Syllabus for T.Y.BSc Course : Mathematic Semester : V · 1 JAI HIND COLLEGE AUTONOMOUS Syllabus for T.Y.BSc Course : Mathematic Semester : V Credit Based Semester & Grading System

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JAI HIND COLLEGE AUTONOMOUS

Syllabus for T.Y.BSc

Course : Mathematic

Semester : V

Credit Based Semester & Grading System

With effect from Academic Year 2018-19

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List of Courses

Course: Mathematics Semester: V

SR. NO. COURSE CODE COURSE TITLE

NO. OF LECTURES

/ WEEK

NO. OF CREDITS

TYBSc

1 SMAT 501 Integral Calculus 3 4

2 SMAT 502 Algebra 3 4

3 SMAT 503 Topology of metric spaces 3 4

4 SMAT 504 Numerical Analysis I 3 4

5 SMAT 5 PR1 Practical-I(Based on

SMAT 501,SMAT 502)

6 4

6 SMAT 5 PR2 Practical-II(Based on

SMAT 503,SMAT 504 )

6 4

7 SMAT 5 AC Applied Component Theory 4 2.5

8 SMAT 5 AC PR Applied Component Practical

4 2.5

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SEMESTER V

COURSE : INTEGRAL CALCULUS Course Description: This course is an extension of integration theory of one variable to integration theory of multiple variable over different type of domains in Rn .

Course Ob jective:

(1) This course has a wide variety of application in physics and engineering. The main objective of the course is to make students competent in solving real world maths problem.

(2) This course can help students to pursue research in applied mathematics.

Unit 1: Multiple integrals (15L)

(1) Definition of double (respectively: triple) integral of a function bounded on a rectangle (respectively: box), Geometric interpretation as area and volume.

(2) Fubini’s Theorem over rectangles and any closed bounded sets, Iterated Integrals. Basic

properties of double and triple integrals proved using the Fubini’s theorem such as; Integrability of the sums, scalar multiples, products, and (under suitable conditions) quotients of integrable functions, Formulae for the integrals of sums and scalar multiples of integrable functions

(3) Integrability of continuous functions,integrability of bounded functions having finite

number of points of discontinuity, Domain additivity of the integral. Integrability and the integral over arbitrary bounded domains

(4) Change of variables formula (Statement only), Polar, cylindrical and spherical coordi-

nates and integration using these coordinates.

(5) Differentiation under the integral sign. Applications to finding the center of gravity and moments of inertia.

Unit 2: Line Integrals (15L)

(1) Review of Scalar and Vector fields on Rn , Vector Differential Operators, Gradient Paths (parametrized curves) in Rn (emphasis on R2 and R3), Smooth and piecewise smooth paths, Closed paths.

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(2) Equivalence and orientation preserving equivalence of paths. Definition of the line integral of a vector field over a piecewise smooth path.

(3) Basic properties of line integrals including linearity, path-additivity and behavior under

a change of parameters, Examples.

(4) Line integrals of the gradient vector field, Fundamental Theorem of Calculus for Line Integrals, Necessary and sufficient conditions for a vector field to be conservative.

(5) Green’s Theorem (proof in the case of rectangular domains). Applications to evaluation

of line integrals.

Unit 3: Surface Integrals (15L)

(1) Parameterized surfaces. Smoothly equivalent parameterizations, Area of such surfaces.

(2) Definition of surface integrals of scalar-valued functions as well as of vector fields defined on a surface.

(3) Curl and divergence of a vector field, Elementary identities involving gradient, curl and

divergence.

(4) Stoke’s Theorem (proof assuming the general form of Green’s Theorem), Examples.

(5) Gauss’ Divergence Theorem (proof only in the case of cubical domains), Examples. References:

(1) Unit I:

(i) Apostol(1969), Calculus, Vol. 2, Section1.1 to 11.8, Second Ed., John Wiley,Newyork.

(ii) James Stewart , Calculus with early transcendental Functions, Section 15 . (iii) J. E. Marsden and A.J. Tromba(1996), Vector Calculus, Section 5.2 to 5.6, Fourth

Ed. W.H. Freeman and Co., New York.

