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UNIVERSITY OF MUMBAI
Syllabus
for
S.Y.B.A./S.Y.B.Sc. (CBCS)
Program: B.A/B.Sc.
Course: Mathematics
with e ect from the academic year 2017-2018
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SYBA SYBSc Mathematics Sem III IV_1 Syllabus Mumbai University by munotes
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Preamble
The Board of Studies in Mathematics has prepared the syllabus of S.Y.B.A./S.Y. B.Sc. (w.e.f.
2017-18) and T.Y.B.A./T.Y. B.Sc. (w.e.f. 2018-19) in the subject of Mathematics under the
Choice Based Credit System (CBCS).
The syllabus provides best learning experience to the students as well as to the teachers by
o ering
1. two interdisciplinary courses in Semesters III and IV of S.Y.B.A./S.Y. B.Sc. and
2. two projects based courses in Semesters V and VI of T.Y.B.A./T.Y. B.Sc.
The interdisciplinary course o ered in Semesters III is INTRODUCTION TO COMPUTING
AND PROBLEM SOLVING - I. The Aim of this course is to develop Algorithm thinking towards
problem solving.
The interdisciplinary course o ered in Semesters IV is INTRODUCTION TO COMPUTING
AND PROBLEM SOLVING - II. In this course students are enabled to write their own Programs
in Python.
In the two project courses o ered in Semesters V and VI of T.Y.B.A./T.Y. B.Sc. a student
can do a project on a topic from Mathematics, Financial Mathematics, Statistics, and Computer
programming. Nearly fty topics are listed for this project courses in this syllabus.
By this syllabus, the quality of education o ered to students is enhanced. This curriculum
creates positive improvements in the educational system.
The curriculum retains the current workload of Mathematics Departments.
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S.Y.B.A./S.Y.B.Sc. (CBCS)
Semester III
CALCULUS III
Course Code Unit Topics Credits L/W
USMT301,UAMT301Unit I Functions of several variables
Unit II Di erentiation 2 3
Unit III Applications
ALGEBRA III
USMT302, UAMT302Unit I Linear transformations and Matrices
Unit II Determinants 2 3
Unit III Groups, Subgroups
INTRODUCTION TO COMPUTING AND PROBLEM SOLVING - I
USMT303Unit I Algorithms
Unit II Graphs & The shortest path algorithm 2 3
Unit III Trees & Traversal algorithm
PRACTICALS
USMTP03Practicals based on
USMT301,USMT302 3 5
and USMT303
UAMTP03Practicals based on
UAMT301,UAMT302 2 4
Teaching Pattern
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,USMT302 combined & 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.
3. One Practical (2L) per week per batch for each of the courses UAMT301,UAMT302, (the
batches to be formed as prescribed by the University). Each practical session is of 48
minutes duration.
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S.Y.B.A./S.Y.B.Sc. (CBCS)
Semester IV
CALCULUS IV
Course Code Unit Topics Credits L/W
USMT401,UAMT401Unit I Nested Interval Theorem & Applications
USMT401,UAMT401Unit II Riemann integration
Unit III Inde nite and improper Riemann integrals, 2 3
Double integrals
ORDINARY DIFFERENTIAL EQUATIONS
Unit I First order rst degree di erential equations
USMT402, UAMT402 Unit II Second order linear di erential equations 2 3
Unit III Linear system of ODEs
INTRODUCTION TO COMPUTING AND PROBLEM SOLVING - II
Unit I Problem solving strategies
USMT403 Unit II Python programming language 2 3
Unit III Iterations, Strings & File handling in Python
PRACTICALS
USMTP04Practicals based on
USMT401,USMT402 3 5
and USMT403
UAMTP04Practicals based on
UAMT401,UAMT402 2 4
Teaching Pattern
1. Three lectures per week per course. Each lecture is of 48 minutes duration.
2. One Practical (2L) per week per batch for courses USMT401,USMT402 combined & one
Practical (3L) per week for course USMT403 (the batches to be formed as prescribed by
the University). Each practical session is of 48 minutes duration.
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3. One Practical (2L) per week per batch for each of the courses UAMT401,UAMT402, (the
batches to be formed as prescribed by the University). Each practical session is of 48
minutes duration.
Syllabus for Semester III & IV
SEMESTER III
Note : All topics have to be covered with proof in details (unless mentioned otherwise) and
with examples.
USMT301/UAMT301 CALCULUS III
Unit I: Functions of several variables (15 Lectures)
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 in Rn;
sequences in Rn;convergence of sequences- these concepts should be speci cally discussed for
R2andR3:
Functions from RntoR(scalar elds) and functions from Rn !Rm(vector elds), limits and
continuity of functions, basic results on limits and continuity of sum, di erence, scalar multiples
of vector elds, continuity and components of a vector eld.
