## S Y BSc Mathematics syllabus2018 32 Syllabus Mumbai University by munotes

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(UNIVERSITY OF MUMBAI)

Syllabus for: S.Y.B.Sc./S.Y.B.A.

Program: B.Sc./B/A.

Course: Mathematics

Choice based Credit System (CBCS)

with eect from the

academic year 2018-19

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SEMESTER III

CALCULUS III

Course Code UNIT TOPICS Credits L/Week

USMT 301, UAMT 301I Functions of several variables

2 3 II Dierentiation

III Applications

ALGEBRA III

USMT 302 ,UAMT 302I Linear Transformations and Matrices

2 3 II Determinants

III Inner Product Spaces

DISCRETE MATHEMATICS

USMT 303I Permutations and Recurrence Relation

2 3 II Preliminary Counting

III Advanced Counting

PRACTICALS

USMTP03Practicals based on3 5USMT301, USMT 302 and USMT 303

UAMTP03Practicals based on2 4UAMT301, UAMT 302

SEMESTER IV

CALCULUS IV

Course Code UNIT TOPICS Credits L/Week

USMT 401, UAMT 401I Riemann Integration

2 3II Indenite Integrals and Improper Integrals

III Beta and Gamma Functions

And Applications

ALGEBRA IV

USMT 402 ,UAMT 402I Groups and Subgroups

2 3 II Cyclic Groups and Cyclic subgroups

III Lagrange's Theorem and Group

Homomorphism

ORDINARY DIFFERENTIAL EQUATIONS

USMT 403I First order First degree

2 3 Dierential equations

II Second order Linear

Dierential equations

III Linear System of Ordinary

Dierential Equations

PRACTICALS

USMTP04Practicals based on3 5USMT401, USMT 402 and USMT 403

UAMTP04Practicals based on2 4UAMT401, UAMT 402

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Teaching Pattern for Semester III

1. Three lectures per week per course. Each lecture is of 48 minutes duration.

2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and

one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed

by the University. Each practical session is of 48 minutes duration.)

Teaching Pattern for Semester IV

1. Three lectures per week per course. Each lecture is of 48 minutes duration.

2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and

one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed

by the University. Each practical session is of 48 minutes duration.)

S.Y.B.Sc. / S.Y.B.A. Mathematics

SEMESTER III

USMT 301, UAMT 301: CALCULUS III

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

examples.

Unit I: Functions of several variables (15 Lectures)

1. The Euclidean inner product on Rnand Euclidean norm function on Rn, distance between

two points, open ball in Rn;denition of an open subset of Rn;neighbourhood of a point

inRn;sequences in Rn, convergence of sequences- these concepts should be specically

discussed for n= 3 andn= 3:

2. Functions from Rn!R(scalar elds) and from Rn!Rm(vector elds), limits,

continuity of functions, basic results on limits and continuity of sum, dierence, scalar

multiples of vector elds, continuity and components of a vector elds.

3. Directional derivatives and partial derivatives of scalar elds.

4. Mean value theorem for derivatives of scalar elds.

Reference for Unit I:

Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10 of Calculus, Vol. 2 (Second Edition) by

Apostol.

Unit II: Dierentiation (15 Lectures)

1. Dierentiability of a scalar eld at a point of Rn(in terms of linear transformation) and

on an open subset of Rn;the total derivative, uniqueness of total derivative of a dier-

entiable function at a point, simple examples of nding total derivative of functions such

asf(x;y) =x2+y2;f(x;y;z ) =x+y+z;, dierentiability at a point of a function f

implies continuity and existence of direction derivatives of fat the point, the existence of

continuous partial derivatives in a neighbourhood of a point implies dierentiability at the

point.

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2. Gradient of a scalar eld, geometric properties of gradient, level sets and tangent planes.

3. Chain rule for scalar elds.

4. Higher order partial derivatives, mixed partial derivatives, sucient condition for equality

of mixed partial derivative.

Reference for Unit II:

Sections 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, 8.23 of Calculus, Vol.2 (Second Edition) by T.

Apostol, John Wiley.

