AAMS UG 97 Mathematics CBCS 1 Syllabus Mumbai University by munotes
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AC - ___________
Item No. ____________
UNIVERSITY OF MUMBAI
Syllabus
for the
Program : S.Y.B.Sc. / S.Y.B.A Sem. III
& IV (CBCS)
Course : Mathematics
(Choice Based and Credit System w ith effect from
the academic year 202 1-22)
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AC _________
Item No.
UNIVERSITY OF MUMBAI
Syllabus for Approval
Sr.
No.
Heading
Particulars
1 Title of the
Course S. Y. B. Sc. /B. A. Mathematics , Sem III & IV
2 Eligibility for
Admission As per university regulations
3 Passing
Marks 40%
( Internal 10/25 Marks and External 30/75)
4 Ordinances /
Regulations ( if any) -
5 No. of Years /
Semesters Three Years / Six Semesters Programme
( Syllabus for sem III & IV)
6 Level UG
7 Pattern Semester
8 Status Revised
9 To be implemented
from Academic Year From Academic Year : 2021-2022
Date: Signature:
Name : Prof. R. P. Deore Chairman of BoS of Mathematics
19.05.2021
19.05.2021
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Dr. Anuradha Majumdar (Dean, Science and Technology)
Prof. Shivram Garje (Associate Dean, Science)
Prof. R. P. Deore , Chairman (BoS) Member(BoS)
Prof. P. Veeramani, Member
Prof. S. R. Ghorpade , Member
Prof. Ajit Diwan, Member
Dr. Sushil Kulkarni, Member
Dr. S. A. Shende, Member
Prof. V. S. Kulkarni
Dr. Sanjeevani Gharge, Member
Dr. Mittu Bhattacharya, Member
Dr. Abhaya Chitre, Member
Dr. S. Aggarwal, Member
Dr. Amul Desai, Member
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CONTENTS
1. Preamble
2. Programme Outcomes
3. Course Outcomes
4. Course structure with minimum credits and Lectures/ Week
5. Teaching Pattern for semester III & IV
6. Consolidated Syllabus for semester III& IV
7. Scheme of Evaluation
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1. Preamble
The University of Mumbai has brought into force the revised syllabi as per the Choice Based
Credit System (CBCS) for the Second year B. Sc / B. A. Programme in Mathematics from the
academic year 202 1-2022. Mathematics has been fundamental to the development of science
and technology. In recent decades, the extent of application of Mathematics to real world
problems has increased by leaps and bounds. Taking into consideration the rapid changes in
scien ce and technology and new approaches in di fferent areas of mathematics and related
subjects like Physics, Statistics and Computer Sciences, the board of studies in Mathematics
with concern of teachers of Mathematics from di fferent colleges a ffiliated to Un iversity of
Mumbai has prepared the syllabus of S.Y.B. Sc. / S. Y. B. A. Mathematics. The present syllabi
of S. Y. B. Sc. for Semester III and Semester I V has been designed as per U. G. C. Model
curriculum so that the students learn Mathematics needed for these branches, learn basic
concepts of Mathematics and are exposed to rigorous methods gently and slowly. The syllabi of
S. Y. B. Sc. / S. Y. B. A. would consist of two semesters and each semester would comprise of
three courses and one practical course for S. Y. B. Sc Mathematics and two course s and one
practical course for each semester for S. Y. B. A. Mathematics.
Aims and Objectives :
(1) Give the students a su fficient knowledge of fundamental principles, methods and a clear
perception of innumerous power of mathematical ideas and tools and know how to use them by
modeling, solving and interpreting.
(2) Re flecting the broad nature of the subject and developing mathematical tools for continuing
further study in various fields of science.
(3) Enhanci ng students' overall development and to equip them with mathematical modeling
abilities, problem solving skills, creative talent and power of communication necessary for
various kinds of employment.
(4) A student should get adequate exposure to global an d local concerns that explore them
many aspects of Mathematical Sciences
2. Programme Outcomes :
(1) Enabling students to develop positive attitude towards mathematics as an interesting
and valuable subject
(2) Enhancing students overall development and to equip them with mathematical
modeling , abilities, problem solving skills, creative talent and power of communication.
(3) Acquire good knowledge and understanding in advanced areas of mathematics and
statistics.
3. Course outcomes:
1. Calculus (Sem III) & Multivariable Calculus I(Sem I V): This course gives introduction to
basic concepts of Analysis with rigor and prepares students to study further courses in
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Analysis. Formal proofs are given lot of emphasis in this course which also enhances
understanding of the subject of Mathematics as a whole.
2. Linear Algebra I ( Sem III) & Linear Algebra II (Sem IV): This course gives expositions to
system of linear equations and matrices , Vector spaces, Basis and dimension, Linear
Transformation, Inner product space, Eigen values and eigenvectors.
3. Ordinary Differential Equations ( Sem III) prepares learner to get solutions of so many
kinds of problems in all subjects of Science and also prepares learner for further studies
of differential equations and related fields.