Directional derivatives and partial derivatives of scalar elds.
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 T.
Apostol .
Unit II: Di erentiation (15 Lectures)
Di erentiability 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 di eren-
tiable function at a point, simple examples of nding total derivative of functions such as
f(x;y) =x2+y2;f(x;y;z ) =x+y+z;di erentiability at a point of a function fimplies
continuity and existence of direction derivatives of fat the point, the existence of continuous
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partial derivatives in a neighbourhood of a point implies di erentiability at the point.
Gradient of a scalar eld, geometric properties of gradient, level sets and tangent planes.
Chain rule for scalar elds.
Higher order partial derivatives, mixed partial derivatives, sucient condition for equality of
mixed partial derivatives.
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 .
Unit III: Applications (15 Lectures)
Second order Taylors formula for scalar elds.
Di erentiability of vector elds, de nition of di erentiability of a vector eld at a point, Jacobian
matrix, di erentiability of a vector eld at a point implies continuity. The chain rule for derivative
of vector elds (statements only).
Mean value inequality.
Hessian matrix, Maxima, minima and saddle points.
Second derivative test for extrema of functions of two variables.
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 of
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. Thomas andR. L. Finney ,Calculus and Analytic Geometry , Ninth Edition,
Addison-Wesley, 1998.
2.Sudhir. R. Ghorpade andBalmohan V. Limaye ,A Course in Calculus and Real
Analysis , Springer International Edition.
3.Howard Anton ,Calculus - A new Horizon , Sixth Edition, John Wiley and Sons Inc,
1999.
USMT302/UAMT302 ALGEBRA III
Note: Revision of relevant concepts is necessary.
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Unit I: Linear transformations and Matrices (15 Lectures)
Linear transformations, representation of linear maps by matrices and e ect under a change of
basis, examples.
Kernel and image of a linear transformation, examples. Rank-Nullity theorem and applications.
CompositeSTof linear maps T:V !W&S:W !Uof f.d. real vector spaces V;W;U
and matrix representation of ST:
Linear isomorphisms, inverse of a linear isomorphism. Any n-dimensional real vector space is
isomorphic to Rn:
The following are equivalent for a linear map T:V !Vof a nite dimensional real vector
space:
1.Tis an isomorphism.
2. kerT=f0g:
3. Im (T) =V:
Recommended Text Book for Unit I :
S. Kumaresan ,Linear Algebra A Geometric Approach , PHI, 2014 ( sections 4.1,4.2, 4.3,
4.4)
Unit II: Matrices and Determinants (15 Lectures)
The matrix units, row operations, elementary matrices. Elementary matrices are invertible and
an invertible matrix is a product of elementary matrices.
Row space and column space of a matrix, 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.
Rnis the space of column vectors x=0
@x1
x2
...
xn1
Awhere each xj2R;equivalence of rank of an
nn-matrix and rank of the linear transformation LA:Rn !Rm(LA(x) =Ax8x2Rn);
the dimension of solution space of the system of linear equations Ax= 0 equalsn rank(A):
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 solutions of the system is the sum of a
particular solution of the system and the solutions of the associated homogeneous system.
Determinant D(A1;A2)of order 2and its properties:
1. As a function of column vectors, the determinant is linear.
2. If the two columns are equal, then the determinant is equal to 0:
3. IfIis the unit matrix, I= (E1;E2);thenD(E1;E2) = 1:
Results on Determinants of order 2 :
1. If one adds a scalar multiple of one column to the other, then the value of the determinant
does not change.
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2. The determinant of Ais equal to the determinant of its transpose.
3. Two vectors A1;A2ofR2are linearly dependent if and only if the determinant D(A1;A2) =
0:
4. Let be a function of two variables A1;A22R2such that is bilinear (i.e. is linear
in each variable), (A1;A1) = 08A12R2and (E1;E2) = 1 whereE1=1
0
;E2=
0
1
are the standard unit vectors of R2;then (A1;A2)is the determinant D(A1;A2):
Determinants of order 33;nn;expansion of the determinant according to i-th row, properties
of the determinant function.
Results (without proof): For two nnmatricesA&B;Det(A) = Det(tA);Det(AB) =
Det(A)Det(B):
Linear dependence and independence of vectors in Rnusing determinants, the existence and
uniqueness of the system Ax=bwhereAis annn-matrix with det (A)6= 0:
Cofactors and minors, adjoint adj (A)of annn-matrixA; A adj(A) = det(A)Id (without
proof). Ann-real matrix is invertible if and only if det (A)6= 0 andA 1=1
det(A)adj(A)for
an invertible matrix A:Cramer's rule.
Determinant as area and volume.
Recommended text book for Unit II :
Introduction to Linear Algebra bySerge Lang .
