# VIDYASAGAR UNIVERSITY Mathematics

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VIDYASAGAR UNIVERSITY

Curriculum for 3-Year BSc (General) in

Mathematics

Under Choice Based Credit System (CBCS) [w.e.f 2018-2019]

1

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Year 1

Semester I

Course Type

Core-1 (DSC-1A) Core-2 (DSC-2A) Core-3 (DSC-3A) AECC-1 (Elective)

Course Code

VIDYASAGAR UNIVERSITY

B Sc (General) in Mathematics

[Choice Based Credit System]

Course Title

Credit

SEMESTER-I

Differential Calculus

6

Other Discipline/TBD

6

Other Discipline/TBD

6

English/MIL

2

Semester - I : Total

20

II

SEMESTER-II

Core-4

Differential Equations

6

(DSC-1B)

Core-5

Other Discipline/TBD

6

(DSC-2B)

Core-6

Other Discipline/TBD

6

(DSC-3B)

AECC-2

Environmental Studies

4

(Elective)

Semester - 2 : Total

22

L-T-P

5-1-0

4-0-4/ 5-1-0 4-0-4/ 5-1-0

1-1-0

5-1-0

4-0-4/ 5-1-0 4-0-4/ 5-1-0

Marks CA ESE TOTAL 15 60 75 15 60 75 15 60 75 10 40 50

275

15 60 75 15 60 75 15 60 75 20 80 100

325

2

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Year 2

Semester III

Course Type

Core-7 (DSC-1C) Core-8 (DSC-2C) Core-9 (DSC-3C) SEC-1

IV

Core-10 (DSC-1D) Core-11 (DSC-2D) Core-12 (DSC-3D) SEC-2

Course Code

Course Title

SEMESTER-III Real Analysis Other Discipline/TBD Other Discipline/TBD TBD

Semester - 3 : Total

Algebra

SEMESTER-IV

Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 4 : Total

Credit L-T-P

Marks

CA ESE TOTAL 6 5-1-0 15 60 75

6

4-0-4/ 15 60

75

5-1-0

6

4-0-4/ 15 60

75

5-1-0

2 1-1-0 10 40 50

20

275

6 5-1-0 15 60 75

6

4-0-4/ 15 60

75

5-1-0

6

4-0-4/ 15 60

75

5-1-0

2 1-1-0 10 40 50

20

275

3

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Year 3

Semester V

Course Type

DSE-1A DSE-2A

DSE-3A

SEC-3

Course Code

Course Title

SEMESTER-V Discipline-1(Mathematics) Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 5 : Total

Credit L-T-P

Marks

CA ESE TOTAL

6 5-1-0 15 60 75

6 4-0-4/ 15 60 75

5-1-0

6 4-0-4/ 15 60 75

5-1-0

2 1-1-0 10 40 50

20

275

VI

DSE-1B DSE-2B

DSE-3B

SEC-4

SEMESTER-VI Discipline-1(Mathematics) Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 6 : Total

6 5-1-0 15 60 75

6 4-0-4/ 15 60 75

5-1-0

6 4-0-4/ 15 60 75

5-1-0

2 1-1-0 10 40 50

20

275

Total in all semester:

122

1700

CC = Core Course , AECC = Ability Enhancement Compulsory Course , GE = Generic Elective , SEC = Skill Enhancement Course , DSE = Discipline

Specific Elective , CA= Continuous Assessment , ESE= End Semester Examination , TBD=To be decided , CT = Core Theory, CP=Core Practical , L =

Lecture, T = Tutorial ,P = Practical , MIL = Modern Indian Language , ENVS = Environmental Studies ,

4

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List of Core and Elective Courses Core Courses (CC)

DSC-1A: Differential Calculus DSC-1B: Differential Equations DSC-1C: Real Analysis DSC-1D: Algebra

Discipline Specific Electives (DSE) DSE-1: Complex Analysis Or DSE-1: Matrices Or DSE-1: Linear Algebra Or DSE-1: Vector Calculus and Analytical Geometry

