies continuity and existence of direction derivatives of fat the point, the existence of
continuous partial derivatives in a neighbourhood of a point implies dierentiability at the
point.

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2. Gradient of a scalar eld, geometric properties of gradient, level sets and tangent planes.
3. Chain rule for scalar elds.
4. Higher order partial derivatives, mixed partial derivatives, sucient condition for equality
of mixed partial derivative.
Reference for Unit II:
Sections 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, 8.23 of Calculus, Vol.2 (Second Edition) by T.
Apostol, John Wiley.
Unit III: Applications (15 lectures)
1. Second order Taylor's formula for scalar elds.
2. Dierentiability of vector elds, denition of dierentiability of a vector eld at a point,
Jacobian matrix, dierentiability of a vector eld at a point implies continuity. The chain
rule for derivative of vector elds (statements only)
3. Mean value inequality.
4. Hessian matrix, Maxima, minima and saddle points.
5. Second derivative test for extrema of functions of two variables.
6. Method of Lagrange Multipliers.
Reference for Unit III:
Sections 8.18, 8.19, 8.20, 8.21, 8.22, 9.9, 9.10, 9.11, 9.12, 9.13, 9.14 9.13, 9.14 from Apostol,
Calculus Vol. 2, (Second Edition) by T. Apostol.
Recommended Text Books:
1. T. Apostol: Calculus, Vol. 2, John Wiley.
2. J. Stewart, Calculus, Brooke/ Cole Publishing Co.
Additional Reference Books
(1) G.B. Thoman and R. L. Finney, Calculus and Analytic Geometry, Ninth Edition, Addison-
Wesley, 1998.
(2) Sudhir R. Ghorpade and Balmohan V. Limaye, A Course in Multivariable Calculus and
Analysis, Springer International Edition.
(3) Howard Anton, Calculus- A new Horizon, Sixth Edition, John Wiley and Sons Inc, 1999.
USMT 302/UAMT 302: ALGEBRA III
Note: Revision of relevant concepts is necessary.
Unit 1: Linear Transformations and Matrices (15 lectures)

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1. Review of linear transformations: Kernel and image of a linear transformation, Rank-
Nullity theorem (with proof), Linear isomorphisms, inverse of a linear isomorphism, Any
ndimensional real vector space is isomorphic to Rn:
2. The matrix units, row operations, elementary matrices, elementary matrices are invertible
and an invertible matrix is a product of elementary matrices.
3. Row space, column space of an mnmatrix, row rank and column rank of a matrix,
Equivalence of the row and the column rank, Invariance of rank upon elementary row or
column operations.
4. Equivalence of rank of an mnmatrixAand rank of the linear transformation LA:
Rn!Rm(LA(X) =AX). The dimension of solution space of the system of linear
equationsAX= 0 equals nrank(A).
5. The solutions of non-homogeneous systems of linear equations represented by AX=B;
Existence of a solution when rank( A)= rank(A;B), The general solution of the system is
the sum of a particular solution of the system and the solution of the associated homoge-
neous system.
Reference for Unit 1: Chapter VIII, Sections 1, 2 of Introduction to Linear Algebra,
Serge Lang, Springer Verlag and Chapter 4, of Linear Algebra A Geometric Approach, S.
Kumaresan, Prentice-Hall of India Private Limited, New Delhi.
Unit II: Determinants (15 Lectures)
1. Denition of determinant as an nlinear skew-symmetric function from RnRn:::
Rn!Rsuch that determinant of ( E1;E2;:::;En) is 1, where Ejdenotes the jthcolumn
of thennidentity matrix In:Determinant of a matrix as determinant of its column
vectors (or row vectors). Determinant as area and volume.
2. Existence and uniqueness of determinant function via permutations, Computation of de-
terminant of 22;33 matrices, diagonal matrices, Basic results on determinants such
as det(At) = det(A);det(AB) = det(A) det(B);Laplace expansion of a determinant, Van-
dermonde determinant, determinant of upper triangular and lower triangular matrices.
3. Linear dependence and independence of vectors in Rnusing determinants, The existence
and uniqueness of the system AX=B, whereAis annnmatrix wither det( A)6= 0, Co-
factors and minors, Adjoint of an nnmatrixA, Basic results such as Aadj(A) = det(A)In:
Annnreal matrix Ais invertible if and only if det( A)6= 0;A1=1
det(A)adj(A) for an
invertible matrix A, Cramer's rule.
4. Determinant as area and volume.
References for Unit 2: Chapter VI of Linear Algebra A geometric approach, S. Kumaresan,
Prentice Hall of India Private Limited, 2001 and Chapter VII Introduction to Linear Algebra,
Serge Lang, Springer Verlag.
Unit III: Inner Product Spaces (15 Lectures)
1. Dot product in Rn;Denition of general inner product on a vector space over R:Examples
of inner product including the inner product < f;g > =Z
f(t)g(t)dtonC[;];the
space of continuous real valued functions on [ ;].