Unit III: Groups & subgroups (15 Lectures)
De nition of a group, Abelian group, Order of a group, nite groups, in nite groups.
Examples of groups including:
1.Z;R;Cunder addition.
2.Q(=Qnf0g);R(Rnf0g);Q+(=positive rational numbers );Cunder multiplication.
3.Zn(=the group of residue classes modulo n) under addition.
4.Un(= the group of prime residue classes modulo n) under multiplication
5.Sn(=the group of all permutations of f1;2;;ng).
6. Klein 4-group.
7. The group of symmetries of a plane gure. The Dihedral group Dn(= the group of
symmetries of a regular polygon of nsides in the plane R2(n=3,4)) under composition.
8.Mmn(R)(=the group of all mn-matrices with real entries) under addition of matrices.
9.GLn(R)(=the group of invertible nnmatrices with real entries) under multiplication
of matrices.
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Subgroups, Cyclic groups:
1.S1is a subgroup of C; nis a subgroup of S1:
2. Cyclic groups (examples of Z;Zn;n) and cyclic subgroups.
3. The center Z(G)of a groupGas a subgroup of G:
4. Cosets, Lagrange's theorem.
Group homomorphisms, and isomorphisms. Examples and properties. Automorphism of a group
and inner automorphisms.
Recommended Text Books for Unit III :
1.I.N. Herstein ,Topics in Algebra , Vikas Publishing House.
2.J.B. Fraleigh ,A rst course in Abstract Algebra , third edition, Narosa, New Delhi.
Additional Reference Books :
1.M. Artin :Algebra, Prentice Hall of India Private Limited.
2.K. Hoffman and R. Kunze ,Linear Algebra, Tata McGraw-Hill, New Delhi.
3.G. 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. Banchoff and J. Wermer ,Linear Algebra through Geometry, Springer Verlag
New York, 1984.
7.Sheldon Axler ,Linear Algebra done right, Springer Verlag, New York.
8.Klaus Janich ,Linear Algebra .
9.O. Bretcher ,Linear Algebra with Applications, Pearson Education.
10.G. Williams ,Linear Algebra with Applications, Narosa Publication.
USMT303 INTRODUCTION TO COMPUTING
AND PROBLEM SOLVING - I
Aim of this course:
1. to develop Algorithm thinking towards problem solving and
2. to study Problem solving strategies like divide and concur, recursive thinking etc.
Unit I: Algorithms (15 Lectures)
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A. De nition of an algorithm, characteristics of an algorithm, Selection and iterative con-
structs in pseudocode, simple examples such as
(a) Exchanging values of variables,
(b) Sum of ngiven numbers.
B. Searching and sorting algorithms including the following:
(a) Finding maximum and/or minimum element in a nite sequence of integers,
(b) The linear search and binary search algorithms of an integer xin a nite sequence
of distinct integers,
(c) Sorting of a nite sequence of integers in ascending order,selection sort.
C. Algorithms on integers:
(a) Modular exponent,
(b) Euclidean algorithm to nd the g.c.d of two non-zero integers.
D. Complexity of algorithm: Big O notation, Growth of functions, Time complexity, Best
case, Average case, Worst Case complexity. Using big O notation to express the best,
average and worst case behaviour for sorting and searching algorithms.
E. Recursion, Examples including:
(a) Tower of Hanoi
(b) Fibonacci sequence
Reference for Unit I :
Chapter 3 of Discrete Mathematics and Its Applications byKenneth H. Rosen , (McGraw
Hill, seventh Edition).
Unit II: Graphs (15 Lectures)
A. Introduction to graphs: Types of graphs: Simple graph, directed graph, (One exam-
ple/graph model of each type to be discussed).
B. (a) Graph Terminology: Adjacent vertices, degree of a vertex, isolated vertex, pendant
vertex in a undirected graph.
(b) The handshaking Theorem for an undirected graph (statement only), Theorem: An
undirected graph has an even number odd vertices (statement only).
C. Some special simple graphs (by simple examples): Complete graph, cycle, wheel in a
graph, Bipartite graph, regular graph.
D. Representing graphs and graph isomorphism:
(a) Adjacency matrix of a simple graph.
(b) Incidence matrix of an undirected graph.
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E. Connectivity:
(a) Paths, circuits, simple paths, simple circuits in a graph (simple examples).
(b) Connecting paths between vertices (simple examples).
(c) Euler paths and circuits, Hamilton paths and circuits, Diracs Theorem (statement
only), Ores Theorem (statement only)
(d) Planar graphs, planar representation of graphs, Eulers formula. Kuratowskis Theorem
(statement only).
F.Algorithms :
Shortest path problem: Construction of Eulerian path by Fleury's Algorithm, The shortest
path algorithm - Dijkstras Algorithm, Floyd's Algorithm to nd the length of the shortest
path.