DSE-2: Mechanics Or DSE-2: Linear Programming Or DSE-2: Numerical Methods Or DSE-2: Integer Programming and Theory of Games

Skill Enhancement Course (SEC) SEC-1: Theory of Equation Or SEC-1: Logic and Sets Or SEC-1: Boolean Algebra

SEC-2: Graph Theory Or SEC-2: Integral Calculus Or SEC-2: Mathematical Finance

SEC-2: Number Theory Or SEC-3: Bio-Mathematics Or SEC-3: Mathematical Modeling

SEC-4: Probability and Statistics Or SEC-4: Understanding Probability and Statistics through practical Or SEC-4: Forecasting Or SEC-4: Portfolio Optimization

5

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Core Courses (CC)

DSC-1A(CC-1): Differential Calculus

Credit : 06

DSC-1AT(CC-1): Differential Calculus

Course Contents:

Limit and Continuity (ε and δ definition), Types of discontinuities, Differentiability of functions, Successive differentiation, Leibnitz’s theorem, Partial differentiation, Euler’s theorem on homogeneous functions. Tangents and normals, Curvature, Asymptotes, Singular points, Tracing of curves. Parametric representation of curves and tracing of parametric curves, Polar coordinates and tracing of curves in polar coordinates. Rolle’s theorem, Mean Value theorems, Lagrange and cauchy theorems. Taylor’s theorem with Lagrange’s and Cauchy’s forms of remainder, Power series and its convergences. Taylor’s series, Maclaurin’s series of sin x, cos x, ex, log(l+x), (l+x)m, Maxima and Minima, Indeterminate forms.

Suggested Readings: H. Anton, I. Birens and S. Davis, Calculus, John Wiley and Sons, Inc., 2002. G.B. Thomas and R.L. Finney, Calculus, Pearson Education, 2007.

DSC-1B(CC-2): Differential Equations

Credit : 06

DSC1BT(CC-2) : Differential Equations

Course Contents:

First order exact differential equations. Integrating factors, rules to find an integrating factor. First order higher degree equations solvable for x, y, p. Methods for solving higherorder differential equations. Basic theory of linear differential equations, Wronskian, and its properties. Solving a differential equation by reducing its order. Linear homogenous equations with constant coefficients, Linear non-homogenous equations, The method of variation of parameters, The Cauchy-Euler equation, Simultaneous differential equations, Total differential equations. Order and degree of partial differential equations, Concept of linear and non-linear partial differential equations, Formation of first order partial differential equations, Linear partial differential equation of first order, Lagrange’s method, Charpit’s method. Classification of second order partial differential equations into elliptic, parabolic and hyperbolic through illustrations only.

Suggested Readings: Shepley L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, 1984. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, International

Edition, 1967.

6

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DSC-1C(CC-3): Real Analysis

Credit : 06

DSC1CT(CC-3): Real Analysis

Course Contents:

Finite and infinite sets, examples of countable and uncountable sets. Real line, bounded sets, suprema and infima, completeness property of R, Archimedean property of R, intervals. Concept of cluster points and statement of Bolzano-Weierstrass theorem.Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences. Cauchy’s theorem on limits, order preservation and squeeze theorem, monotone sequences and their convergence (monotone convergence theorem without proof). Infinite series. Cauchy convergence criterion for series, positive term series, geometric series, comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test (Tests of Convergence without proof). Definition and examples of absolute and conditional Convergence Series. Sequences and series of functions, Pointwise and uniform convergence. µ-test, M-test, Statements of the results about uniform convergence and integrability and differentiability of functions, Power series and radius of convergence.

Suggested Readings: T.M. Apostol, Calculus (Vol. I), John Wiley and Sons (Asia) P. Ltd., 2002. R.G. Bartle and D. R Sherbert, Introduction to Real Analysis, John Wiley and Sons

(Asia) P. Ltd., 2000. E. Fischer, Intermediate Real Analysis, Springer Verlag, 1983. K.A. Ross,(2003). Elementary Analysis - The Theory of Calculus Series-

Undergraduate Texts in Mathematics, Springer Verlag. S.K.Mapa Introduction to real analysis, Sarat Book House.

