PHYSICS/CHEMISTRY – Second

PHYSICS/CHEMISTRY – Third

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only), Partial derivatives, The Chain Rule.

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Functions, inverse circular and hyperbolic function,Separation Into Real And Imaginary Parts.

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Functions

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PHYSICS/CHEMISTRY

MM2CMT01: INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS

Volumes using Cross-Sections, Volumes using Cylindrical shells, Arc lengths, Areas of surfaces of Revolution.

Double and iterated integrals over rectangles, Double integrals over general regions, Area by

double integration, Triple integrals in rectangular co-ordinates.

double integration

Equations, Solutions by Substitutions, Homogeneous equations and Bernoulli’s Equations.

Surfaces and Curves in three dimensions, Solution of equations of the form

Origin of first order and second order partial differential equations, Linear equations of the first order, Lagrange’s method.

PHYSICS/CHEMISTRY

MM3CMT01:VECTOR CALCULUS, ANALYTIC GEOMETRY AND ABSTRACT ALGEBRA

Curves in space and their tangents, Arc length in space, Curvature and Normal Vectors of a

curve, Directional Derivatives and Gradient Vectors.

curve

Line Integrals, Vector fields and line integrals: Work, Circulation and Flux. Path independence,

Conservation Fields and Potential Functions , Green’s theorem in Plane (Statement and problems only),

Surface area and Surface integral, Stoke’s theorem( Statement and Problems only), the Divergence

theorem and a Unified theory ( Statement and simple problems only).

theorem and a Unified theory

Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism.

Periodic Functions, Trigonometric Series, Fourier Series, Functions of any period p = 2L, Even and Odd functions, Half range Expansions. A brief introduction to power series and power series method for solving Differential equations, Legendre equation and Legendre polynomials.

Laplace Transform, Inverse Laplace transform, Linearity, Shifting, transforms of Derivatives and Integrals, Differential Equations, Differentiation and Integration of Transforms, Laplace transform general Formula (relevant formulae only), Table of Laplace Transforms (relevant part only)

Complex Numbers, Complex Plane, Polar form of Complex Numbers, Powers and Roots, Derivative, Analytic Functions, Cauchy-Riemann Equations, Laplace’s Equation, Exponential Function, Trigonometric Functions, Hyperbolic Functions, Logarithm, General Power.

Line Integral in the Complex Plane, Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Derivatives of Analytic functions.

Review – (Exponents, polynomials, factoring, fractions, radicals, order of mathematical operations.) Cartesian Co-ordinate system, linear equations and graphs slopes intercepts. The slope intercept form. Determining the equation of a straight line. Applications of line equations in business and economics.

Functions Concepts and definitions- graphing functions. The algebra of functions. Applications of linear functions for business and economics. Solving quadratic equations Facilitating non-linear graphing. Application of non-linear functions in business and economics. System of equations Introduction, graphical solutions. Supply-demand analysis. Break-even analysis. Elimination and substitution methods. IS-LM analysis. Economic and mathematical modeling. Implicit functions and inverse functions.

Break-even analysis.

Elimination and substitution methods

IS-LM analysis.

Economic and mathematical modeling

Implicit functions and inverse functions.

Limits and continuity. Evaluation of limit of a function. Algebraic limit. The derivative and the rules of differentiation: The slope of curvilinear function. Derivative notation. Rules of differentiation. Higher order derivatives. Derivative of Implicit functions. Applications of derivatives. Increasing and decreasing functions. Concavity and convexity. Relative extrema. 147 Inflection points. Curve sketching. Optimization of functions. The successive derivative test. Marginal concepts in economics. Optimizing economic functions of business. Relation among total, marginal and average functions.

Exponential functions. Logarithmic functions. Properties of exponents and logarithms. Natural exponential and logarithmic functions. Solving natural exponential and logarithmic functions. Logarithmic transformation of non-linear functions. Derivatives of natural exponential and logarithmic functions. Interest compounding. Estimating growth rates from data points.