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2. Norm of a vector in an inner product space. Cauchy-Schwartz inequality, Triangle in-
equality, Orthogonality of vectors, Pythagoras theorem and geometric applications in R2;
Projections on a line, The projection being the closest approximation, Orthogonal com-
plements of a subspace, Orthogonal complements in R2andR3. Orthogonal sets and
orthonormal sets in an inner product space, Orthogonal and orthonormal bases. Gram-
Schmidt orthogonalization process, Simple examples in R3;R4.
Reference of Unit 3: Chapter VI, Sections 1,2 of Introduction to Linear Algebra, Serge
Lang, Springer Verlag and Chapter 5, of Linear Algebra A Geometric Approach, S. Kumaresan,
Prentice-Hall of India Private Limited, New Delhi.
Recommended Books:
1. Serge Lang: Introduction to Linear Algebra, Springer Verlag.
2. S. Kumaresan: Linear Algebra A geometric approach, Prentice Hall of India Private Lim-
ited.
Additional Reference Books:
1. M. Artin: Algebra, Prentice Hall of India Private Limited.
2. K. Homan and R. Kunze: Linear Algebra, Tata McGraw-Hill, New Delhi.
3. Gilbert Strang: Linear Algebra and its applications, International Student Edition.
4. L. Smith: Linear Algebra, Springer Verlag.
5. A. Ramachandra Rao and P. Bhima Sankaran: Linear Algebra, Tata McGraw-Hill, New
Delhi.
6. T. Bancho and J. Wermer: Linear Algebra through Geometry, Springer Verlag Newyork,
1984.
7. Sheldon Axler: Linear Algebra done right, Springer Verlag, Newyork.
8. Klaus Janich: Linear Algebra.
9. Otto Bretcher: Linear Algebra with Applications, Pearson Education.
10. Gareth Williams: Linear Algebra with Applications, Narosa Publication.
USMT 303: Discrete Mathematics
Unit I: Permutations and Recurrence relation (15 lectures)
1. Permutation of objects, Sn, composition of permutations, results such as every permutation
is a product of disjoint cycles, every cycle is a product of transpositions, even and odd
permutation, rank and signature of a permutation, cardinality of Sn;An
2. Recurrence Relations, denition of non-homogeneous, non-homogeneous, linear , non-
linear recurrence relation, obtaining recurrence relation in counting problems, solving
homogeneous as well as non homogeneous recurrence relations by using iterative meth-
ods, solving a homogeneous recurrence relation of second degree using algebraic method
proving the necessary result.

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Recommended Books:
1. Norman Biggs: Discrete Mathematics, Oxford University Press.
2. Richard Brualdi: Introductory Combinatorics, John Wiley and sons.
3. V. Krishnamurthy: Combinatorics-Theory and Applications, Aliated East West Press.
4. Discrete Mathematics and its Applications, Tata McGraw Hills.
5. Schaum's outline series: Discrete mathematics,
6. Applied Combinatorics: Allen Tucker, John Wiley and Sons.
Unit II: Preliminary Counting (15 Lectures)
1. Finite and innite sets, countable and uncountable sets examples such as N;Z;NN;Q;(0;1);R
2. Addition and multiplication Principle, counting sets of pairs, two ways counting.
3. Stirling numbers of second kind. Simple recursion formulae satised by S(n;k) fork=
1;2;


;n1;n
4. Pigeonhole principle and its strong form, its applications to geometry, monotonic sequences
etc.
Unit III: Advanced Counting (15 Lectures)
1. Binomial and Multinomial Theorem, Pascal identity, examples of standard identities such
as the following with emphasis on combinatorial proofs.
ˆrX
k=0m
kn
rk
=m+n
r
ˆnX
i=ri
r
=n+ 1
r+ 1
ˆkX
i=0k
i2
=2k
k
ˆnX
i=0n
i
= 2n
2. Permutation and combination of sets and multi-sets, circular permutations, emphasis on
solving problems.
3. Non-negative and positive solutions of equation x1+x2+


+xk=n
4. Principal of inclusion and exclusion, its applications, derangements, explicit formula for
dn, deriving formula for Euler's function (n).