(2) Unit II

(i) Lawrence Corwin and Robert Szczarba, Multivariable Calculus, Chapter 12. (ii) Apostol(1969), Calculus, Vol. 2, Section 10.1 to 10.5 and 10.10 to 10.18, Second

Ed., John Wiley, Newyork. (iii) J. E. Marsden and A.J. Tromba(1996),Vector Calculus, Section 6.1 ,7.1.7.4, Fourth

Ed. W.H. Freeman and Co., New York. (iv) James Stewart, Calculus with early transcendental Functions, Section 16.1-16.4.

(3) Unit III

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(i) Apostol(1969), Calculus, Vol. 2, Section 11.1 to 11.8, Second Ed., John Wiley, Newyork.

(ii) J. E. Marsden and A.J. Tromba(1996), Vector Calculus, Section 6.2 to 6.4, Fourth Ed. W.H. Freeman and Co., , New York.

(iii) James Stewart, Calculus with early transcendental Functions, Section 16.5-16.9. Additional References:

(1) T Apostol(1974), Mathematical Analysis, Second Ed., Narosa, New Delhi.

(2) R. Courant and F. John(1989), Introduction to Calculus and Analysis, Vol.2, Springer

Verlag, New York.

(3) W. Fleming(1977), Functions of Several Variables, Second Ed., Springer-Verlag, New York.

(4) M. H. Protter and C. B. Morrey, Jr.(1995), Intermediate Calculus, Second Ed., Springer-

Verlag, New York.

(5) G. B. Thomas and R. L. Finney(1998), Calculus and Analytic Geometry, Ninth Ed. (ISE Reprint), Addison- Wesley, Reading Mass.

(6) D. V. Widder(1989), Advanced Calculus, Second Ed., Dover Pub., New York.

(7) Sudhir R. Ghorpade and Balmohan Limaye, A course in Multivariable Calculus and

Analysis, Springer International Edition.

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COURSE : ALGEBRA I Course Description: This course aims to introduce the basic ideas and techniques of linear algebra which will be useful for learning future courses in linear algebra.

Course Ob jective:

(1) Students will get familiar with the notion of orthogonal transformation and isometries.

(2) They will be able to find eigen values and eigen vectors and will learn how to diaganilise a matrix.

Unit I: Quotient Spaces and Orthogonal Linear Transformations (15L)

(1) Review of vector spaces over R , sub spaces and linear transformation. Quotient Spaces: For a real vector space V and a subspace V /W , the cosets v + W and the quotient space V /W.

(2) First Isomorphism theorem of real vector spaces (fundamental theorem of homomor-

phism of vector spaces), Dimension and basis of the quotient space V /W , when V is finite dimensional.

(3) Orthogonal transformations: Isometries of a real finite dimensional inner product space,

Translations and Reflections with respect to a hyperplane, Orthogonal matrices over R. Equivalence of orthogonal transformations and isometries fixing origin on a finite dimensional inner product space.

(4) Any orthogonal transformation in R2 is a reflection or a rotation, Characterization of isometries as composites of orthogonal transformations and translation.

(5) Characteristic polynomial of an n × n real matrix. Cayley Hamilton Theorem and its

Applications (Proof assuming the result A(AdjA) = det(A)In for an n × n matrix over the polynomial ring R[t].

Unit II: Eigenvalues and eigen vectors (15L)

(1) Eigen values and eigen vectors of a linear transformation T : V → V , where V is a finite dimensional real vector space and examples, Eigen values and Eigen vectors of n × n real matrices.

(2) The linear independence of eigenvectors corresponding to distinct eigenvalues of a linear

transformation and a Matrix.

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(3) The characteristic polynomial of a n × n real matrix and a linear transformation of a finite dimensional real vector space to itself.

(4) characteristic roots, Similar matrices, Relation with change of basis, Invariance of the

characteristic polynomial and (hence of the) eigen values of similar matrices, Every square matrix is similar to an upper triangular matrix.

(5) Minimal Polynomial of a matrix, Examples like minimal polynomial of scalar matrix,

diagonal matrix, similar matrix, Invariant subspaces. Unit III: Diagonalisation (15L)

(1) Geometric multiplicity and Algebraic multiplicity of eigen values of an n×n real matrix.