Reference for Unit II :
Sections 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7 of Chapter 10 of Discrete Mathematics and
Its Applications byKenneth H. Rosen , (McGraw Hill Edition, seventh Edition).
Unit III: Trees (15 Lectures)
A. (a) Trees: De nition and Examples.
(b) Forests, binary trees
(c) Trees as models.
(d) Properties of Trees (no proofs).
B. Application of Trees:
(a) Binary Search Trees, Algorithm for locating an item in or adding an item to a Binary
Search Tree.
(b) Decision Trees (simple examples).
(c) Algorithm for Hu man's coding, construction of Hu man's code by examples.
C. Minimum Spanning Trees, Prims Algorithm, Kruskals Algorithm (The Proofs of the results
in this unit are not required and may be omitted).
Reference for Unit 3 :
Chapter 9, Sections 9.1, 9.2, 9.3, 9.4, 9.5 of Discrete Mathematics and Its Applications by
Kenneth H. Rosen ( McGraw Hill Edition).
Recommended Text Books :
1.R.G. Dromey ,How to Solve it by computers , Prentice-Hall India.
2.R. Wilson ,Introduction to Graph theory , Fourth Edition, Prentice Hall.
3.T. H. Cormen ,Charles E. Leisenon andRonald L. Rivest :Introduction to
Algorithms , Prentice Hall of India, New Delhi, 1998 Edition.
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4.K. H. Rosen ,Discrete Mathematics and Its Applications , McGraw Hill Edition.
5.B. Kolman ,Robert Busby ,Sharon Ross :Discrete Mathematical Structures ,
Prentice-Hall India.
6.N. Biggs ,Discrete Mathematics , Oxford.
Additional Reference Books :
1. D. B. West, Introduction to graph Theory , Pearson.
2. F. Harary, Graph Theory , Narosa Publication.
3. Graham, Knuth and Patashnik, Concrete Mathematics , Pearson Education Asia Low
Price Edition.
USMTP03/UAMTP03 Practicals
A. Practical for USMT301/UAMT301 :
(1) Sequences in R2;R3;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, di erentiation 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.
1. Miscellaneous Theoretical Questions based on full paper.
B. Practical for USMT302/UAMT302 :
(1) Rank-Nullity Theorem.
(2) System of linear equations.
(3) Computation of row rank and column rank of 33matrices.
(4) Calculating determinants of matrices, triangular matrices using de nition and Laplace
expansion.
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(5) Finding inverses of matrices using adjoint.
1. Groups, Subgroups, Lagranges Theorem, Cyclic groups and Groups of Symmetry.
2. Group homomorphisms, isomorphisms.
3. Miscellaneous Theoretical Questions based on full paper.
C. Practical for USMT303 :
1. Describe an algorithm to count total number of positive and negative values from the set
f23; 632;325;63; 63;0; 55; 652:23;65:21;98:235; 1g.
2. Describe an algorithm to accept the values of A and B and swap them.
3. Describe an algorithm to accept the highest number from the user.
4. List all the steps used to search for 9in the sequence 1;3;4;5;6;8;9;11using
a) a linear search. b) a binary search.
5. Describe an algorithm that inserts an integer xin the appropriate position into the list
a1;a2;;an of integers that are in increasing order.
6. Describe an algorithm based on the Selection sort for sorting the list 3;2;5;4;1;9;6;8;7;2.
7. Describe an algorithm that prints rst nterms of the Fibonacci sequence.
8. Describe an algorithm that nds factorial of a non-negative integer.
9. Describe Euclidean algorithm to nd GCD of given two integers.
10. Show that f(x) =x2+ 2x+ 1 isO(x2).
11. Show that n2is notO(n).
12. Show that x2 + 4x+ 17 isO(x3)but thatx3is notO(x2+ 4x+ 17) .
13. Letkbe a positive integer. Show that 1k+ 2k++nkisO(nk+1).
14. Towers of Hanoi for recursion.
15. What are the worst-case, average-case, and best-case time complexities, in terms of com-
parisons, of the algorithm that nds the smallest integer in a list of nintegers by comparing
each of the integers with the smallest integer found so far?
16. Drawing a graph, counting the degree of vertices and number of edges.
17. Representing a given graph by an adjacency matrix and drawing a graph having given
matrix as adjacency matrix.
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18. Determining whether the given graph is connected or not. Finding connected components
of a graph. Finding strongly connected components of a graph. Finding cut vertices.
19. To determine whether the given graph is a tree. Construction of Binary Search Tree and
applications to sorting and searching.
20. Spanning Trees. Finding Spanning Tree using Breadth First Search and/or Depth First
Search.
21. Convert messages in to binary sequence using Ho man's Algorithm.
SEMESTER IV
Note : All topics have to be covered with proof in details (unless mentioned otherwise) and
with examples.