Unit III: Applications (15 lectures)

1. Second order Taylor's formula for scalar elds.

2. Dierentiability of vector elds, denition of dierentiability of a vector eld at a point,

Jacobian matrix, dierentiability of a vector eld at a point implies continuity. The chain

rule for derivative of vector elds (statements only)

3. Mean value inequality.

4. Hessian matrix, Maxima, minima and saddle points.

5. Second derivative test for extrema of functions of two variables.

6. Method of Lagrange Multipliers.

Reference for Unit III:

Sections 8.18, 8.19, 8.20, 8.21, 8.22, 9.9, 9.10, 9.11, 9.12, 9.13, 9.14 9.13, 9.14 from Apostol,

Calculus Vol. 2, (Second Edition) by T. Apostol.

Recommended Text Books:

1. T. Apostol: Calculus, Vol. 2, John Wiley.

2. J. Stewart, Calculus, Brooke/ Cole Publishing Co.

Additional Reference Books

(1) G.B. Thoman and R. L. Finney, Calculus and Analytic Geometry, Ninth Edition, Addison-

Wesley, 1998.

(2) Sudhir R. Ghorpade and Balmohan V. Limaye, A Course in Multivariable Calculus and

Analysis, Springer International Edition.

(3) Howard Anton, Calculus- A new Horizon, Sixth Edition, John Wiley and Sons Inc, 1999.

USMT 302/UAMT 302: ALGEBRA III

Note: Revision of relevant concepts is necessary.

Unit 1: Linear Transformations and Matrices (15 lectures)

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1. Review of linear transformations: Kernel and image of a linear transformation, Rank-

Nullity theorem (with proof), Linear isomorphisms, inverse of a linear isomorphism, Any

ndimensional real vector space is isomorphic to Rn:

2. The matrix units, row operations, elementary matrices, elementary matrices are invertible

and an invertible matrix is a product of elementary matrices.

3. Row space, column space of an mnmatrix, row rank and column rank of a matrix,

Equivalence of the row and the column rank, Invariance of rank upon elementary row or

column operations.

4. Equivalence of rank of an mnmatrixAand rank of the linear transformation LA:

Rn!Rm(LA(X) =AX). The dimension of solution space of the system of linear

equationsAX= 0 equals nrank(A).

5. The solutions of non-homogeneous systems of linear equations represented by AX=B;

Existence of a solution when rank( A)= rank(A;B), The general solution of the system is

the sum of a particular solution of the system and the solution of the associated homoge-

neous system.

Reference for Unit 1: Chapter VIII, Sections 1, 2 of Introduction to Linear Algebra,

Serge Lang, Springer Verlag and Chapter 4, of Linear Algebra A Geometric Approach, S.

Kumaresan, Prentice-Hall of India Private Limited, New Delhi.

Unit II: Determinants (15 Lectures)

1. Denition of determinant as an nlinear skew-symmetric function from RnRn:::

Rn!Rsuch that determinant of ( E1;E2;:::;En) is 1, where Ejdenotes the jthcolumn

of thennidentity matrix In:Determinant of a matrix as determinant of its column

vectors (or row vectors). Determinant as area and volume.

2. Existence and uniqueness of determinant function via permutations, Computation of de-

terminant of 22;33 matrices, diagonal matrices, Basic results on determinants such

as det(At) = det(A);det(AB) = det(A) det(B);Laplace expansion of a determinant, Van-

dermonde determinant, determinant of upper triangular and lower triangular matrices.

3. Linear dependence and independence of vectors in Rnusing determinants, The existence

and uniqueness of the system AX=B, whereAis annnmatrix wither det( A)6= 0, Co-

factors and minors, Adjoint of an nnmatrixA, Basic results such as Aadj(A) = det(A)In:

Annnreal matrix Ais invertible if and only if det( A)6= 0;A1=1

det(A)adj(A) for an

invertible matrix A, Cramer's rule.

4. Determinant as area and volume.

References for Unit 2: Chapter VI of Linear Algebra A geometric approach, S. Kumaresan,

Prentice Hall of India Private Limited, 2001 and Chapter VII Introduction to Linear Algebra,

Serge Lang, Springer Verlag.

Unit III: Inner Product Spaces (15 Lectures)

1. Dot product in Rn;Denition of general inner product on a vector space over R:Examples

of inner product including the inner product < f;g > =Z

f(t)g(t)dtonC[;];the

space of continuous real valued functions on [ ;].