(2) An n × n matrix A is diagonalizable if and only if has a basis of eigenvectors of A if and only if the sum of dimension of eigen spaces of A is n if and only if the algebraic and geometric multiplicities of eigen values of A coincide.

(3) Examples of non diagonalizable matrices, Diagonalisation of a linear transformation

T : V → V , where V is a finite dimensional real vector space and examples.

(4) Orthogonal diagonalisation and Quadratic Forms. Diagonalisation of real Symmetric matrices, Examples, Applications to real Quadratic forms.

(5) Rank and Signature of a Real Quadratic form, Classification of conics in R2 and quadric

surfaces in R3 . Positive definite and semi definite matrices, Characterization of positive definite matrices in terms of principal minors.

References

[1] S. Kumaresan, Linear Algebra: A Geometric Approach.

[2] Ramachandra Rao and P. Bhimasankaram, Tata McGraw Hill Publishing Company.

Additional References:

(1) T. Banchoff and J. Wermer, Linear Algebra through Geometry, Springer.

(2) L. Smith, Linear Algebra, Springer.

(3) M. R. Adhikari and Avishek Adhikari,Introduction to linear Algebra, Asian Books Pri- vate Ltd.

(4) K Hoffman and Kunze, Linear Algebra, Prentice Hall of India, New Delhi.

(5) Inder K Rana, Introduction to Linear Algebra, Ane Books Pvt. Ltd.

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COURSE : TOPOLOGY OF METRIC SPACES Course Description: This course will begin with defining different ways of measuring distance between points in euclidean spaces. Basic notion of real analysis will be extended over any metric spaces.

Course Ob jective:

(1) Students will get handle in solving problems of continuity over different metric spaces.

(2) This course will enhance the ability to analyze mathematical problems geometrically.

(3) In unit 3 we introduce compact set which has nice properties. These type of sets are used in proving important theorems of topology.

Unit 1: Metric spaces (15L)

(1) Definition, examples of metric spaces R, R2, Euclidean space Rn with its Euclidean sup and sum metric.

(2) The spaces `1 and `2 of sequences and the space C [a, b] of real valued continuous functions on [a, b], Discrete metric space.

(3) Distance metric induced by the norm, translation invariance of the metric induced by

the norm. Metric subspaces, Product of two metric spaces.

(4) Open balls and open set in a metric space, examples of open sets in various metric spaces, Hausdorff property, Interior of a set, Properties of open sets, Structure of an open set in R.

(5) Equivalent metrics, distance of a point from a set, between sets, diameter of a set in a

metric space and bounded sets.

Unit 2: Closed sets, Limit Points and Sequences (15L)

(1) Closed ball in a metric space, Closed sets- definition, examples. Limit point of a set, Isolated point.

(2) A closed set contains all its limit points, Closure of a set and boundary, Sequences in

a metric space.

(3) Convergent sequence in a metric space, Cauchy sequence in a metric space, subse- quences, examples of convergent sequences

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(4) Cauchy sequence in finite metric spaces, Rn with different metrics and other metric spaces. Characterization of limit points and closure points in terms of sequences.

(5) Definition and examples of relative openness/closeness in subspaces, Dense subsets in

a metric space and Separability

Unit 3: Compact sets (15L)

(1) Definition of compact metric space using open cover, examples of compact sets in different metric spaces R, R2, Rn and other metric spaces.

(2) Properties of compact sets?, compact set is closed and bounded, every infinite bounded

subset of a compact metric space has a limit point

(3) Heine Borel theorem-every subset of Euclidean metric space Rn is compact if and only if it is closed and bounded.

(4) Equivalent statements for compact sets in Rn ; Heine-Borel property, Closed and bound- edness property, Bolzano-Weierstrass property, Sequentially compactness property.

References

[1] W. Rudin, Principles of Mathematical Analysis.

[2] T. Apostol(1974),Mathematical Analysis, Second edition, Narosa, New Delhi.

[3] E. T. Copson(1996),Metric Spaces. Universal Book Stall, New Delhi.

[4] R. R. Goldberg(1970), Methods of Real Analysis, Oxford and IBH Pub. Co., New Delhi.