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USMT P03/UAMTP03 Practicals
Suggested Practicals for USMT 301/UAMT303
1. Sequences in R2andR3;limits and continuity of scalar elds and vector elds, using
\denition and otherwise" , iterated limits.
2. Computing directional derivatives, partial derivatives and mean value theorem of scalar
elds.
3. Total derivative, gradient, level sets and tangent planes.
4. Chain rule, higher order derivatives and mixed partial derivatives of scalar elds.
5. Taylor's formula, dierentiation of a vector eld at a point, nding Hessian/Jacobean
matrix, Mean Value Inequality.
6. Finding maxima, minima and saddle points, second derivative test for extrema of functions
of two variables and method of Lagrange multipliers.
7. Miscellaneous Theoretical Questions based on full paper
Suggested Practicals for USMT302/UAMT302:
1. Rank-Nullity Theorem.
2. System of linear equations.
3. Determinants , calculating determinants of 2 2 matrices, nndiagonal, upper triangular
matrices using denition and Laplace expansion.
4. Finding inverses of nnmatrices using adjoint.
5. Inner product spaces, examples. Orthogonal complements in R2andR3.
6. Gram-Schmidt method.
7. Miscellaneous Theoretical Questions based on full paper
Suggested Practicals for USMT 303:
1. Derangement and rank signature of permutation.
2. Recurrence relation.
3. Problems based on counting principles, Two way counting.
4. Stirling numbers of second kind, Pigeon hole principle.
5. Multinomial theorem, identities, permutation and combination of multi-set.
6. Inclusion-Exclusion principle. Euler phi function.
7. Miscellaneous theory quesitons from all units.

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SEMESTER IV
USMT 401/UAMT 401: CALCULUS IV
Note: All topics have to be covered with proof in details (unless mentioned otherwise) and
examples.
Unit I: Riemann Integration (15 Lectures)
Approximation of area, Upper/Lower Riemann sums and properties, Upper/Lower integrals,
Denition of Riemann integral on a closed and bounded interval, Criterion of Riemann in-
tegrability, if a < c < b thenf2R[a;b];if and only if f2R[a;c] andf2R[c;b] andZb
af=Zc
af+Zb
cf.
Properties:
(i)f;g2R[a;b] =)f+g;f2R[a;b].
(ii)Zb
a(f+g) =Zb
af+Zb
ag:
(iii)Zb
af=Zb
af:
(iv)f2R[a;b] =)jfj2R[a;b] andjZb
afjZb
ajfj,
(v)f0;f2C[a;b] =)f2R[a;b].
(vi) Iffis bounded with nite number of discontinuities then f2R[a;b], generalize this if f
is monotone then f2R[a;b]:
Unit II: Indenite and improper integrals (15 lectures)
Continuity of F(x) =Zx
af(t)dtwheref2R[a;b], Fundamental theorem of calculus, Mean
value theorem, Integration by parts, Leibnitz rule, Improper integrals-type 1 and type 2, Abso-
lute convergence of improper integrals, Comparison tests, Abel's and Dirichlet's tests.
Unit III: Applications (15 lectures)
(1)and functions and their properties, relationship between and functions (without
pro).
(2) Applications of denite Integras: Area between curves, nding volumes by sicing, volumes
of solids of revolution-Disks and Washers, Cylindrical Shells, Lengths of plane curves, Ar-
eas of surfaces of revolution.
References:
(1) Calculus Thomas Finney, ninth edition section 5.1, 5.2, 5.3, 5.4, 5.5, 5.6.
(2) R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