4. Numerical Methods and Statistical Methods: Lerner will learn different types of Numerical
methods and statistical methods to apply in different fields of Mathematics.
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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 e ect from the
academic year 2021-22
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2
4. Course structure with minimum Credits and Lectures/ Week
SEMESTER III
Calculus III
Course Code UNIT TOPICS Credits L/Week
USMT 301, UAMT 301I In nite Series
2 3 II Riemann Integration
III Applications of Integrations and
Improper Integrals
Linear Algebra I
USMT 302 ,UAMT 302I System of Equations and Matrices
2 3 II Vector Spaces over IR
III Determinants, Linear Equations (Revisited)
ORDINARY DIFFERENTIAL EQUATIONS
USMT 303I Higher Order linear Di erential Equations
2 3 II Systems of First Order
Linear di erential equations
III Numerical Solutions of Ordinary
Di erential Equations
PRACTICALS
USMTP03Practicals based on3 5USMT301, USMT 302 and USMT 303
UAMTP03Practicals based on2 4UAMT301, UAMT 302
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SEMESTER IV
Multivariable Calculus I
Course Code UNIT TOPICS Credits L/Week
USMT 401, UAMT 401I Functions of several variables
2 3II Di erentiation of Scalar Fields
III Applications of Di erentiation of
Scalar Fields and Di erentiation of
Vector Fields
Linear Algebra II
USMT 402 ,UAMT 402I Linear transformation, Isomorphism,
2 3 Matrix associated with L.T.
II Inner product spaces
III Eigen values, eigen vectors,
diagonalizable matrix
Numerical methods (Elective A)
USMT 403AI Solutions of algebraic and
2 3 transcendental equations
II Interpolation, Curve tting,
Numerical integration
III Solutions of linear system
of Equations and eigen value problems
Statistical methods an their applications(Elective B)
USMT 403BI Descriptive Statistics and
2 3 random variables
II Probability Distribution and
Correlation
III Inferential Statistics
PRACTICALS
USMTP04Practicals based on3 5USMT401, USMT 402 and USMT 403
UAMTP04Practicals based on2 4UAMT401, UAMT 402
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5. Teaching Pattern for Semester III & IV
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 to be formed as prescribed
by the University. Each practical session is of 48 minutes duration.)
6. Consolidated Syllabus for Semester III & IV
Semester-III
Note: Unless indicated otherwise, proofs of the results mentioned in the syllabus should be
covered.
USMT301/ UAMT301: Calculus III
Unit I. In nite Series (15 Lectures)
1. In nite series in R. De nition of convergence and divergence. Basic examples including
geometric series. Elementary results such as if1X
n=1anis convergent, then an !0 but
converse not true. Cauchy Criterion. Algebra of convergent series.
2. Tests for convergence: Comparison Test, Limit Comparison Test, Ratio Test (without
proof), Root Test (without proof), Abel Test (without proof) and Dirichlet Test (without
proof). Examples. The decimal expansion of real numbers. Convergence of1X
n=11
np(p>1):
Divergence of harmonic series1X
n=11
n.
3. Alternating series. Leibnitz's Test. Examples. Absolute convergence, absolute conver-
gence implies convergence but not conversely. Conditional Convergence.
Unit II. Riemann Integration (15 Lectures)
1. Idea of approximating the area under a curve by inscribed and circumscribed rectangles.
Partitions of an interval. Re nement of a partition. Upper and Lower sums for a bounded
real valued function on a closed and bounded interval. Riemann integrability and the
Riemann integral.
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2. Criterion for Riemann integrability. Characterization of the Riemann integral as the limit
of a sum. Examples.
3. Algebra of Riemann integrable functions. Also, basic results such as if f: [a;b] !R
is integrable, then (i)Zb
af(x)dx=Zc
af(x)dx+Zb
cf(x)dx. (ii)jfjis integrable and
Zb
af(x)dx Zb
ajfj(x)dx(iii) Iff(x)0 for allx2[a;b] thenZb
af(x)dx0:
4. Riemann integrability of a continuous function, and more generally of a bounded function
whose set of discontinuities has only nitely many points. Riemann integrability of mono-
tone functions.
Unit III. Applications of Integrations and Improper Integrals (15 lectures)
1. Area between the two curves. Lengths of plane curves. Surface area of surfaces of revolu-
tion.
2. Continuity of the function F(x) =Zx
af(t)dt;x2[a;b];whenf: [a;b] !Ris Riemann
integrable. First and Second Fundamental Theorems of Calculus.
3. Mean value theorem. Integration by parts formula. Leibnitz's Rule.
4. De nition of two types of improper integrals. Necessary and sucient conditions for
convergence.
5. Absolute convergence. Comparison and limit comparison tests for convergence.
6. Gamma and Beta functions and their properties. Relationship between them (without
proof).