Additional References:

(1) S. Kumaresan, Topology of Metric spaces.

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COURSE : NUMERICAL ANALYSIS I Course Description: In this course we study various numerical approximation methods(As apposed to general symbolic manipulations) to solve the mathematical problems. The overall goal of this course is to design techniques to give approximate but accurate solutions to hard problems.

Course Ob jective:

(1) In this course students will learn different numerical methods to solve algebraic and

transcendental equations.

(2) In last unit students will learn to solve system of equations that is used in all areas of science.

Unit I: Errors Analysis and Transcendental & Polynomial Equations (15L)

(1) Measures of Errors: Relative, absolute and percentage errors. Types of errors: Inherent error, Round-off error and Truncation error.

(2) Taylor series example. Significant digits and numerical stability.

(3) Concept of simple and multiple roots. Iterative methods, error tolerance, use of inter-

mediate value theorem.

(4) Iteration methods based on first degree equation: Newton-Raphson method, Secant method, Regula-Falsi method, Iteration Method.

(5) Condition of convergence and Rate of convergence of all above methods.

Unit II: Transcendental and Polynomial Equations (15L)

(1) Iteration methods based on second degree equation: Muller method, Chebyshev method, Multipoint iteration method.

(2) Iterative methods for polynomial equations; Descart? rule of signs, Birge-Vieta method,

Bairstrow method.

(3) Newton-Raphson method. System of non-linear equations by Newton- Raphson method. Methods for complex roots.

(4) Condition of convergence and Rate of convergence of all above methods.

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Unit III : Linear System of Equations (15L)

(1) Matrix representation of linear system of equations. Direct methods: Gauss elimination method.

(2) Pivot element, Forward and backward substitution method.

(3) Triangularization methods-Doolittle and Crouts method, Cholesky method. Error anal-

ysis of direct methods.

(4) Iteration methods: Jacobi iteration method, Gauss-Siedal method. Convergence anal- ysis of iterative method.

(5) Eigen value problem, Jacobi method for symmetric matrices Power method to deter-

mine largest eigenvalue and eigenvector. References

[1] Kendall E. and Atkinson,An Introduction to Numerical Analysis, Wiley.

[2] M. K. Jain, S. R. K. Iyengar and R. K. Jain,Numerical Methods for Scientific and

Engineering Computation, New Age International Publications.

[3] S.D. Comte and Carl de Boor, Elementary Numerical Analysis, An algorithmic ap- proach, McGraw Hill International Book Company.

[4] S. Sastry, Introductory methods of Numerical Analysis, PHI Learning.

[5] Hildebrand F.B.,Introduction to Numerical Analysis, Dover Publication, NY.

[6] Scarborough James B.,Numerical Mathematical Analysis, Oxford University Press,

New Delhi.

Exam pattern

(1) Semester End Exam (100 marks) for all 4 theory papers.

(2) Semester End Exam (50 marks) for practical papers.

(3) Semester End Exam (100 marks) for Applied component theory paper.

(4) Semester End Exam (100 marks) for Applied component practical paper.

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T.Y.B.Sc. End Semester

Theory Question Paper Pattern (1) All Questions are compulsory

(2) Question (1), (2) and (3) are based on Unit 1, Unit 2 and Unit 3 respectively. The

scheme of Question is as Follows:

(A) Attempt any 2 out of 3. Each Question is of 8 Marks. (B) Attempt any 2 out of 4. Each Question is of 6 Marks.

(3) Question 4 is Based on Unit 1, 2 and 3. Attempt any 4 out of 6. Each Question is of 4 Marks.

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T.Y.B.Sc. Practical Exam Pattern (1) At the end of the Semesters V, Practical examinations of three hours duration and 100

marks shall be conducted for the courses SMAT 5 PR 1 and SMAT 5 PR 2.

Practical Paper Pattern (1) Practical Paper I and Practical Paper II are divided into two parts:

(A) Objective type Question of 3 Marks each. Students have to attempt 8 out of 12 (B) Descriptive type Questions of 8 Marks each. Students have to attempt 2 out of 3.

(2) Practical Paper I and Practical Paper II : Journal 5 marks and Viva 5 marks.