USMT401/UAMT401 CALCULUS IV
Unit I: Nested Interval theorem & Applications (15 Lectures)
Nested Interval theorem in R:Applications of Nested Interval Theorem:
1. The set of real numbers is uncountable.
2. Decimal representation of a real number.
3. Bolzano Weierstrass Theorem: Every bounded sequence of real numbers has a convergent
subsequence.
4. Intermediate Value theorem: Let f: [a;b] !Rbe a function continuous with f(a)f(b)<
0:Then9c2(a;b)such thatf(c) = 0:
5. Heine-Borel theorem: Let [a;b]be a closed and bounded interval and let fJ : 2g
be a family of open intervals such that [a;b][ 2J : Then there exists a nite subset
such thatFsuch that [a;b][ 2FJ :
Recommended Text Book for Unit I :
R.G. Bartle - D.R. Sherbet ,Introduction to Real analysis , John Wiley & Sons.
Unit II: Riemann Integration (15 Lectures)
De nition of uniform continuity of a real valued function on a subset of R:A continuous function
on a closed and bounded interval is uniformly continuous (only statement).
Approximation of area; Upper/Lower Riemann sums and properties; Upper/Lower Riemann
integrals; De nition of Riemann integral on a closed and bounded interval; Riemann's Criterion
for Riemann integrability.
Fora
af+Rb
cf:
Properties of Riemann integrals:
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i)2R&f;g2R[a;b])f+g;f2R[a;b] &Rb
a(f+g) =Rb
af+Rb
ag;Rb
af=Rb
af
ii)f2R[a;b])jfj2R[a;b]andjRb
afjRb
ajfj
iii)f0)Rb
af0:
f2C[a;b])f2R[a;b];every bounded function with nitely many discontinuities is Riemann
integrable; monotone functions are Riemann integrable.
Recommended Text Books for Unit II :
1.R.G. Bartle -D.R. Sherbet ,Real analysis , John Wiley & Sons.
2.T. Apostol ,Calculus Vol.2, John Wiley.
Unit III: Inde nite and improper Riemann integrals, double integrals (15 Lec-
tures)
1. Continuity of F(x) =Rx
af(t)dtwheref2R[a;b]:First and second Fundamental theorem
of Calculus.
2. Mean value theorem for integrals. Integration by parts, Change of variable formula (statement
only).
3. Improper integrals- type 1 and type 2; Absolute convergence of improper integrals; Compar-
ison tests; Abels and Dirichlets tests (without proof).
4. functions and their properties; function (x;y);and relationship between and
functions.
5. Double integrals: De nition of double integrals over rectangles, properties, double integrals
over a bounded region.
Fubini theorem (without proof) - iterated integrals, double integrals as volume.
Application of double integrals: average value, area, moment, center of mass.
Double integral in polar form.
Reference for para 4 of unit III: W. Rudin ,Principles of Mathematical Analysis , McGraw
Hill.
Recommended Text Books for Unit III :
1.R.G. Bartle - D.R. Sherbet ,Introduction to Real analysis , John Wiley & Sons.
2.J. E. Marsden- A. J. Tromba- A. Weinstein ,Basic multi-variable calculus ,
Springer.
Additional reference books :
1.R. R. Goldberg ,Methods of Real Analysis , Oxford and IBH, 1964.
2.Ajit Kumar & S. Kumaresan ,A Basic Course in Real Analysis, CRC Press, 2014.
3.J. Stewart ,Calculus, Brooke/Cole Publishing Co, 1994.
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4.W. Rudin ,Principles of Mathematical Analysis , McGraw Hill.
USMT402/UAMT402 Ordinary Di erential Equations
Unit I: First order rst degree di erential equations (15 Lectures)
De nitions of: Di erential Equation, Order and Degree of a Di erential Equation, Ordinary
Di erential Equation (ODE), Partial Di erential Equation, Linear ODE, non-linear ODE.
De nition of Lipschitz function, examples. Existence and Uniqueness Theorem for the di erential
equationy0=f(x;y);y(x0) =y0wheref(x;y)is a continuous function satisfying Lipschitz
conditionjf(x;y1) f(x;y2)jKjy1 y2jon the strip axb&y2R(statement only).
Solve examples verifying the conditions of existence and uniqueness theorem.
Existence and Uniqueness Theorem for the solutions of a second order linear ODE:
d2y
dx2+P(x)dy
dx+Q(x)y=R(x)
with initial conditions y(x0) =y0&y0(x0) =y1whereP(x);Q(x);R(x)are continuous functions
on[a;b](statement only). Solve examples verifying the conditions of existence and uniqueness
theorem.
Review of solution of homogeneous and non-homogeneous di erential equations of rst order
and rst degree.