Reference Books
1. Sudhir Ghorpade, Balmohan Limaye; A Course in Calculus and Real Analysis (second
edition); Springer.
2. R.R. Goldberg; Methods of Real Analysis; Oxford and IBH Pub. Co., New Delhi, 1970.
3. Calculus and Analytic Geometry (Ninth Edition); Thomas and Finney; Addison-Wesley,
Reading Mass., 1998.
4. T. Apostol; Calculus Vol. 2; John Wiley.
Additional Reference Books
1. Ajit Kumar, S.Kumaresan; A Basic Course in Real Analysis; CRC Press, 2014
2. D. Somasundaram and B.Choudhary; A First Course in Mathematical Analysis, Narosa,
New Delhi, 1996.
3. K. Stewart; Calculus, Booke/Cole Publishing Co, 1994.
4. J. E. Marsden, A.J. Tromba and A. Weinstein; Basic Multivariable Calculus; Springer.
Page 14
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5. R.G. Brtle and D. R. Sherbert; Introduction to Real Analysis Second Ed. ; John Wiley,
New Yorm, 1992.
6. M. H. Protter; Basic Elements of Real Analysis; Springer-Verlag, New York, 1998.
USMT/UAMT 302: Linear Algebra I
Unit I. System of Equations, Matrices (15 Lectures)
1. Systems of homogeneous and non-homogeneous linear equations, Simple examples of nd-
ing solutions of such systems. Geometric and algebraic understanding of the solutions.
Matrices (with real entries), Matrix representation of system of homogeneous and non-
homogeneous linear equations. Algebra of solutions of systems of homogeneous linear
equations. A system of homogeneous linear equations with number of unknowns more
than the number of equations has in nitely many solutions.
2. Elementary row and column operations. Row equivalent matrices. Row reduction (of a
matrix to its row echelon form). Gaussian elimination. Applications to solving systems of
linear equations. Examples.
3. Elementary matrices. Relation of elementary row operations with elementary matrices.
Invertibility of elementary matrices. Consequences such as (i) a square matrix is invertible
if and only if its row echelon form is invertible. (ii) invertible matrices are products of
elementary matrices. Examples of the computation of the inverse of a matrix using Gauss
elimination method.
Unit II. Vector space over R(15 Lectures)
1. De nition of a vector space over R. Subspaces; criterion for a nonempty subset to be
a subspace of a vector space. Examples of vector spaces, including the Euclidean space
Rn, lines, planes and hyperplanes in Rnpassing through the origin, space of systems of
homogeneous linear equations, space of polynomials, space of various types of matrices,
space of real valued functions on a set.
2. Intersections and sums of subspaces. Direct sums of vector spaces. Quotient space of a
vector space by its subspace.
3. Linear combination of vectors. Linear span of a subset of a vector space. De nition of a
nitely generated vector space. Linear dependence and independence of subsets of a vector
space.
4. Basis of a vector space. Basic results that any two bases of a nitely generated vector
space have the same number of elements. Dimension of a vector space. Examples. Bases
of a vector space as a maximal linearly independent sets and as minimal generating sets.
Unit III. Determinants, Linear Equations (Revisited) (15 Lectures)
1. Inductive de nition of the determinant of a nnmatrix ( e. g. in terms of expansion
along the rst row). Example of a lower triangular matrix. Laplace expansions along an
arbitrary row or column. Determinant expansions using permutations
det(A) =X
2Snsign()nY
i=1a(i);i
.
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2. Basic properties of determinants (Statements only); (i) det A= detAT. (ii) Multilinearity
and alternating property for columns and rows. (iii) A square matrix Ais invertible if
and only if det A6= 0. (iv) Minors and cofactors. Formula for A 1when detA6= 0. (v)
det(AB) = detAdetB.
3. Row space and the column space of a matrix as examples of vector space. Notion of row
rank and the column rank. Equivalence of the row rank and the column rank. Invariance
of rank upon elementary row or column operations. Examples of computing the rank using
row reduction.
4. Relation between the solutions of a system of non-homogeneous linear equations and the
associated system of homogeneous linear equations. Necessary and sucient condition
for a system of non-homogeneous linear equations to have a solution [viz., the rank of
the coecient matrix equals the rank of the augmented matrix [ AjB]]. Equivalence of
statements (in which Adenotes an nnmatrix) such as the following.
(i) The system Axxx=bbbof non-homogeneous linear equations has a unique solution.
(ii) The system Axxx= 000 of homogeneous linear equations has no nontrivial solution.
(iii)Ais invertible.
(iv) detA6= 0:
(v) rank(A) =n.
5. Cramers Rule. LUDecomposition. If a square matrix Ais a matrix that can be reduced
to row echelon form Uby Gauss elimination without row interchanges, then Acan be
factored as A=LUwhereLis a lower triangular matrix.