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2. Norm of a vector in an inner product space. Cauchy-Schwartz inequality, Triangle in-

equality, Orthogonality of vectors, Pythagoras theorem and geometric applications in R2;

Projections on a line, The projection being the closest approximation, Orthogonal com-

plements of a subspace, Orthogonal complements in R2andR3. Orthogonal sets and

orthonormal sets in an inner product space, Orthogonal and orthonormal bases. Gram-

Schmidt orthogonalization process, Simple examples in R3;R4.

Reference of Unit 3: Chapter VI, Sections 1,2 of Introduction to Linear Algebra, Serge

Lang, Springer Verlag and Chapter 5, of Linear Algebra A Geometric Approach, S. Kumaresan,

Prentice-Hall of India Private Limited, New Delhi.

Recommended Books:

1. Serge Lang: Introduction to Linear Algebra, Springer Verlag.

2. S. Kumaresan: Linear Algebra A geometric approach, Prentice Hall of India Private Lim-

ited.

Additional Reference Books:

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

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

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

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

5. A. Ramachandra Rao and P. Bhima Sankaran: Linear Algebra, Tata McGraw-Hill, New

Delhi.

6. T. Bancho and J. Wermer: Linear Algebra through Geometry, Springer Verlag Newyork,

1984.

7. Sheldon Axler: Linear Algebra done right, Springer Verlag, Newyork.

8. Klaus Janich: Linear Algebra.

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

10. Gareth Williams: Linear Algebra with Applications, Narosa Publication.

USMT 303: Discrete Mathematics

Unit I: Permutations and Recurrence relation (15 lectures)

1. Permutation of objects, Sn, composition of permutations, results such as every permutation

is a product of disjoint cycles, every cycle is a product of transpositions, even and odd

permutation, rank and signature of a permutation, cardinality of Sn;An

2. Recurrence Relations, denition of non-homogeneous, non-homogeneous, linear , non-

linear recurrence relation, obtaining recurrence relation in counting problems, solving

homogeneous as well as non homogeneous recurrence relations by using iterative meth-

ods, solving a homogeneous recurrence relation of second degree using algebraic method

proving the necessary result.

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Recommended Books:

1. Norman Biggs: Discrete Mathematics, Oxford University Press.

2. Richard Brualdi: Introductory Combinatorics, John Wiley and sons.

3. V. Krishnamurthy: Combinatorics-Theory and Applications, Aliated East West Press.

4. Discrete Mathematics and its Applications, Tata McGraw Hills.

5. Schaum's outline series: Discrete mathematics,

6. Applied Combinatorics: Allen Tucker, John Wiley and Sons.

Unit II: Preliminary Counting (15 Lectures)

1. Finite and innite sets, countable and uncountable sets examples such as N;Z;NN;Q;(0;1);R

2. Addition and multiplication Principle, counting sets of pairs, two ways counting.

3. Stirling numbers of second kind. Simple recursion formulae satised by S(n;k) fork=

1;2;;n1;n

4. Pigeonhole principle and its strong form, its applications to geometry, monotonic sequences

etc.

Unit III: Advanced Counting (15 Lectures)

1. Binomial and Multinomial Theorem, Pascal identity, examples of standard identities such

as the following with emphasis on combinatorial proofs.

rX

k=0m

kn

rk

=m+n

r

nX

i=ri

r

=n+ 1

r+ 1

kX

i=0k

i2

=2k

k

nX

i=0n

i

= 2n

2. Permutation and combination of sets and multi-sets, circular permutations, emphasis on

solving problems.

3. Non-negative and positive solutions of equation x1+x2++xk=n

4. Principal of inclusion and exclusion, its applications, derangements, explicit formula for

dn, deriving formula for Euler's function (n).

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USMT P03/UAMTP03 Practicals

Suggested Practicals for USMT 301/UAMT303

1. Sequences in R2andR3;limits and continuity of scalar elds and vector elds, using

\denition and otherwise" , iterated limits.

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

elds.

3. Total derivative, gradient, level sets and tangent planes.

4. Chain rule, higher order derivatives and mixed partial derivatives of scalar elds.

5. Taylor's formula, dierentiation of a vector eld at a point, nding Hessian/Jacobean

matrix, Mean Value Inequality.