DSC-1D (CC-4): Algebra

Credits 06

DSC1DT(CC-4): Algebra

Course Contents:

Definition and examples of groups, examples of abelian and non-abelian groups, the group Zn of integers under addition modulo n and the group U(n) of units under multiplication modulo n. Cyclic groups from number systems, complex roots of unity, circle group, the general linear group GLn (R), groups of symmetries of (i) an isosceles triangle, (ii) an equilateral triangle, (iii) a rectangle, and (iv) a square, the permutation group Sym (n), Group of quaternions. Subgroups, cyclic subgroups, the concept of a subgroup generated by a subset and the commutator subgroup of group, examples of subgroups including the center of a group. Cosets, Index of subgroup, Lagrange’s theorem, order of an element, Normal subgroups: their definition, examples, and characterizations, Quotient groups.Definition and examples of rings, examples of commutative and non-commutative rings: rings from number systems, Zn the ring of integers modulo n, ring of real quaternions, rings of matrices, polynomial rings, and rings of continuous functions. Subrings and ideals, Integral domains and fields, examples of fields: Zp, Q, R, and C. Field of rational functions.

7

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Suggested Readings: John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. Joseph A Gallian, Contemporary Abstract Algebra, Narosa, 1999. George E Andrews, Number Theory, Hindustan Publishing Corporation, 1984. S.K.Mapa. Higher Algebra : Abstract and Linear, Sarat Book House, Calcutta.

Discipline Specific Electives (DSE)

DSE-1 : Complex Analysis

Credit : 06

DSE1T : Complex Analysis

Course Contents:

Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy - Goursat theorem, Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and its examples, absolute and uniform convergence of power series. Singular points of all kind, residue, Cauchy residue theorem and example.

Suggested Readings: James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed., McGraw – Hill International Edition, 2009. Joseph Bak and Donald J. Newman, Complex analysis, 2nd Ed., Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., New York, 1997.

Or

DSE - 1: Matrices

Credit : 06

DSE1T: Matrices

Course Contents:

R, R2, R3 as vector spaces over R. Standard basis for each of them. Concept of Linear Independence and examples of different bases. Subspaces of R2, R3. Translation, Dilation, Rotation, Reflection in a point, line and plane. Matrix form of basic geometric transformations. Interpretation of eigen values and eigenvectors for such transformations and eigen spaces as invariant subspaces. Matrices in diagonal form. Reduction to diagonal form upto matrices of order 3. Computation of matrix inverses using elementary row operations. Rank of matrix. Solutions of a system of linear equations using matrices.

8

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Illustrative examples of above concepts from Geometry, Physics, Chemistry, Combinatorics and Statistics.

Suggested Readings: A.I. Kostrikin, Introduction to Algebra, Springer Verlag, 1984. S. H. Friedberg, A. L. Insel and L.E. Spence, Linear Algebra, Prentice Hall of India Pvt. Ltd., New Delhi, 2004. Richard Bronson, Theory and Problems of Matrix Operations, Tata McGraw Hill, 1989. M. Pal (2013). Higher Algebra, PHI Learning Pvt. Ltd.

Or

DSE-1 : Linear Algebra

Credit : 06

DSE1T: Linear Algebra

Course Contents:

Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces. Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations. Isomorphisms, Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix.

Suggested Readings: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th

Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education

Asia, Indian Reprint, 2007. S. Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005. Gilbert Strang, Linear Algebra and its Applications, Thomson, 2007. M.Pal. Higher Algebra. PHI Pvt. Ltd. 2013.

Or

DSE - 1: Vector Calculus and Analytical Geometry

Credit : 06

DSE1T: Vector Calculus and Analytical Geometry

Course Contents:

Algebra of vectors, Differentiation and partial differentiation of a vector function. Derivative of sum, dot product and cross product of two vectors. Gradient, divergence and curl. Techniques for sketching parabola, ellipse and hyperbola. Reflection properties of parabola, ellipse and hyperbola. Classification of quadratic equations representing lines, parabola, ellipse and hyperbola. Spheres, Cylindrical surfaces. Illustrations of graphing standard quadric surfaces like cone, ellipsoid.