Reference books
1 Howard Anton, Chris Rorres, Elementary Linear Algebra, Wiley Student Edition).
2 Serge Lang, Introduction to Linear Algebra, Springer.
3 S Kumaresan, Linear Algebra - A Geometric Approach, PHI Learning.
4 Sheldon Axler, Linear Algebra done right, Springer.
5 Gareth Williams, Linear Algebra with Applications, Jones and Bartlett Publishers.
6 David W. Lewis, Matrix theory.
USMT303: Ordinary Di erential Equations
Unit I. Higher order Linear Di erential equations (15 Lectures)
1. The general n th order linear di erential equations, Linear independence, An existence
and uniqueness theorem, the Wronskian, Classi cation: homogeneous and non-homogeneous,
General solution of homogeneous and non-homogeneous LDE, The Di erential operator
and its properties.
2. Higher order homogeneous linear di erential equations with constant coecients, the aux-
iliary equations, Roots of the auxiliary equations: real and distinct, real and repeated,
complex and complex repeated.
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8
3. Higher order homogeneous linear di erential equations with constant coecients, the
method of undermined coecients, method of variation of parameters.
4. The inverse di erential operator and particular integral, Evaluation of1
f(D)for the func-
tions likeeax, sinax, cosax,xm,xmsinax,xmcosax,eaxVandxVwhereVis any function
ofx,
5. Higher order linear di erential equations with variable coecients:
The Cauchy's equation: x3d3y
dx3+x2d2y
dx2+xdy
dx+y=f(x) and
The Legendre's equation: ( ax+b)3d3y
dx3+ (ax+b)2d2y
dx2+ (ax+b)dy
dx+y=f(x).
Reference Books
1. Units 5, 6, 7 and 8 of E.D. Rainville and P.E. Bedient; Elementary Di erential Equations;
Macmillan.
2. Units 5, 6 and 7 of M.D. Raisinghania; Ordinary and Partial Di erential Equations; S.
Chand.
Unit II. Systems of First Order Linear Di erential Equations (15 Lectures)
(a) Existence and uniqueness theorem for the solutions of initial value problems for a system
of two rst order linear di erential equations in two unknown functions x;yof a single
independent variable t, of the form8
><
>:dx
dt=F(t;x;y )
dy
dt=G(t;x;y )(Statement only).
(b) Homogeneous linear system of two rst order di erential equations in two unknown func-
tions of a single independent variable t, of the form8
><
>:dx
dt=a1(t)x+b1(t)y;
dy
dt=a2(t)x+b2(t)y:.
(c) Wronskian for a homogeneous linear system of rst order linear di erential equations in
two functions x;yof a single independent variable t:Vanishing properties of the Wronskian.
Relation with linear independence of solutions.
(d) Homogeneous linear systems with constant coecients in two unknown functions x;yof
a single independent variable t. Auxiliary equation associated to a homogenous system
of equations with constant coecients. Description fo the general solution depending on
the roots and their multiplicities of the auxiliary equation, proof of independence of the
solutions. Real form of solutions in case the auxiliary equation has complex roots.
(e) Non-homogeneous linear system of linear system of two rst order di erential equations
in two unknown functions of a single independent variable t, of the form 8
><
>:dx
dt=a1(t)x+b1(t)y+f1(t);
dy
dt=a2(t)x+b2(t)y+f2(t):
General Solution of non-homogeneous system. Relation between the solutions of a system
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of non-homogeneous linear di erential equations and the associated system of homoge-
neous linear di erential equations.
Reference Books
1. G.F. Simmons; Di erential Equations with Applications and Historical Notes; Taylor's
and Francis.
Unit III. Numerical Solution of Ordinary Di erential Equations (15 lectures)
1. Numerical Solution of initial value problem of rst order ordinary di erential equation
using:
(i) Taylor's series method,
(ii) Picard's method for successive approximation and its convergence,
(iii) Euler's method and error estimates for Euler's method,
(iv) Modi ed Euler's Method,
(v) Runge-Kutta method of second order and its error estimates,
(vi) Runge-Kutta fourth order method.
2. Numerical solution of simultaneous and higher order ordinary di erential equation using:
(i) Runge-Kutta fourth order method for solving simultaneous ordinary di erential equa-
tion,
(ii) Finite di erence method for the solution of two point linear boundary value problem.
Reference Books
1. Units 8 of S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
Additional Reference Books
1. E.D. Rainville and P.E. Bedient, Elementary Di erential Equations, Macmillan.
2. M.D. Raisinghania, Ordinary and Partial Di erential Equations, S. Chand.
3. G.F. Simmons, Di erential Equations with Applications and Historical Notes, Taylor's
and Francis.
4. S. S. Sastry, Introductory Methods of Numerical Analysis, PHI.
5. K. Atkinson, W.Han and D Stewart, Numerical Solution of Ordinary Di erential Equa-
tions, Wiley.
xxxxxx
USMT P03 / UAMT P03: Practicals
Suggested Practicals for USMT 301/ UAMT 301
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1. Examples of convergent / divergent series and algebra of convergent series.