6. Finding maxima, minima and saddle points, second derivative test for extrema of functions

of two variables and method of Lagrange multipliers.

7. Miscellaneous Theoretical Questions based on full paper

Suggested Practicals for USMT302/UAMT302:

1. Rank-Nullity Theorem.

2. System of linear equations.

3. Determinants , calculating determinants of 2 2 matrices, nndiagonal, upper triangular

matrices using denition and Laplace expansion.

4. Finding inverses of nnmatrices using adjoint.

5. Inner product spaces, examples. Orthogonal complements in R2andR3.

6. Gram-Schmidt method.

7. Miscellaneous Theoretical Questions based on full paper

Suggested Practicals for USMT 303:

1. Derangement and rank signature of permutation.

2. Recurrence relation.

3. Problems based on counting principles, Two way counting.

4. Stirling numbers of second kind, Pigeon hole principle.

5. Multinomial theorem, identities, permutation and combination of multi-set.

6. Inclusion-Exclusion principle. Euler phi function.

7. Miscellaneous theory quesitons from all units.

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SEMESTER IV

USMT 401/UAMT 401: CALCULUS IV

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

examples.

Unit I: Riemann Integration (15 Lectures)

Approximation of area, Upper/Lower Riemann sums and properties, Upper/Lower integrals,

Denition of Riemann integral on a closed and bounded interval, Criterion of Riemann in-

tegrability, if a < c < b thenf2R[a;b];if and only if f2R[a;c] andf2R[c;b] andZb

af=Zc

af+Zb

cf.

Properties:

(i)f;g2R[a;b] =)f+g;f2R[a;b].

(ii)Zb

a(f+g) =Zb

af+Zb

ag:

(iii)Zb

af=Zb

af:

(iv)f2R[a;b] =)jfj2R[a;b] andjZb

afjZb

ajfj,

(v)f0;f2C[a;b] =)f2R[a;b].

(vi) Iffis bounded with nite number of discontinuities then f2R[a;b], generalize this if f

is monotone then f2R[a;b]:

Unit II: Indenite and improper integrals (15 lectures)

Continuity of F(x) =Zx

af(t)dtwheref2R[a;b], Fundamental theorem of calculus, Mean

value theorem, Integration by parts, Leibnitz rule, Improper integrals-type 1 and type 2, Abso-

lute convergence of improper integrals, Comparison tests, Abel's and Dirichlet's tests.

Unit III: Applications (15 lectures)

(1)and functions and their properties, relationship between and functions (without

pro).

(2) Applications of denite Integras: Area between curves, nding volumes by sicing, volumes

of solids of revolution-Disks and Washers, Cylindrical Shells, Lengths of plane curves, Ar-

eas of surfaces of revolution.

References:

(1) Calculus Thomas Finney, ninth edition section 5.1, 5.2, 5.3, 5.4, 5.5, 5.6.

(2) R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

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(3) Ajit Kumar, S.Kumaresan, A Basic Course in Real Analysis, CRC Press, 2014.

(4) T. Apostol, Calculus Vol.2, John Wiley.

(5) K. Stewart, Calculus, Booke/Cole Publishing Co, 1994.

(6) J. E. Marsden, A.J. Tromba and A. Weinstein, Basic multivariable calculus.

(7) Bartle and Sherbet, Real analysis.

USMT 402/ UAMT 402: ALGEBRA IV

Unit I: Groups and Subgroups (15 Lectures)

(a) Denition of a group, abelian group, order of a group, nite and innite groups.

Examples of groups including:

i)Z;Q;R;Cunder addition.

ii)Q(=Qnf0g);R(=Rnf0g);C(=Cnf0g):Q+(= positive rational numbers)

under multiplication.

iii)Zn;the set of residue classes modulo nunder addition.

iv)U(n);the group of prime residue classes modulo nunder multiplication.

v) The symmetric group Sn:

vi) The group of symmetries of a plane gure. The Dihedral group Dnas the group

of symmetries of a regular polygon of nsides (forn= 3;4).

vii) Klein 4-group.

viii) Matrix groups Mnn(R) under addition of matrices, GLn(R);the set of invertible

real matrices, under multiplication of matrices.

ix) Examples such as S1as subgroup of C;nthe subgroup of nth roots of unity.