Matrix Algebra

Introduction. Definition and terms. Addition and subtraction of matrices. Scalar multiplication. Vector multiplication. Multiplication of matrices. Matrix expression of a system of linear equations. Augmented matrix. Row operation. Gaussian method of solving linear equations. Solving linear equations with. Matrix algebra Determinants and linear independence. Third order determinants. Cramer’s rule for solving linear equations. Inverse matrices. Gaussian method of finding an inverse matrix. Solving linear equations with an inverse matrix. Business and Economic applications. Special determinants.

Linear programming

Linear programming problem (LPP), Mathematical Formulation of LPP. Basic solution, Feasible solution and Region of feasible solution of an LPP. The extreme point theorem. Solving Maximisation and Minimisation problems using graphical method.

Integral calculus

Integration rules for indefinite integrals. Integration by substitution. Integration by parts. The definite integral. The fundamental theorems of calculus. Properties of definite integrals. Area under a curve. Area between curves. Present value of cash flow consumers and producers surplus.

Calculus of Multivariable functions

Functions of several independent variables. Partial derivatives. Rules of partial differentiation. Second order partial derivatives. Optimization of multivariable functions. Constrained optimization with Lagrange Multipliers. Income determination Multipliers. Optimization of multivariable functions in business and economics constrained optimization of multivariable economic functions. Constrained optimization of Cobb Douglas production functions.

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Propositional Logic, Propositional Equivalence, Predicates and Quantifiers and Rules of

Inference

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Basic Structures

Sets, Set Operations, Functions, Sequences and Summations

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Basic Structures

Sets, Set Operations, Functions, Sequences and Summations

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The Integers and Division, Primes and Greatest Common Divisors, Applications of Number

Theory

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Relations

Relations and Their Properties, Representing Relations, Equivalence Relations, Partial

Ordering

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Graphs and Graph Models, Graph Terminology and Special types of Graphs,Representing

Graphs and Graph Isomorphism, Connectivity, Euler and Hamilton Paths.

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Introduction to Trees, Application of Trees, Tree Traversal, and Spanning Trees.

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Boolean Function, Representing Boolean Functions and Logic Gates

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Matrices

Definitions and examples of Symmetric, Skew-symmetric, Conjugate, Hermitian, Skew

hermitian matrices. Rank of Matrix , Determination of rank by Row Canonical form and Normal

form , Linear Equations, Solution of non homogenous equations using Augmented matrix and

by Cramers Rule , Homogenous Equations, Characteristic Equation, Characteristic roots and

Characteristic vectors of matix , Cayley Hamilton theorem and applications.

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Propositional logic, Propositional equivalences, Predicates and quantifiers, Rules of inference,

Introduction to proofs

Sets, Set operations, Functions

Relations and their properties, Representing relations, Equivalence relations, Partial orderings

Roots of Equations, Relation Connecting the roots and coefficients of an equation,

Transformation of equations, Special Cases, The Cubic equation, The Biquadratic Equation,

Character and Position of the Roots of an Equation, Some General Theorems, Descartes’s Rule of

Signs, Corollaries, Reciprocal Equations

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Expansion of functions using Maclaurin’s theorem and Taylor’s theorem, Concavity and points of

inflexion. Curvature and Evolutes. Length of arc as a function derivative of arc, radius of curvature –

Cartesian equations only. (Parametric, Polar, Pedal equation and Newtonian Method are excluded)

Centre of curvature, Evolutes and Involutes, properties of evolutes. Asymptotes and Envelopes.

inflexion

Partial derivatives, The Chain rule, Extreme values and saddle points, Lagrange multipliers.

Volumes using Cross-sections, Volumes using cylindrical shells, Arc lengths, Areas of surfaces of

Revolution.

Double and iterated integrals over rectangles, Double integrals over general regions, Area by double

integration, Triple integrals in rectangular coordinates, Triple integrals in cylindrical and spherical

coordinates, Substitutions in multiple integrals.

Vector Differentiation

A vector equation and Parametric equations for lines and equation for a plane in space, Vector functions, Arc length and Unit tangent vector, Curvature and the Unit normal vector, Tangential and Normal Components of Acceleration, Directional derivatives and Gradient vectors, tangent planes and Normal lines.

Vector Integration

Line integrals, Vector fields, Work, Circulation and Flux, Path Independence, Conservative Fields and Potential Functions, Green’s theorem in the plane, Surfaces and Area: Parameterisations of surfaces, Implicit surfaces, Surface integrals, Stokes’ theorem, Divergence theorem.