Exact Equations: General Solution of Exact equations of rst order and rst degree, Necessary
and sucient condition for Mdx +Ndy = 0 to be exact. Non-exact equations. Rules for
nding integrating factors (without proof) for non exact equations such as:
i)1
Mx+Nyis an integrating factor if Mx+Ny6= 0 &Mdx +Ndy is homogeneous
ii)1
Mx Nyis an integrating factor if Mx Ny6= 0 &Mdx Ndy is of the form
f1(x;y)ydx+f2(x;y)xdy
iii) a)eR
f(x)dxis an integrating factor if N6= 0 &1
N@M
@y @N
@Nis a function of xalone, say
f(x)
b)eR
g(y)dyis an integrating factor if M6= 0 &1
M@M
@y @N
@x
is a function of y
alone, sayg(y):
Linear and reducible to linear equations, nding solutions of rst order di erential equations of
the type for applications to orthogonal trajectories, population growth, and nding the current
at a given time.
Unit II: Second order Linear Di erential equations (15 Lectures) Existence and
uniqueness theorems to be stated clearly when needed in the sequel.
Homogeneous and non-homogeneous second order linear di erentiable equations: The space of
solutions of the homogeneous equation as a vector space. Wronskian and linear independence
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of the solutions. The general solution of homogeneous di erential equation. The use of known
solutions to nd the general solution of homogeneous equations. The general solution of a
non-homogeneous second order equation. Complementary functions and particular integrals.
The homogeneous equation which constant coecient, auxiliary equation. The general solution
corresponding to real and distinct roots, real and equal roots and complex roots of the auxiliary
equation.
Non-homogeneous equations: The method of undetermined coecients. The method of varia-
tion of parameters.
Unit III: Linear system of ODEs (15 Lectures)
Existence and uniqueness theorems to be stated clearly when needed in the sequel.
Study of homogeneous linear system of ODEs in two variables: Let a1(t); a2(t); b1(t); b2(t)be
continuous real valued functions de ned on [a;b]:Fixt02[a;b]:Then there exists a unique
solutionx=x(t);y=y(t)valid throughout [a;b]of the following system
dx
dt=a1(t)x+b1(t)y;
dy
dt=a2(t)x+b2(t)y:
satisfying the initial conditions x(t0) =x0&y(t0) =y0:
The Wronskian W(t)of two solutions of a homogeneous linear system of ODEs in two variables,
result:W(t)is identically zero or nowhere zero on [a;b]:Two linearly independent solutions and
the general solution of a homogeneous linear system of ODEs in two variables.
Explicit solutions of Homogeneous linear systems with constant coecients in two variables,
examples.
Recommended Text Books for Unit I and II :
1.G. F. Simmons ,Di erential equations with applications and historical notes , McGraw
Hill.
2.E. A. Coddington ,An introduction to ordinary di erential equations , Dover Books.
Recommended Text Book for Unit III :
G. F. Simmons ,Di erential equations with applications and historical notes , McGraw Hill.
USMT403 INTRODUCTION TO COMPUTING AND PROBLEM
SOLVING-II
Aim of this course:
to introduce Programming as a vehicle to test Algorithms
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and enable the students to write their own Programs.
Unit-I Problem solving strategies (15 lectures)
A. Problem Solving strategies
Problem analysis, formal de nition of problem, Solution, top-down design, breaking a prob-
lem into sub problems, overview of the solution to the sub problems by writing step by step
procedure (algorithm),
owcharts, pseudocodes
B.Python programming language :
1. Variables, expressions and statements Values and types:int,
oat and str
Variables: assignment statements, printing variable values, types of variables.
2. Operators, operands and precedence:+, -, /, *, **, %
PEMDAS(Rules of precedence)
3. String operations: + : Concatenation, * : Repetition
4. Boolean, Comparison and Logical operators:Boolean operator: ==
Comparison operators: ==, !=, >,<,>=,<=
Logical operators: and, or, not
Mathematical functions: sin, cos, tan, log, sqrt etc.
Keyboard input: input() statement
Unit-II: Iterations and Conditional statements (15 lectures)
A. Conditional and alternative statements, Chained and Nested Conditionals:
if, if-else, if-elif-else, nested if, nested if-else
looping statements such as while, for etc
Tables using while.