(b) Properties such as

1) In a group ( G;:) the following indices rules are true for all integers n;m:

i)anam=an+mfor allainG:

ii) (an)m=anmfor allainG:

iii) (ab)n=anbnfor allabinGwheneverab=ba:

2) In a group ( G;:) the following are true:

i) The identity element eofGis unique.

ii) The inverse of every element in Gis unique.

iii) (a1)1=afor allainG:

iv) (a:b)1=b1a1for alla;binG:

v) Ifa2=efor everyainGthen (G;:) is an abelian group.

vi) (aba1)n=abna1for everya;binGand for every integer n:

vii) If (a:b)2=a2:b2for everya;binGthen (G;:) is an abelian group.

viii) (Z

n;:) is a group if and only if nis a prime.

3) Properties of order of an element such as: ( nandmare integers.)

i) Ifo(a) =nthenam=eif and only if n=m.

ii) Ifo(a) =nmtheno(an) =m.

iii) Ifo(a) =ntheno(am) =n

(n;m);. where (n;m) is the GCD of nandm.

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iv)o(aba1) =o(b) ando(ab) =o(ba).

v) Ifo(a) =mando(b) =m;ab =ba;(n;m) = 1 theno(ab) =nm.

(c) Subgroups

i) Denition, necessary and sucient condition for a non-empty set to be a Sub-

group.

ii) The center Z(G) of a group is a subgroup.

iii) Intersection of two (or a family of ) subgroups is a subgroup.

iv) Union of two subgroups is not a subgroup in general. Union of two subgroups is

a subgroup if and only if one is contained in the other.

v) IfHandKare subgroups of a group GthenHK is a subgroup of Gif and only

ifHK =KH.

Reference for Unit I:

(1) I.N. Herstein, Topics in Algebra.

(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.

Unit II: Cyclic groups and cyclic subgroups (15 Lectures)

(a) Cyclic subgroup of a group, cyclic groups, (examples including Z;Znandn).

(b) Properties such as:

(i) Every cyclic group is abelian.

(ii) Finite cyclic groups, innite cyclic groups and their generators.

(iii) A nite cyclic group has a unique subgroup for each divisor of the order of the group.

(iv) Subgroup of a cyclic group is cyclic.

(v) In a nite group G;G = if and only if o(G) =o(a).

(vi) IfG= ando(a) =nthenG=

(vii) IfGis a cyclic group of order pnandH

References for Unit II:

(1) I.N. Herstein, Topics in Algebra.

(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.

Unit III: Lagrange's Theorem and Group homomorphism (15 Lectures)

(a) Denition of Coset and properties such as :

1) IFHis a subgroup of a group Gandx2Gthen

(i)xH=Hif and only if x2H.

(ii)Hx=Hif and only if x2H:

2) IfHis a subgroup of a group Gandx;y2Gthen

(i)xH=yHif and only if x1y2H:

(ii)Hx=Hyif and only if xy12H:

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3) Lagrange's theorem and consequences such as Fermat's Little theorem, Euler's theo-

rem and if a group Ghas no nontrivial subgroups then order of Gis a prime and G

is Cyclic.

(b) Group homomorphisms and isomorphisms, automorphisms

i) Denition.

ii) Kernel and image of a group homomorphism.

iii) Examples including inner automorphism.

Properties such as:

(1)f:G!G0is a group homomorphism then ker f

(3)f:G!G0is a group homomorphism then

(i)Gis abelian if and only if G0is abelian.

(ii)Gis cyclic if and only if G0is cyclic.

Reference for Unit III:

1. I.N. Herstein, Topics in Algebra.

2. P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.

Recommended Books:

1. I.N. Herstein, Topics in Algebra, Wiley Eastern Limied, Second edition.

2. N.S. Gopalkrishnan, University Algebra, Wiley Eastern Limited.

3. M. Artin, Algebra, Prentice Hall of India, New Delhi.

4. P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra, Second edition, Foundation

Books, New Delhi, 1995.

5. J.B. Fraleigh, A rst course in Abstract Algebra, Third edition, Narosa, New Delhi.

6. J. Gallian. Contemporary Abstract Algebra. Narosa, New Delhi.

7. COmbinatroial Techniques by Sharad S. Sane, Hindustan Book Agency.

Additional Reference Books:

1. S. Adhikari. An introduction to Commutative Algebra and Number theory. Narosa Pub-

lishing House.