2. Tests for convergence of series.
3. Calculation of upper sum, lower sum and Riemann integral.
4. Problems on properties of Riemann integral.
5. Problems on fundamental theorem of calculus, mean value theorems, integration by parts,
Leibnitz rule.
6. Convergence of improper integrals, di erent tests for convergence. Beta Gamma Functions.
7. Miscellaneous Theoretical Questions based on full paper.
Suggested Practicals for USMT302 / UAMT 302
1. Systems of homogeneous and non-homogeneous linear equations.
2. Elementary row/column operations and Elementary matrices.
3. Vector spaces, Subspaces.
4. Linear Dependence/independence, Basis, Dimension.
5. Determinant and Rank of a matrix.
6. Solution to a system of linear equations, LU decomposition
7. Miscellaneous Theory Questions.
8. Miscellaneous theory questions from units I, II and III.
Suggested Practicals For USMT 303
1. Finding the general solution of homogeneous and non-homogeneous higher order linear
di erential equations.
2. Solving higher order linear di erential equations using method of undetermined coecients
and method of variation of parameters.
3. Solving a system of rst order linear ODES have auxiliary equations with real and complex
roots.
4. Finding the numerical solution of initial value problems using Taylor's series method,
Picard's method, modi ed Euler's method, Runge-Kutta method of fourth order and cal-
culating their accuracy.
5. Finding the numerical solution of simultaneous ordinary di erential equation using fourth
order Runge-Kutta method.
6. Finding the numerical solution of two point linear boundary value problem using Finite
di erence method.
xxxxxx
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Semester-IV
Note: Unless indicated otherwise, proofs of the results mentioned in the syllabus should be
covered.
USMT 401/ UAMT 401: Multivariable Calculus I
UNIT I. Functions of Several Variables (15 Lectures)
1. Review of vectors in Rn[with emphasis on R2andR3] and basic notions such as addition
and scalar multiplication, inner product, length (norm), and distance between two points.
2. Real-valued functions of several variables (Scalar elds). Graph of a function. Level sets
(level curves, level surfaces, etc). Examples. Vector valued functions of several variables
(Vector elds). Component functions. Examples.
3. Sequences, Limits and Continuity: Sequence in Rn[with emphasis on R2andR3] and
their limits. Neighbourhoods in Rn:Limits and continuity of scalar elds. Composition
of continuous functions. Sequential characterizations. Algebra of limits and continuity
(Results with proofs). Iterated limits.
Limits and continuity of vector elds. Algebra of limits and continuity vector elds.
(without proofs).
4. Partial and Directional Derivatives of scalar elds: De nitions of partial derivative and
directional derivative of scalar elds (with emphasis on R2andR3). Mean Value Theorem
of scalar elds.
UNIT II. Di erentiation of Scalar Fields (15 Lectures)
1. Di erentiability of scalar elds (in terms of linear transformation). The concept of (total)
derivative. Uniqueness of total derivative of a di erentiable function at a point. Examples
of functions of two or three variables. Increment Theorem. Basic properties including
(i) continuity at a point of di erentiability, (ii)existence of partial derivatives at a point
of di erentiability, and (iii) di erentiability when the partial derivatives exist and are
continuous.
2. Gradient. Relation between total derivative and gradient of a function. Chain rule.
Geometric properties of gradient. Tangent planes.
3. Euler's Theorem.
4. Higher order partial derivatives. Mixed Partial Theorem (n=2).
UNIT III. Applications of Di erentiation of Scalar Fields and Di erentiation of
Vector Fields (15 lectures)
1. Applications of Di erentiation of Scalar Fields: The maximum and minimum rate of
change of scalar elds. Taylor's Theorem for twice continuously di erentiable functions.
Notions of local maxima, local minima and saddle points. First Derivative Test. Examples.
Hessian matrix. Second Derivative Test for functions of two variables. Examples. Method
of Lagrange Multipliers.
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2. Di erentiation of Vector Fields: Di erentiability and the notion of (total) derivative. Dif-
ferentiability of a vector eld implies continuity, Jacobian matrix. Relationship between
total derivative and Jacobian matrix. The chain rule for derivative of vector elds (state-
ments only).
Reference books
1. T. Apostol; Calculus, Vol. 2 (Second Edition); John Wiley.
2. Sudhir Ghorpade, Balmohan Limaye; A Course in Multivariable Calculus and Analysis
(Second Edition); Springer.
3. Walter Rudin; Principles of Mathematical Analysis; McGraw-Hill, Inc.
4. J. E. Marsden, A.J. Tromba and A. Weinstein, Basic Multivariable Calculus; Springer.
5. D.Somasundaram and B.Choudhary; A First Course in Mathematical Analysis, Narosa,
New Delhi, 1996.