9

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Suggested Readings:

G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. H. Anton, I. Bivens and S. Davis, Calculus, John Wiley and Sons (Asia) P. Ltd.

2002. P.C. Matthew’s, Vector Calculus, Springer Verlag London Limited, 1998. S.L. Loney, The Elements of Coordinate Geometry, McMillan and Company,

London. R.Khan. Analytical Geometry and Vector Algebra, New Central Book Agency. R.J.T. Bill, Elementary Treatise on Coordinate Geometry of Three Dimensions,

McMillan India Ltd., 1994.

DSE-2 : Mechanics

Credit : 6

DSE2T: Mechanics

Course Contents:

Conditions of equilibrium of a particle and of coplanar forces acting on a rigid Body, Laws of friction, Problems of equilibrium under forces including friction, Centre of gravity, Work and potential energy. Velocity and acceleration of a particle along a curve: radial and transverse components (plane curve), tangential and normal components (space curve), Newton’s Laws of motion, Simple harmonic motion, Simple Pendulum, Projectile Motion.

Suggested Readings: A.S. Ramsay, Statics, CBS Publishers and Distributors (Indian Reprint), 1998. A.P. Roberts, Statics and Dynamics with Background in Mathematics, Cambridge University Press, 2003. S.L.Loney. An elementary treaties on the dynamics of a particle and rigid body, New Age International Pvt. Ltd.

Or

DSE-2: Linear Programming

Credit: 06

DSE2T: Linear Programming

Course Contents:

Linear Programming: Definition and formation Problems, Graphical Approach for solving some Linear Programming problems. Convex Sets, Supporting and Separating Hyperplanes. Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex method in tableau format, introduction to artificial variables, twophase method, Big-M method and their comparison. Duality, formulation of the dual problem, primal- dual relationships, economic interpretation of the dual.

10

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Curriculum for 3-Year BSc (General) in

Mathematics

Under Choice Based Credit System (CBCS) [w.e.f 2018-2019]

1

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Year 1

Semester I

Course Type

Core-1 (DSC-1A) Core-2 (DSC-2A) Core-3 (DSC-3A) AECC-1 (Elective)

Course Code

VIDYASAGAR UNIVERSITY

B Sc (General) in Mathematics

[Choice Based Credit System]

Course Title

Credit

SEMESTER-I

Differential Calculus

6

Other Discipline/TBD

6

Other Discipline/TBD

6

English/MIL

2

Semester - I : Total

20

II

SEMESTER-II

Core-4

Differential Equations

6

(DSC-1B)

Core-5

Other Discipline/TBD

6

(DSC-2B)

Core-6

Other Discipline/TBD

6

(DSC-3B)

AECC-2

Environmental Studies

4

(Elective)

Semester - 2 : Total

22

L-T-P

5-1-0

4-0-4/ 5-1-0 4-0-4/ 5-1-0

1-1-0

5-1-0

4-0-4/ 5-1-0 4-0-4/ 5-1-0

Marks CA ESE TOTAL 15 60 75 15 60 75 15 60 75 10 40 50

275

15 60 75 15 60 75 15 60 75 20 80 100

325

2

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Year 2

Semester III

Course Type

Core-7 (DSC-1C) Core-8 (DSC-2C) Core-9 (DSC-3C) SEC-1

IV

Core-10 (DSC-1D) Core-11 (DSC-2D) Core-12 (DSC-3D) SEC-2

Course Code

Course Title

SEMESTER-III Real Analysis Other Discipline/TBD Other Discipline/TBD TBD

Semester - 3 : Total

Algebra

SEMESTER-IV

Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 4 : Total

Credit L-T-P

Marks

CA ESE TOTAL 6 5-1-0 15 60 75

6

4-0-4/ 15 60

75

5-1-0

6

4-0-4/ 15 60

75

5-1-0

2 1-1-0 10 40 50

20

275

6 5-1-0 15 60 75

6

4-0-4/ 15 60

75

5-1-0

6

4-0-4/ 15 60

75

5-1-0

2 1-1-0 10 40 50

20

275

3

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Year 3

Semester V

Course Type

DSE-1A DSE-2A

DSE-3A

SEC-3

Course Code

Course Title

SEMESTER-V Discipline-1(Mathematics) Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 5 : Total

