## AAMS UG 97 Mathematics CBCS 1 Syllabus Mumbai University by munotes

## Page 2

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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)

## Page 4

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

## Page 5

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

## Page 6

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.

## Page 9

(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 2021-22

## Page 10

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 Innite 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 Dierential Equations

2 3 II Systems of First Order

Linear dierential equations

III Numerical Solutions of Ordinary

Dierential Equations

PRACTICALS

USMTP03Practicals based on3 5USMT301, USMT 302 and USMT 303

UAMTP03Practicals based on2 4UAMT301, UAMT 302

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3

SEMESTER IV

Multivariable Calculus I

Course Code UNIT TOPICS Credits L/Week

USMT 401, UAMT 401I Functions of several variables

2 3II Dierentiation of Scalar Fields

III Applications of Dierentiation of

Scalar Fields and Dierentiation 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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4

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. Innite Series (15 Lectures)

1. Innite series in R. Denition 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. Renement 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.

## Page 13

5

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)dxZb

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. Denition 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 innitely 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. Denition 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. Denition 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 denition 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

.

## Page 15

7

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 A1when 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 Dierential Equations

Unit I. Higher order Linear Dierential equations (15 Lectures)

1. The general nth order linear dierential equations, Linear independence, An existence

and uniqueness theorem, the Wronskian, Classication: homogeneous and non-homogeneous,

General solution of homogeneous and non-homogeneous LDE, The Dierential operator

and its properties.

2. Higher order homogeneous linear dierential 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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3. Higher order homogeneous linear dierential equations with constant coecients, the

method of undermined coecients, method of variation of parameters.

4. The inverse dierential 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 dierential 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 Dierential Equations;

Macmillan.

2. Units 5, 6 and 7 of M.D. Raisinghania; Ordinary and Partial Dierential Equations; S.

Chand.

Unit II. Systems of First Order Linear Dierential Equations (15 Lectures)

(a) Existence and uniqueness theorem for the solutions of initial value problems for a system

of two rst order linear dierential 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 dierential 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 dierential 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 dierential 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 dierential equations and the associated system of homoge-

neous linear dierential equations.

Reference Books

1. G.F. Simmons; Dierential Equations with Applications and Historical Notes; Taylor's

and Francis.

Unit III. Numerical Solution of Ordinary Dierential Equations (15 lectures)

1. Numerical Solution of initial value problem of rst order ordinary dierential 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) Modied 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 dierential equation using:

(i) Runge-Kutta fourth order method for solving simultaneous ordinary dierential equa-

tion,

(ii) Finite dierence 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 Dierential Equations, Macmillan.

2. M.D. Raisinghania, Ordinary and Partial Dierential Equations, S. Chand.

3. G.F. Simmons, Dierential 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 Dierential Equa-

tions, Wiley.

xxxxxx

USMT P03 / UAMT P03: Practicals

Suggested Practicals for USMT 301/ UAMT 301

## Page 18

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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, dierent 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

dierential equations.

2. Solving higher order linear dierential 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, modied Euler's method, Runge-Kutta method of fourth order and cal-

culating their accuracy.

5. Finding the numerical solution of simultaneous ordinary dierential equation using fourth

order Runge-Kutta method.

6. Finding the numerical solution of two point linear boundary value problem using Finite

dierence method.

xxxxxx

## Page 19

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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: Denitions of partial derivative and

directional derivative of scalar elds (with emphasis on R2andR3). Mean Value Theorem

of scalar elds.

UNIT II. Dierentiation of Scalar Fields (15 Lectures)

1. Dierentiability of scalar elds (in terms of linear transformation). The concept of (total)

derivative. Uniqueness of total derivative of a dierentiable function at a point. Examples

of functions of two or three variables. Increment Theorem. Basic properties including

(i) continuity at a point of dierentiability, (ii)existence of partial derivatives at a point

of dierentiability, and (iii) dierentiability 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 Dierentiation of Scalar Fields and Dierentiation of

Vector Fields (15 lectures)

1. Applications of Dierentiation of Scalar Fields: The maximum and minimum rate of

change of scalar elds. Taylor's Theorem for twice continuously dierentiable 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.

## Page 20

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2. Dierentiation of Vector Fields: Dierentiability 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. Denition 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, Eect of change of bases

on matrices of linear operator.

## Page 21

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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 nsignicant digits or signicant 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 dierence operators: Forward Dierence

operator, Backward Dierence operator. Shift operator. Newton's forward dierence

interpolation formula. Newton's backward dierence 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 Scientic 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 signicance, con-

dence interval and two types of errors, Tests of signicance (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 "denition and otherwise\ ,

iterated limits.

2. Computing directional derivatives, partial derivatives and mean value theorem of scalar

elds.

3. Dierentiability 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. Dierentiation 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 scientic

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.

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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: Denitions/ Fill in the blanks/ True or False with Justication (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: Denitions/ Fill in the blanks/ True or False with Justication (04 Marks).

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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 certied journal before appearing for the practical examination.

In case a student does not posses a certied journal he/she will be evaluated for 120 =80 marks.

He/she is not qualied for Journal + Viva marks.

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