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(3) Ajit Kumar, S.Kumaresan, A Basic Course in Real Analysis, CRC Press, 2014.
(4) T. Apostol, Calculus Vol.2, John Wiley.
(5) K. Stewart, Calculus, Booke/Cole Publishing Co, 1994.
(6) J. E. Marsden, A.J. Tromba and A. Weinstein, Basic multivariable calculus.
(7) Bartle and Sherbet, Real analysis.
USMT 402/ UAMT 402: ALGEBRA IV
Unit I: Groups and Subgroups (15 Lectures)
(a) Denition of a group, abelian group, order of a group, nite and innite groups.
Examples of groups including:
i)Z;Q;R;Cunder addition.
ii)Q(=Qnf0g);R(=Rnf0g);C(=Cnf0g):Q+(= positive rational numbers)
under multiplication.
iii)Zn;the set of residue classes modulo nunder addition.
iv)U(n);the group of prime residue classes modulo nunder multiplication.
v) The symmetric group Sn:
vi) The group of symmetries of a plane gure. The Dihedral group Dnas the group
of symmetries of a regular polygon of nsides (forn= 3;4).
vii) Klein 4-group.
viii) Matrix groups Mnn(R) under addition of matrices, GLn(R);the set of invertible
real matrices, under multiplication of matrices.
ix) Examples such as S1as subgroup of C;nthe subgroup of nth roots of unity.
(b) Properties such as
1) In a group ( G;:) the following indices rules are true for all integers n;m:
i)anam=an+mfor allainG:
ii) (an)m=anmfor allainG:
iii) (ab)n=anbnfor allabinGwheneverab=ba:
2) In a group ( G;:) the following are true:
i) The identity element eofGis unique.
ii) The inverse of every element in Gis unique.
iii) (a1)1=afor allainG:
iv) (a:b)1=b1a1for alla;binG:
v) Ifa2=efor everyainGthen (G;:) is an abelian group.
vi) (aba1)n=abna1for everya;binGand for every integer n:
vii) If (a:b)2=a2:b2for everya;binGthen (G;:) is an abelian group.
viii) (Z
n;:) is a group if and only if nis a prime.
3) Properties of order of an element such as: ( nandmare integers.)
i) Ifo(a) =nthenam=eif and only if n=m.
ii) Ifo(a) =nmtheno(an) =m.
iii) Ifo(a) =ntheno(am) =n
(n;m);. where (n;m) is the GCD of nandm.

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iv)o(aba1) =o(b) ando(ab) =o(ba).
v) Ifo(a) =mando(b) =m;ab =ba;(n;m) = 1 theno(ab) =nm.
(c) Subgroups
i) Denition, necessary and sucient condition for a non-empty set to be a Sub-
group.
ii) The center Z(G) of a group is a subgroup.
iii) Intersection of two (or a family of ) subgroups is a subgroup.
iv) Union of two subgroups is not a subgroup in general. Union of two subgroups is
a subgroup if and only if one is contained in the other.
v) IfHandKare subgroups of a group GthenHK is a subgroup of Gif and only
ifHK =KH.
Reference for Unit I:
(1) I.N. Herstein, Topics in Algebra.
(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.
Unit II: Cyclic groups and cyclic subgroups (15 Lectures)
(a) Cyclic subgroup of a group, cyclic groups, (examples including Z;Znandn).
(b) Properties such as:
(i) Every cyclic group is abelian.
(ii) Finite cyclic groups, innite cyclic groups and their generators.
(iii) A nite cyclic group has a unique subgroup for each divisor of the order of the group.
(iv) Subgroup of a cyclic group is cyclic.
(v) In a nite group G;G = if and only if o(G) =o(a).
(vi) IfG= ando(a) =nthenG=if and only if ( n;m) = 1:
(vii) IfGis a cyclic group of order pnandH orKH:
References for Unit II:
(1) I.N. Herstein, Topics in Algebra.
(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.
Unit III: Lagrange's Theorem and Group homomorphism (15 Lectures)
(a) Denition of Coset and properties such as :
1) IFHis a subgroup of a group Gandx2Gthen
(i)xH=Hif and only if x2H.
(ii)Hx=Hif and only if x2H:
2) IfHis a subgroup of a group Gandx;y2Gthen
(i)xH=yHif and only if x1y2H:
(ii)Hx=Hyif and only if xy12H: