# Syllabus of USMT301401

## Unit 1 : Real Numbers

(a) (i) Statements of algebraic and order properties of $\mathbb{R}$ (ii) Elementary consequences of these properties including the A.M. – G.M. inequality Cauchy-Schwarz inequality, and Bernoulli inequality (without proof).

(b) (i) Review of absolute value and neighbourhood of a real number. (ii) Hausdroff Property

(c) Supremum (lub) and infimum (glb) of a subset of $\mathbb{R}$, lub axion of $\mathbb{R}$. Consequences of lub axiom of $\mathbb{R}$ including (i) Archimedian property (ii) Density of rational numbers (iii) Existence of $n^{th}$ root of a positive real number (in particular root). (iv) Decimal representation of a real number.

(d) (i) Nested Interval Theorem (ii) Open sets in $\mathbb{R}$ and closed sets as complements of open sets. (iii) Limit points of a subset of $\mathbb{R}$, examples, characterization of a closed set as a set containing all its limit points.

(e) Open cover of a susbset of $\mathbb{R}$, Compact susbset of $\mathbb{R}$, Definition and examples. A closed and bounded interval [a, b] is compact.

Reference for Unit 1 : Chapter II, Sections 1, 2, 4, 5, 6 and Chaper X, Sections 1, 2 of Introduction to Real Analysis, ROBERT G. BARTLE and DONALD R. SHERBET, Spriger Verlag.

## Unit 2 : Sequences, Limits and Continuity

(a) Sequence of real numbers, Definition and examples. Sum, difference, product, quotient and scalar multiple of sequences. (b) Limit of a sequence. Convergent and divergent sequences. Uniqueness of limit of a convergent sequence. Algebra of convergent sequences. Sandwich Theorem of sequences. Limits of standard sequences such as (i) $\{\dfrac{1}{n^a}\}, a \gt 0$, (ii) $\{a^n\}, |a| \lt 1$, (iii) ${n^{1/n}}.{a^{1/n}}, a \gt 0$, (iv)

Examples of divergent sequences. (c) (i) Bounded sequences – A convergent sequence is bounded. (ii) Monotone sequences – Convergence of bounded monotone sequences. The number as a limit of a sequence, Calculation of square root of a positive real number. (d) (i) Subsequences (ii) Limit inferior and limit superior of a sequence (iii) Bolzano – Weierstrass Theorem of sequences (iv) Sequential characterization of limit points of a set (e) Cauchy sequences, Cauchy completeness of (f) Limit of a real valued function at a point (i) Review of the definition of limit of functions at a point, uniqueness of limits of a function at a point whenever it exists. (ii) Sequential characterization for limits of functions at a point. Theorems of limits (Limits of sum, difference, product, quotient, scalar multiple and sandwich theorem). (iii) Continuity of function at a point, definition, sequential criterion. Theorems about continuity of sum, difference, product, quotient and scalar multiple of functions at a point in the domain using definition or sequential criterion. Continuity of composite functions. Examples of limits and continuity of a function at a point using sequential criterion. (iv) A continuous function on closed and bounded interval is bounded and attains bounds.

Reference for Unit 2 : Chapter III. Sections 1, 2, 3, 4, 5 Chapter IV Sections 1, 2 and Chapter V, Sections 1, 2, 3 of Introduction to Real Analysis, ROBERT G. BARTLE and DONALD R. SHERBET, Springer Verlag.

Unit 3 : Infinite Series (a) Infinite series of real numbers. The sequence of partial terms of an infinite series, convergence and divergence of series, sum, difference and multiple of convergent series and again convergent. (b) Cauchy criterion of convergence of series. Absolute convergence of a series. Geometric series. (c) Alternating series, Leibnitz Theorem, Conditional convergence, An absolutely convergent series is conditionally convergent, but the converse is not true. (d) Rearrangement of series (without proof), Cauchy condensation test (Statement only), application to convergence of p-series . Divergence of Harmonic series . (e) Tests for absolute convergence, Comparison test, Ratio test, Root test. (f) Power series, Radius of convergence of power series, The exponential, sine and cosine series. (g) Fourier series, Computing Fourier Coefficients of simple functions such as piecewise continuous functions on . Reference for Unit 3: Chapter IX, Sections, 1,2,3,4 and Chapter VIII, Sections 3, 4 of Introduction to Real Analysis, ROBERT G. BARTLE AND DONALD R. SHERBET, Springer Verlag.

Unit 4 : Differential Equations (a) First Order Differential Equations : (i) Review of separable differential equations, homogeneous and non-homogeneous differential equations. (ii) Exact differential equations and integrating factors. Rules for finding integrating factors of (without proof) when: • , • , • (iii) Linear differential equations and Bernoulli differential equations.

(iv) Modeling with first order equations. Examples from Financial Mathematics, Chemistry, Environmental Science, Population growth and decay.

(b) Second order Linear Differential Equations : (i) The general second order linear differential equation. Existence and Uniqueness Theorem for the solutions of a second order initial value problem (statement only). (ii) Homogeneous and non-homogeneous second order linear differential equations : • The space of solutions of the homogeneous equations as a vector space. • Wronskian and linear independence of the solutions. • The general solution of homogeneous differential equation. The use of known solutions to find the general solution of a homogeneous equations. • The general solution of a non-homogeneous second order equation, complementary functions and particular integrals. (iii) The homogeneous equation with constant coefficients, auxiliary equation, the general solution corresponding to real and distinct roots, real and distinct roots, real and equal roots and complex roots of the auxiliary equation. (iv) Non-homogeneous equations : The method of undetermined coefficients. The method of variation of parameters.

Reference for Unit 4 : Chaper 2, Sections 7, 8, 9, 10 and Chapter 3, Sections 14, 15, 16, 17, 18, 19, 20 of Differential Equations with Applications and Historical Notes, G.F. SIMMONS, McGraw Hill and Chapter 1, Sections 1, 2, 3 of Elements of Partial Differential Equations, I. SNEDDON McGraw Hill.

Unit 5 : Multiple Integrals Review of functions of two and three variables, partial derivatives and gradient of two or three variables.

(a) Double integrals : (i) Definition of double integrals over rectangles (ii) Properties of double integrals (iii) Double integrals over bounded regions. (b) Statement of Fubini’s Theorem, Double integrals as volumes. (c) Applications of Double integrals : Average value, Areas, Moments, Center of Mass. (d) Double integrals in polar form (e) Triple integrals in Rectangular coordinates, Average, volumes (f) Applications of Triple integrals : Mass, Moments, Parallel axis Theorem (g) Triple integrals in Spherical and Cylindrical coordinates.

Reference for Unit 5 : Chapter 13, Sections 13.1, 13.2, 13.3, 13.4, 13.5, 13.6 of Calculus and Analytic Geometry, G.B. THOMAS and R.L. FINNEY, Ninth Edition, Addison – Wesley, 1998.

Unit 6 : Integration of Vector Fields

(a) Line Integrals, Definition, Evaluation for smooth curves Mass and moments for coils, springs, thin rods (b) Vector fields, Gradient fields, Work done by a force over a curve in space, Evaluation of work integrals. (c) Flow integrals and circulation around a curve. (d) Flux across a plane curve (e) Path independence of the integral in an open region. F being a vector field over the region and A, B points in the region. Conservative fields, potential function. (f) The Fundamental theorems of line integrals (without proof). (g) Flux density (divergence), Circulation density (curl) at a point. (h) Green’s Theorem in plane (without proof), Evaluation of line integrals using Green Theorem.

Reference for Unit 6 : Chapter 14 of 14.1, 14.2, 14.3, 14.4 Calculus and Analytic Geometry, G.B. THOMAS and R.L. FINNEY, Ninth Edition, Addison – Wesley, 1998.

The proofs of the results mentioned in the syllabus to be covered unless indicated otherwise.

Recommended Books :

1. ROBERT G. BARTLE and DONALD R. SHERBET : Introduction to Real Analysis, Springer Verlag. 2. R. COURANT and F. JOHN : Introduction to Calculus and Analysis Vol I Reprint of First Edition, Springer Verlag, New York, 1999. 3. R.R. GOLDBERG : Methods of Real Analysis, Oxford and IBH Publication Company, New Delhi. 4. T. APOSTOL : Calculus Vol I, Second Edition, John Wiley. 5. M.H. PROTTER : Basic elements of Real Analysis, Springer Verlag, New York, 1998. 6. G.B. THOMAS and R.L. FINNEY, Calculus and Analytic Geometry, Ninth Edition Addison – Wesley, 1998. 7. G.F. SIMMONS : Differential Equations with Applications and Historical Notes, McGraw Hill. 8. I. SNEDDON : Elements of Partial Differential Equations, McGraw Hill.