Credit L-T-P

Marks

CA ESE TOTAL

6 5-1-0 15 60 75

6 4-0-4/ 15 60 75

5-1-0

6 4-0-4/ 15 60 75

5-1-0

2 1-1-0 10 40 50

20

275

VI

DSE-1B DSE-2B

DSE-3B

SEC-4

SEMESTER-VI Discipline-1(Mathematics) Other Discipline/TBD

Other Discipline/TBD

TBD Semester - 6 : Total

6 5-1-0 15 60 75

6 4-0-4/ 15 60 75

5-1-0

6 4-0-4/ 15 60 75

5-1-0

2 1-1-0 10 40 50

20

275

Total in all semester:

122

1700

CC = Core Course , AECC = Ability Enhancement Compulsory Course , GE = Generic Elective , SEC = Skill Enhancement Course , DSE = Discipline

Specific Elective , CA= Continuous Assessment , ESE= End Semester Examination , TBD=To be decided , CT = Core Theory, CP=Core Practical , L =

Lecture, T = Tutorial ,P = Practical , MIL = Modern Indian Language , ENVS = Environmental Studies ,

4

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List of Core and Elective Courses Core Courses (CC)

DSC-1A: Differential Calculus DSC-1B: Differential Equations DSC-1C: Real Analysis DSC-1D: Algebra

Discipline Specific Electives (DSE) DSE-1: Complex Analysis Or DSE-1: Matrices Or DSE-1: Linear Algebra Or DSE-1: Vector Calculus and Analytical Geometry

DSE-2: Mechanics Or DSE-2: Linear Programming Or DSE-2: Numerical Methods Or DSE-2: Integer Programming and Theory of Games

Skill Enhancement Course (SEC) SEC-1: Theory of Equation Or SEC-1: Logic and Sets Or SEC-1: Boolean Algebra

SEC-2: Graph Theory Or SEC-2: Integral Calculus Or SEC-2: Mathematical Finance

SEC-2: Number Theory Or SEC-3: Bio-Mathematics Or SEC-3: Mathematical Modeling

SEC-4: Probability and Statistics Or SEC-4: Understanding Probability and Statistics through practical Or SEC-4: Forecasting Or SEC-4: Portfolio Optimization

5

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Core Courses (CC)

DSC-1A(CC-1): Differential Calculus

Credit : 06

DSC-1AT(CC-1): Differential Calculus

Course Contents:

Limit and Continuity (ε and δ definition), Types of discontinuities, Differentiability of functions, Successive differentiation, Leibnitz’s theorem, Partial differentiation, Euler’s theorem on homogeneous functions. Tangents and normals, Curvature, Asymptotes, Singular points, Tracing of curves. Parametric representation of curves and tracing of parametric curves, Polar coordinates and tracing of curves in polar coordinates. Rolle’s theorem, Mean Value theorems, Lagrange and cauchy theorems. Taylor’s theorem with Lagrange’s and Cauchy’s forms of remainder, Power series and its convergences. Taylor’s series, Maclaurin’s series of sin x, cos x, ex, log(l+x), (l+x)m, Maxima and Minima, Indeterminate forms.

Suggested Readings: H. Anton, I. Birens and S. Davis, Calculus, John Wiley and Sons, Inc., 2002. G.B. Thomas and R.L. Finney, Calculus, Pearson Education, 2007.