2. T. W. Hungerford. Algebra, Springer.

3. D. Dummit, R. Foote. Abstract Algebra, John Wiley & Sons, Inc.

4. I.S. Luther, I.B.S. Passi. Algebra. Vol. I and II.

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USMT 403: ORDINARY DIFFERENTIAL EQUATIONS

Unit I: First order First degree Dierential equations (15 Lectures)

(1) Denition of a dierential equation, order, degree, ordinary dierential equation and partial

dierential equation, linear and non linear ODE.

(2) Existence and Uniqueness Theorem for the solution of a second order initial value prob-

lem (statement only), Denition of Lipschitz function, Examples based on verifying the

conditions of existence and uniqueness theorem

(3) Review of Solution of homogeneous and non-homogeneous dierential equations of rst

order and rst degree. Notion of partial derivatives. 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

M x +N yis an I.F. if M x +N y6= 0 andMdx +Ndy = 0 is homogeneous.

ii)1

M xN yis an I.F. if M xN y6= 0 andMdx +Ndy = 0 is of the form

f1(x;y)y dx +f2(x;y)x dy = 0:

iii)eR

f(x)dx(respeR

g(y)dy) is an I.F. if N6= 0 (respM6= 0) and1

N@M

@y@N

@x

resp1

M@M

@y@N

@x

is a function of x(respy) alone, say f(x) (respg(y)).

iv) Linear and reducible linear equations of rst order, nding solutions of rst order dif-

ferential equations of the type for applications to orthogonal trajectories, population

growth, and nding the current at a given time.

Unit II: Second order Linear Dierential equations (15 Lectures)

1. Homogeneous and non-homogeneous second order linear dierentiable equations: The

space of solutions of the homogeneous equation as a vector space. Wronskian and linear

independence of the solutions. The general solution of homogeneous dierential equa-

tions. The general solution of a non-homogeneous second order equation. Complementary

functions and particular integrals.

2. The homogeneous equation with constant coecients. auxiliary equation. The general

solution corresponding to real and distinct roots, real and equal roots and complex roots

of the auxiliary equation.

3. Non-homogeneous equations: The method of undetermined coecients. The method of

variation 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 dened on [ a;b]. Fixt02[a;b]. Then there exists a unique solution

x=x(t);y=y(t) valid throughout [ a;b] of the following system:

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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, Dierential equations with applications and historical notes, McGraw Hill.

2. E. A. Coddington, An introduction to ordinary dierential equations, Dover Books.

Recommended Text Book for Unit III:

G. F. Simmons, Dierential equations with applications and historical notes, McGraw Hill.

USMT P04/UAMT P04 Practicals.

Suggested Practicals 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.

4. Convergence of improper integrals, applications of comparison tests, Abel's and Dirichlet's

tests, and functions.

5. Beta Gamma Functions

6. Problems on area, volume, length.

7. Miscellaneous Theoretical Questions based on full paper.

Suggested Practicals for USMT402/UAMT 402:

1. Examples and properties of groups.

2. Group of symmetry of equilateral triangle, rectangle, square.

3. Subgroups.

4. Cyclic groups, cyclic subgroups, nding generators of every subgroup of a cyclic group.

5. Left and right cosets of a subgroup, Lagrange's Theorem.

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6. Group homomorphisms, isomorphisms.

7. Miscellaneous Theoretical questions based on full paper.

Suggested Practicals for USMT403:

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.

7. Miscellaneous Theoretical questions from all units.

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 3 Hours 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, Q2, Q3

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.

d) 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 and 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 and IV, Practical examinations of three hours duration

and 150 marks shall be conducted for the courses UAMTP03, UAMTP04.

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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 sepa-

rately 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 sepa-

rately by the college.

Paper pattern: The question paper shall have three parts A, B, C.

Each part shall have two Sections.

Section I Objective in nature: Attempt any Eight out of Twelve multiple choice ques-

tions. (83 = 24 Marks)

Section II Problems: Attempt any Two out of Three. (8 2 = 16 Marks)

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 from UAMT302

USMTP04 Questions Questions Questions 120 3 hours

from USMT401 from USMT402 from USMT403

UAMTP03 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 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.