6. K. Stewart; Calculus; Booke/Cole Publishing Co, 1994.
Additional Reference Books
1. Calculus and Analytic Geometry, G.B. Thomas and R. L. Finney, (Ninth Edition); Addison-
Wesley, 1998.
2. Howard Anton; Calculus- A new Horizon,(Sixth Edition); John Wiley and Sons Inc, 1999.
3. S L Gupta and Nisha Rani; Principles of Real Analysis; Vikas Publishing house PVT LTD.
4. Shabanov, Sergei; Concepts in Calculus, III: Multivariable Calculus; University Press of
Florida, 2012.
5. S C Malik and Savita Arora; Mathematical Analysis; New Age International Publishers.
xxxxxx
USMT402/UAMT402: Linear Algebra II
UNIT I. Linear Transformations
1. De nition of a linear transformation of vector spaces; elementary properties. Examples.
Sums and scalar multiples of linear transformations. Composites of linear transformations.
A Linear transformation of V !W;whereV;W are vector spaces over RandVis a
nite-dimensional vector space is completely determined by its action on an ordered basis
ofV:
2. Null-space (kernel) and the image (range) of a linear transformation. Nullity and rank
of a linear transformation. Rank-Nullity Theorem (Fundamental Theorem of Homomor-
phisms).
3. Matrix associated with linear transformation of V !WwhereVandWare nite
dimensional vector spaces over R:. Matrix of the composite of two linear transformations.
Invertible linear transformations (isomorphisms), Linear operator, E ect of change of bases
on matrices of linear operator.
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4. Equivalence of the rank of a matrix and the rank of the associated linear transformation.
Similar matrices.
UNIT II. Inner Products and Orthogonality
1. Inner product spaces (over R). Examples, including the Euclidean space Rnand the space
of real valued continuous functions on a closed and bounded interval. Norm associated to
an inner product. Cauchy-Schwarz inequality. Triangle inequality.
2. Angle between two vectors. Orthogonality of vectors. Pythagoras theorem and some
geometric applications in R2. Orthogonal sets, Orthonormal sets. Gram-Schmidt orthog-
onalizaton process. Orthogonal basis and orthonormal basis for a nite-dimensional inner
product space.
3. Orthogonal complement of any set of vectors in an inner product space. Orthogonal com-
plement of a set is a vector subspace of the inner product space. Orthogonal decomposition
of an inner product space with respect to its subspace. Orthogonal projection of a vector
onto a line (one dimensional subspace). Orthogonal projection of an inner product space
onto its subspace.
UNIT III. Eigenvalues, Eigenvectors and Diagonalisation
1. Eigenvalues and eigenvectors of a linear transformation of a vector space into itself and of
square matrices. The eigenvectors corresponding to distinct eigenvalues of a linear trans-
formation are linearly independent. Eigen spaces. Algebraic and geometric multiplicity of
an eigenvalue.
2. Characteristic polynomial. Properties of characteristic polynomials (only statements).
Examples. Cayley-Hamilton Theorem. Applications.
3. Invariance of the characteristic polynomial and eigenvalues of similar matrices.
4. Diagonalisable matrix. A real square matrix Ais diagonalisable if and only if there is a
basis of Rnconsisting of eigenvectors of A. (Statement only - Annis diagonalisable if
and only if sum of algebraic multiplicities is equal to sum of geometric multiplicities of all
the eigenvalues of A=n). Procedure for diagonalising a matrix.
5. Spectral Theorem for Real Symmetric Matrices (Statement only ). Examples of orthogonal
diagonalisation of real symmetric matrices. Applications to quadratic forms and classi -
cation of conic sections.
Reference books
1. Howard Anton, Chris Rorres; Elementary Linear Algebra; Wiley Student Edition).
2. Serge Lang; Introduction to Linear Algebra; Springer.
3. S Kumaresan; Linear Algebra - A Geometric Approach; PHI Learning.
4. Sheldon Axler; Linear Algebra done right; Springer.
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5. Gareth Williams; Linear Algebra with Applications; Jones and Bartlett Publishers.
6. David W. Lewis; Matrix theory.
USMT403A: Numerical Methods (Elective A)
Unit I. Solution of Algebraic and Transcendental Equations (15L)
1. Measures of Errors: Relative, absolute and percentage errors, Accuracy and precision: Ac-
curacy tondecimal places, accuracy to nsigni cant digits or signi cant gures, Rounding
and Chopping of a number, Types of Errors: Inherent error, Round-o error and Trunca-
tion error.
2. Iteration methods based on rst degree equation: Newton-Raphson method. Secant
method. Regula-Falsi method.
Derivations and geometrical interpretation and rate of convergence of all above methods
to be covered.
3. General Iteration method: Fixed point iteration method.
Unit II. Interpolation, Curve tting, Numerical Integration(15L)
1. Interpolation: Lagrange's Interpolation. Finite di erence operators: Forward Di erence
operator, Backward Di erence operator. Shift operator. Newton's forward di erence
interpolation formula. Newton's backward di erence interpolation formula.