B. Functions:
Calling functions: type, id
Type conversion:int,
oat, str
Type coercion
Composition of functions
User de ned functions, Parameters and arguments
Unit-III Strings (15 lectures)
A. Elementary Python Graphics such as drawing lines, circles.
B. Strings and Lists in Python
Strings: Compound data types, Length(len function)
String traversal: Using while statement, Using for statement
Comparison operators( >,<. ==)
Lists and List operations
Use of range function Accessing list elements
List membership and for loop
List operations
List updation: addition, removal or updation of elements of a list
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C.Tuples, dictionaries and File handling in Python(5 lectures)
Tuples: De ning a tuple, Index operator, Slice operator, Tuple assignment, Tuple as a return
value
Dictionaries: Creating a Dictionary, Operations on dictionaries (Deletion of elements, addition
of elements, Len function)
Files: Creating a le object, Writing into the le, Closing a le, Opening a le in reading mode,
Reading data from the le
Directories: Opening a le in speci c directory
Exceptions: try and except commands
Recommended Text Books
1.Downey, A. et al., How to think like a Computer Scientist: Learning with Python ,
John Wiley, 2015.
2.Goel, A. ,Computer Fundamentals , Pearson Education.
3.Lambert K. A. ,Fundamentals of Python - First Programs , Cengage Learning India,
2015.
4.Rajaraman, V. ,Computer Basics and C Programming , Prentice-Hall India.
Additional References Books
1.Barry, P. ,Head First Python, O Reilly Publishers.
2.Dromy, R. G. ,How to solve it by Computer , Pearson India.
3.Guzdial, M. J. ,Introduction to Computing and Programming in Python , Pearson
India.
4.Perkovic, L. ,Introduction to Computing Using Python , 2/e, John Wiley, 2015.
5.Sprankle, M. ,Problem Solving & Programming Concepts , Pearson India.
6.Venit, S. and Drake, E. ,Prelude to Programming: Concepts & Design , Pearson
India.
7.Zelle, J. ,Python Programming: An Introduction to Computer Science , Franklin,
Beedle & Associates Inc.
USMTP04/UAMTP04 Practicals
A. Practical for USMT401/UAMT401 :
(1) Calculation of upper sum, lower sum and Riemann integral.
(2) Problems on properties of Riemann integral.
(3) Problems on fundamental theorem of calculus, mean value theorems, integration by parts,
Leibnitz rule.
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(4) Convergence of improper integrals, applications of comparison tests, Abel's and Dirichlet's
tests, and functions.
(5) Sketching of regions in R2andR3;graph of a function, level sets, conversions from one
coordinate system to another.
(6) Double integrals, iterated integrals, applications to compute average value, area, moment,
center of mass and evaluating double integralsRR
f(x;y)dxdy on a rectangle Qsimilar
to following :
(a)Q= [0;1][0;1]andf(x;y) =(
1 x yifx+y1;
0 otherwise:
(b)Q= [0;1][0;1]andf(x;y) =(
x2+y2ifx2+y21;
0 otherwise:
(c)Q= [1;2][1;4]andf(x;y) =(
(x2+y2) 1ifxy2x;
0 otherwise:
(d)Q= [0;1][0;1]andf(x;y) =(
1ifx=y;
0otherwise:
(e)Q= [0;][0;]andf(x;y) =jcos(x+y)j:
(7) Applications of Nested interval theorem.
B. Practical for USMT402/UAMT402 :
1) Solving exact and non exact equations.
2) Linear and reducible to linear equations, applications to orthogonal trajectories, population
growth, and nding the current at a given time.
3) Finding general solution of homogeneous and non-homogeneous equations, use of known
solutions to nd the general solution of homogeneous equations.
4) Solving equations using method of undetermined coecients and method of variation of
parameters.
5) Solving second order linear ODEs.
6) Solving a system of rst order linear ODEs.
C. Practical for USMT403 :
For the Practicals of USMT403, Python version 2.7.9 shall be used by all colleges .
Suggested Practicals for Unit I :
Using Algorithm and
owcharts:
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a) nd factorial of non negative number n
b) nd largest among a given set of numbers
c) generating Fibonacci sequences.
d) arrangennumbers in increasing order.
Installing and setting up of Python IDLE interpreter.
Suggested Practical for Unit II :
Suggested simple exercises:
1. Type Python statements to check the data types of the following:
a) "Hello World!"
b) 1532
c) -265.3665
2. Type Python statements to assign similar to the following values to the variables, check
variable data types and print the variable values:
a) "Hello World!"
b) 1532
c) -265.3665
3. Using Python check output of the statements similar to following :
a)>>> 1+1
b)>>> 17
c)>>> message="Python"
>>> message
d)>>> x=55
>>> x/60
e)>>> 12**2/(5-1)
f)>>> "Hello"*4
g)>>> "Hello "+"World!"
h)>>> id(5)
i)>>> id("Hello")
j)>>> int(-17.3256)
k)>>> str(-17.3256)
l)>>>
oat(55/60)
4. Type a Python code to accept an integer and check whether it's even , odd or prime.
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5. Using Python evaluate Mathematical expressions similar to the following:
a)p
2
b)sin(=2)
c)cos(+=2)whereis entered by the user.
d)e(log10(x))where x is entered by the user.