DSC-1B(CC-2): Differential Equations

Credit : 06

DSC1BT(CC-2) : Differential Equations

Course Contents:

First order exact differential equations. Integrating factors, rules to find an integrating factor. First order higher degree equations solvable for x, y, p. Methods for solving higherorder differential equations. Basic theory of linear differential equations, Wronskian, and its properties. Solving a differential equation by reducing its order. Linear homogenous equations with constant coefficients, Linear non-homogenous equations, The method of variation of parameters, The Cauchy-Euler equation, Simultaneous differential equations, Total differential equations. Order and degree of partial differential equations, Concept of linear and non-linear partial differential equations, Formation of first order partial differential equations, Linear partial differential equation of first order, Lagrange’s method, Charpit’s method. Classification of second order partial differential equations into elliptic, parabolic and hyperbolic through illustrations only.

Suggested Readings: Shepley L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, 1984. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, International

Edition, 1967.

6

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DSC-1C(CC-3): Real Analysis

Credit : 06

DSC1CT(CC-3): Real Analysis

Course Contents:

Finite and infinite sets, examples of countable and uncountable sets. Real line, bounded sets, suprema and infima, completeness property of R, Archimedean property of R, intervals. Concept of cluster points and statement of Bolzano-Weierstrass theorem.Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences. Cauchy’s theorem on limits, order preservation and squeeze theorem, monotone sequences and their convergence (monotone convergence theorem without proof). Infinite series. Cauchy convergence criterion for series, positive term series, geometric series, comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test (Tests of Convergence without proof). Definition and examples of absolute and conditional Convergence Series. Sequences and series of functions, Pointwise and uniform convergence. µ-test, M-test, Statements of the results about uniform convergence and integrability and differentiability of functions, Power series and radius of convergence.

Suggested Readings: T.M. Apostol, Calculus (Vol. I), John Wiley and Sons (Asia) P. Ltd., 2002. R.G. Bartle and D. R Sherbert, Introduction to Real Analysis, John Wiley and Sons

(Asia) P. Ltd., 2000. E. Fischer, Intermediate Real Analysis, Springer Verlag, 1983. K.A. Ross,(2003). Elementary Analysis - The Theory of Calculus Series-

Undergraduate Texts in Mathematics, Springer Verlag. S.K.Mapa Introduction to real analysis, Sarat Book House.

DSC-1D (CC-4): Algebra

Credits 06

DSC1DT(CC-4): Algebra

Course Contents:

Definition and examples of groups, examples of abelian and non-abelian groups, the group Zn of integers under addition modulo n and the group U(n) of units under multiplication modulo n. Cyclic groups from number systems, complex roots of unity, circle group, the general linear group GLn (R), groups of symmetries of (i) an isosceles triangle, (ii) an equilateral triangle, (iii) a rectangle, and (iv) a square, the permutation group Sym (n), Group of quaternions. Subgroups, cyclic subgroups, the concept of a subgroup generated by a subset and the commutator subgroup of group, examples of subgroups including the center of a group. Cosets, Index of subgroup, Lagrange’s theorem, order of an element, Normal subgroups: their definition, examples, and characterizations, Quotient groups.Definition and examples of rings, examples of commutative and non-commutative rings: rings from number systems, Zn the ring of integers modulo n, ring of real quaternions, rings of matrices, polynomial rings, and rings of continuous functions. Subrings and ideals, Integral domains and fields, examples of fields: Zp, Q, R, and C. Field of rational functions.

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Suggested Readings: John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. Joseph A Gallian, Contemporary Abstract Algebra, Narosa, 1999. George E Andrews, Number Theory, Hindustan Publishing Corporation, 1984. S.K.Mapa. Higher Algebra : Abstract and Linear, Sarat Book House, Calcutta.

Discipline Specific Electives (DSE)

DSE-1 : Complex Analysis

Credit : 06

DSE1T : Complex Analysis

Course Contents:

Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy - Goursat theorem, Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and its examples, absolute and uniform convergence of power series. Singular points of all kind, residue, Cauchy residue theorem and example.

Suggested Readings: James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed., McGraw – Hill International Edition, 2009. Joseph Bak and Donald J. Newman, Complex analysis, 2nd Ed., Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., New York, 1997.