Derivations of all above methods to be covered.
2. Curve tting: linear curve tting. Quadratic curve tting.
3. Numerical Integration: Trapezoidal Rule. Simpson's 1/3 rd Rule. Simpson's 3/8th Rule.
Derivations all the above three rules to be covered.
Unit III. Solution Linear Systems of Equations, Eigenvalue problems(15L)
1. Linear Systems of Equations: LU Decomposition Method (Dolittle's Method and Crout's
Method). Gauss-Seidel Iterative method.
2. Eigenvalue problems: Jacobi's method for symmetric matrices. Rutishauser method for
arbitrary matrices.
Reference Books:
1. Kendall E. and Atkinson; An Introduction to Numerical Analysis; Wiley.
2. M. K. Jain, S. R. K. Iyengar and R. K. Jain; Numerical Methods for Scienti c and Engi-
neering Computation; New Age International Publications.
3. S. Sastry; Introductory methods of Numerical Analysis; PHI Learning.
4. An introduction to Scilab-Cse iitb.
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Additional Reference Books
1. S.D. Comte and Carl de Boor; Elementary Numerical Analysis, An algorithmic approach;
McGrawHillll International Book Company.
2. Hildebrand F.B.; Introduction to Numerical Analysis; Dover Publication, NY.
3. Scarborough James B.; Numerical Mathematical Analysis; Oxford University Press, New
Delhi.
USMT403B Statistical Methods and their Applications (Elective B)
Unit I. Descriptive Statistics and random variables (15 Lectures)
Measures of location (mean, median, mode), Partition values and their graphical locations, mea-
sures of dispersion, skewness and kurtosis, Exploratory Data Analysis (Five number summary,
Box Plot, Outliers), Random Variables (discrete and continuous), Expectation and variance of
a random variable.
Unit II. Probability Distributions and Correlation (15 Lectures)
Discrete Probability Distribution (Binomial, Poisson), Continuous Probability Distribution:
(Uniform, Normal), Correlation, Karl Pearson's Coecient of Correlation, Concept of linear
Regression, Fitting of a straight line and curve to the given data by the method of least squares,
relation between correlation coecient and regression coecients.
Unit III. Inferential Statistics (15 lectures)
Population and sample, parameter and statistic, sampling distribution of Sample mean and
Sample Variance, concept of statistical hypothesis, critical region, level of signi cance, con -
dence interval and two types of errors, Tests of signi cance (t-test, Z-test, F-test, Chi-Square
Test (only applications))
Reference Books
1. Fundamentals of Mathematical Statistics,12th Edition, S. C. Gupta and V. K. Kapoor,Sultan
Chand & Sons, 2020.
2. Statistics for Business and Economics, 11th Edition, David R. Anderson, Dennis J. Sweeney
and Thomas A. Williams, Cengage Learning, 2011.
3. Introductory Statistics, 8th Edition, Prem S. Mann, John Wiley & Sons Inc., 2013.
4. A First Course in Statistics, 12th Edition, James McClave and Terry Sincich, Pearson
Education Limited, 2018.
5. Introductory Statistics, Barbara Illowsky, Susan Dean and Laurel Chiappetta, OpenStax,
2013.
6. Hands-On Programming with R, Garrett Grolemund, O'Reilly.
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USMT P04 / UAMT P04: Practicals
Suggested Practical for USMT 401/ UAMT 401
1. 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. Di erentiability of scalar eld,Total derivative, gradient, level sets and tangent planes.
4. Chain rule, higher order derivatives and mixed partial derivatives of scalar elds.
5. Maximum and minimum rate of change of scalar elds. Taylor's Theorem. Finding Hes-
sian/Jacobean matrix. Di erentiation of a vector eld at a point. Chain Rule for vector
elds.
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 USMT402/UAMT 402
1. Linear transformation, Kernel, Rank-Nullity Theorem.
2. Linear Isomorphism, Matrix associated with Linear transformations.
3. Inner product and properties, Projection, Orthogonal complements.
4. Orthogonal, orthonormal sets, Gram-Schmidt orthogonalisation
5. Eigenvalues, Eigenvectors, Characteristic polynomial. Applications of Cayley Hamilton
Theorem.
6. Diagonalisation of matrix, orthogonal diagonalisation of symmetric matrix and application
to quadratic form.
7. Miscellaneous Theoretical Questions based on full paper.
Suggested Practicals for USMT403A
The Practical no. 1 to 6 should be performed either using non-programable scienti c
calculators or by using the software Scilab.
1. Newton-Raphson method, Secant method.
2. Regula-Falsi method, Iteration Method..
3. Interpolating polynomial by Lagrange's Interpolation, Newton forward and backward dif-
ference Interpolation.
4. Curve tting, Trapezoidal Rule, Simpson's 1/3rd Rule, Simpson's 3/8th Rule.
5. LU decomposition method, Gauss-Seidel Interative method.
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6. Jacobi's method, Rutishauser method..