6. Circles, lines drawing.
7. Roots of quadratic equations.
Suggested Practical for Unit III :
1. Type a python code to display integers ranging from 1 to n.(value of n is entered by the
user)
2. Type a Python code that displays integers 1 to 10 and their squares in the table format
using while statement.
3. Type Python statements to assign the value"Hello World!" to the variable message and
print the following:
a) rst letter of the variable message.
b) length of the variable message.
c) last letter of the variable message.
d) each letter from the string using while statement.
e) each letter from the string using for statement.
4. Check output of the following statements with respect to Python:
a)>>> range(1:5)
b)>>> range(10)
c)>>> range(1, 10, 2)
d) horsemen = ["war", "famine", "pestilence", "death" ]
i= 0
whilei<4:
print(horsemen [i])
i=i+ 1
e)>>>a = [1;2;3]
>>>b = [4;5;6]
>>>c =a+b
>>> print(c)
5. Type a Python code to display all odd numbers from 1 to 20 using lists and for statement.
6. De ne a 33matrix using lists in Python. Type a code to display each row of the matrix
and each element of the matrix.
7. Type a Python code to de ne a function for swapping the two variable values using tuples.
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8. Check output of the following with respect to python:
a)
>>> eng2sp =fg
>>> eng2sp ['one']= 'uno'
>>> eng2sp ['two']= 'dos'
>>> print(eng2sp)
b)
>>> eng2sp =f'one': 'uno', 'two': 'dos', 'three': 'tres' g
>>> print eng2sp
c)
>>> inventory=f'apples':430,'bananas':312,'oranges':525,'pears':217 g
>>> del inventory ['pears' ]
>>> print(inventory)
d)
>>> inventory=f'apples':430,'bananas':312,'oranges':525,'pears':217 g
>>> inventory ['pears' ] = 0
>>> print inventory
e)
>>> inventory=f'apples':430,'bananas':312,'oranges':525,'pears':217 g
>>> len(inventory)
9. Type a Python statement to create a le object test.dat.
10. Type a Python statement to put data "Hello World!" in the le test.dat.
11. Type a Python statement to close the le test.dat.
12. Type a Python statement to read the le test.dat.
13. Type a Python statement to read the le test.dat in a directory named test, which resides in share,
which resides in user, which resides in the top-level directory of the system, called /.
14. Type a Python code to prompt the user for the name of a le and then try to open it. If
the le doesn't exist, we don't want the program to crash.(Use try and except statements.)
15. Write Python programs for the following:
(a) To solve quadratic equation.
(b) To nd factorial of non-negative number n.
(c) to nd largest among a given set of nnumbers (n3):
(d) To nd nth term of Fibonacci sequence.
(e) To nd area of circle.
(f) To nd sum, maximum and minimum of nnumbers.
(g) To arrange nnumbers in increasing order.
(h) To arrange nnames in alphabetic order.
(i) to nd g.c.d. of two integers
(j) recursion for Tower of Hanoi
(k) Selection sort, merge sort.
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Scheme of Examination
I. Semester End Theory Examinations :
There will be a Semester-end external Theory examination of 100 marks for each of the courses
USMT301/UAMT301, USMT302/UAMT302, USMT303 of Semester III and USMT401/UAMT401,
USMT402/UAMT402, USMT403 of semester IV to be conducted by the University.
1. Duration: The examinations shall be of 3Hours duration.
2. Theory Question Paper Pattern:
a) There shall be FIVE questions. The rst question Q1 shall be of objective type for
20 marks based on the entire syllabus. The next three questions Q2, Q3, Q4 shall
be of 20 marks, each based on the units I, II , III respectively. The fth question Q5
shall be of 20 marks based on the entire syllabus.
b) All the questions shall be compulsory. The questions Q2,Q3,Q4,Q5 shall have internal
choices within the questions. Including the choices, the marks for each question shall
be 30-32.
c) The questions Q2,Q3,Q4,Q5 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.
(a) The question Q1 may be subdivided into 10 sub-questions of 2 marks each.
II. 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 and USMT302/UAMT302
are held together by the college. The Practical examination for USMT303 is held separately by
the college.
In semester IV, the Practical examinations for USMT401/UAMT401 and USMT402/UAMT402
are held together by the college. The Practical examination for USMT403 is held separately by
the college.
For the Practicals of USMT403, Python version 2.7.9 shall be used by all colleges.
Paper pattern : The question paper shall have three parts A,B, C. Every part shall have three
questions of 20 marks each. Students to attempt any two question from each part.
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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 USMT303
UAMTP03 Questions Questions - 80 2 hours
from UAMT301 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 and USMT403
1. Journals: 5 marks.
2. Viva: 5 marks.
Each Practical of every course of Semester III & IV shall contain 10 (ten) problems out of which
minimum 05 ( ve) have to be written in the journal.
????????????
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