Or

DSE - 1: Matrices

Credit : 06

DSE1T: Matrices

Course Contents:

R, R2, R3 as vector spaces over R. Standard basis for each of them. Concept of Linear Independence and examples of different bases. Subspaces of R2, R3. Translation, Dilation, Rotation, Reflection in a point, line and plane. Matrix form of basic geometric transformations. Interpretation of eigen values and eigenvectors for such transformations and eigen spaces as invariant subspaces. Matrices in diagonal form. Reduction to diagonal form upto matrices of order 3. Computation of matrix inverses using elementary row operations. Rank of matrix. Solutions of a system of linear equations using matrices.

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Illustrative examples of above concepts from Geometry, Physics, Chemistry, Combinatorics and Statistics.

Suggested Readings: A.I. Kostrikin, Introduction to Algebra, Springer Verlag, 1984. S. H. Friedberg, A. L. Insel and L.E. Spence, Linear Algebra, Prentice Hall of India Pvt. Ltd., New Delhi, 2004. Richard Bronson, Theory and Problems of Matrix Operations, Tata McGraw Hill, 1989. M. Pal (2013). Higher Algebra, PHI Learning Pvt. Ltd.

Or

DSE-1 : Linear Algebra

Credit : 06

DSE1T: Linear Algebra

Course Contents:

Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces. Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations. Isomorphisms, Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix.

Suggested Readings: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th

Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education

Asia, Indian Reprint, 2007. S. Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005. Gilbert Strang, Linear Algebra and its Applications, Thomson, 2007. M.Pal. Higher Algebra. PHI Pvt. Ltd. 2013.

Or

DSE - 1: Vector Calculus and Analytical Geometry

Credit : 06

DSE1T: Vector Calculus and Analytical Geometry

Course Contents:

Algebra of vectors, Differentiation and partial differentiation of a vector function. Derivative of sum, dot product and cross product of two vectors. Gradient, divergence and curl. Techniques for sketching parabola, ellipse and hyperbola. Reflection properties of parabola, ellipse and hyperbola. Classification of quadratic equations representing lines, parabola, ellipse and hyperbola. Spheres, Cylindrical surfaces. Illustrations of graphing standard quadric surfaces like cone, ellipsoid.

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Suggested Readings:

G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. H. Anton, I. Bivens and S. Davis, Calculus, John Wiley and Sons (Asia) P. Ltd.

2002. P.C. Matthew’s, Vector Calculus, Springer Verlag London Limited, 1998. S.L. Loney, The Elements of Coordinate Geometry, McMillan and Company,

London. R.Khan. Analytical Geometry and Vector Algebra, New Central Book Agency. R.J.T. Bill, Elementary Treatise on Coordinate Geometry of Three Dimensions,

McMillan India Ltd., 1994.

DSE-2 : Mechanics

Credit : 6

DSE2T: Mechanics

Course Contents:

Conditions of equilibrium of a particle and of coplanar forces acting on a rigid Body, Laws of friction, Problems of equilibrium under forces including friction, Centre of gravity, Work and potential energy. Velocity and acceleration of a particle along a curve: radial and transverse components (plane curve), tangential and normal components (space curve), Newton’s Laws of motion, Simple harmonic motion, Simple Pendulum, Projectile Motion.

Suggested Readings: A.S. Ramsay, Statics, CBS Publishers and Distributors (Indian Reprint), 1998. A.P. Roberts, Statics and Dynamics with Background in Mathematics, Cambridge University Press, 2003. S.L.Loney. An elementary treaties on the dynamics of a particle and rigid body, New Age International Pvt. Ltd.

Or

DSE-2: Linear Programming

Credit: 06

DSE2T: Linear Programming

Course Contents:

Linear Programming: Definition and formation Problems, Graphical Approach for solving some Linear Programming problems. Convex Sets, Supporting and Separating Hyperplanes. Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex method in tableau format, introduction to artificial variables, twophase method, Big-M method and their comparison. Duality, formulation of the dual problem, primal- dual relationships, economic interpretation of the dual.

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