7. Miscellaneous theoretical questions from all units.
Suggested Practicals for USMT403B
All practicals should be performed using any one of the following softwares: MS Excel, R, Strata,
SPSS, Sage Math to carry out data analysis and computations.
1. Descriptive Statistics.
2. Random Variables.
3. Probability Distributions.
4. Correlation and Regression.
5. Testing of hypothesis.
6. Case studies.
7. Miscellaneous Theory questions based on Unit I,II,III.
xxxxxx
7. Scheme of Examination (75:25)
The performance of the learners shall be evaluated into two parts.
Internal Assessment of 25 percent marks.
Semester End Examinations of 75 percent marks.
I.Internal Evaluation of 25 Marks:
S.Y.B.Sc. :
(i) One class Test of 20 marks to be conducted during Practical session.
Paper pattern of the Test:
Q1: De nitions/ Fill in the blanks/ True or False with Justi cation (04 Marks).
Q2: Multiple choice 5 questions. (10 Marks: 5 2)
Q3: Attempt any 2 from 3 descriptive questions. (06 marks: 2 3)
(ii) Active participation in routine class: 05 Marks.
OR
Students who are willing to explore topics related to syllabus, dealing with applica-
tions historical development or some interesting theorems and their applications can
be encouraged to submit a project for 25 marks under the guidance of teachers.
S.Y.B.A. :
(i) One class Test of 20 marks to be conducted during Tutorial session.
Paper pattern of the Test:
Q1: De nitions/ Fill in the blanks/ True or False with Justi cation (04 Marks).
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18
Q2: Multiple choice 5 questions. (10 Marks: 5 2)
Q3: Attempt any 2 from 3 descriptive questions. (06 marks: 2 3)
(ii) Journal : 05 Marks.
OR
Students who are willing to explore topics related to syllabus, dealing with applica-
tions historical development or some interesting theorems and their applications can
be encouraged to submit a project for 25 marks under the guidance of teachers.
II.Semester End Theory Examinations : There will be a Semester-end external Theory
examination of 75 marks for each of the courses USMT301/UAMT301, USMT/USAT
302, USMT 303 of Semester III and USMT/UAMT401, USMT/UAMT 402, USMT 403
of semester IV to be conducted by the college.
1. Duration: The examinations shall be of 2 and1
2hours duration.
2. Theory Question Paper Pattern:
a) There shall be FOUR questions. The rst three questions Q1, Q2, Q3 shall be of
20 marks, each based on the units I, II, III respectively. The question Q4 shall
be of 15 marks based on the entire syllabus.
b) All the questions shall be compulsory. The questions Q1, Q2, Q3, Q4 shall have
internal choices within the questions. Including the choices, the marks for each
question shall be 25-27.
c) The questions Q1, Q2, Q3, Q4 may be subdivided into sub-questions as a, b, c,
d & e, etc and the allocation of marks depends on the weightage of the topic.
III.Semester End Examinations Practicals:
At the end of the Semesters III & IV Practical examinations of three hours duration and
150 marks shall be conducted for the courses USMTP03, USMTP04.
At the end of the Semesters III & IV Practical examinations of two hours duration and
100 marks shall be conducted for the courses UAMTP03, UAMTP04.
In semester III, the Practical examinations for USMT301/UAMT301, USMT302/UAMT302
and USMT303 are held together by the college.
In Semester IV, the Practical examinations for USMT401/UAMT401, USMT402/UAMT402
and USMT403 are held together by the college.
Paper pattern: The question paper shall have two parts A and B.
Each part shall have two Sections.
Section I Objective in nature: Attempt any Eight out of Twelve multiple choice questions
( 04 objective questions from each unit) (8 3 = 24 Marks).
Section II Problems: Attempt any Two out of Three ( 01 descriptive question from each
unit) (82 = 16 Marks).
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Practical Part A Part B Part C Marks duration
Course out of
USMTP03 Questions Questions Questions 120 3 hours
from USMT301 from USMT302 from USMT 303
UAMTP03 Questions Questions || 80 2 hours
from UAMT301 from UAMT302
USMTP04 Questions Questions Questions 120 3 hours
from USMT401 from USMT402 from USMT403
UAMTP04 Questions Questions || 80 2 hours
from UAMT401 from UAMT402
Marks for Journals and Viva:
For each course USMT301/UAMT301, USMT302/UAMT302, USMT303, USMT401/UAMT401,
USMT402/UAMT402, USMT3031:
1. Journal: 10 marks (5 marks for each journal).
2. Viva: 10 marks.
Each Practical of every course of Semester III and IV shall contain 10 (ten) problems out of
which minimum 05 ( ve) have to be written in the journal. .
A student must have a certi ed journal before appearing for the practical examination.
In case a student does not posses a certi ed journal he/she will be evaluated for 120 =80 marks.
He/she is not quali ed for Journal + Viva marks.
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