Theory of Numbers

Basic properties of congruence, Fermat’s theorem, Wilson’s theorem, Euler’s phi function.

Laplace transforms

Laplace transform, Linearity of Laplace transform, First shifting theorem, Existence of Laplace transform, Transforms of derivatives, Solution of ordinary differential equation & initial value problem, Laplace transform of the integral of a function, Convolution and Integral equations.

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What is a differential equation

The nature of solutions, Separable equations, First order linear equations, Exact equations, Orthogonal trajectories and families of curves, Homogeneous equations, Integrating factors, Reduction of order-dependent variable missing-independent variable missing.

Second order linear equations

Second order linear equations with constant coefficients , The method of undetermined coefficients, The method of variation of parameters, The use of a known solution to find another, Higher order linear equations.

Power Series solutions and special functions

Introduction and review of power series, Series solutions of first order differential equations, Second order linear equations: ordinary points, Regular singular points, More on regular singular points.

Partial Differential equations

Methods of solution of = = , origin of first order partial differential equations, Linear equations of the first order, Lagrange’s method, Integral surfaces passing through a given curve.

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Unit 1 :Multidisciplinary nature of environmental studies

Definition, scope and importance

Need for public awareness.

Unit 2 : Natural Resources :

Renewable and non-renewable resources : Natural resources and associated problems.

a) Forest resources : Use and over-exploitation, deforestation, case studies.

Timber extraction, mining, dams and their effects on forest and tribal people.

b) Water resources : Use and over-utilization of surface and ground water,

floods, drought, conflicts over water, dams-benefits and problems.

c) Mineral resources : Use and exploitation, environmental effects of extracting

and using mineral resources, case studies.

d) Food resources : World food problems, changes caused by agriculture and

overgrazing, effects of modern agriculture, fertilizer-pesticide problems, water

logging, salinity, case studies.

e) Energy resources: Growing energy needs, renewable and non renewable energy sources,

use of alternate energy sources, Case studies.

f) Land resources: Land as a resource, land degradation, man induced landslides, soil erosion

and desertification

Role of individual in conservation of natural resources.

Equitable use of resources for sustainable lifestyles.

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Equitable use of resources for sustainable lifestyles

Environmental Pollution

Definition

Causes, effects and control measures of: –

Air pollution ,Water pollution ,Soil pollution, Marine pollution, Noise pollution, Thermal pollution ,Nuclear hazards

Solid waste Management: Causes, effects and control measures of urban and

industrial wastes. 

Role of an individual in prevention of pollution

Pollution case studies

Disaster management: floods, earthquake, cyclone and landslides. (8hrs)  Social Issues and the Environment

Urban problems related to energy

Water conservation, rain water harvesting, watershed management

Resettlement and rehabilitation of people: its problems and concerns, Case studies

Environmental ethics: Issues and possible solutions

Climate change, global warming, acid rain, ozone layer depletion , nuclear accidents and

holocaust, Case studies

Consumerism and waste products

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The rabbit problem, Fibonacci numbers, recursive definition, Lucas numbers, Different types of

Fibonacci and Lucas numbers. Fibonacci numbers in nature : Fibonacci and the earth, Fibonacci

and flowers, Fibonacci and sunflower, Fibonacci, pinecones, artichokes and pineapples,

Fibonacci and bees, Fibonacci and subsets, Fibonacci and sewage treatment, Fibonacci and

atoms, Fibonacci and reflections, Fibonacci, paraffins and cycloparaffins, Fibonacci and music,

Fibonacci and compositions with 1’s and 2’s.

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and flowers, Fibonacci and sunflower, Fibonacci, pinecones, artichokes and pineapples,

Fibonacci and bees, Fibonacci and subsets, Fibonacci and sewage treatment

Euler construction, generation by Newton’s method. The golden ratio revisited, the golden ratio

and human body, golden ratio by origami, Differential equations, Gattei’s discovery of golden

ratio, centroids of circles,

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development, Three Generations of Human Rights (Civil and Political Rights;

Economic, Social and Cultural Rights).

Unit-2 Human Rights and United Nations – contributions, main human rights related

organs – UNESCO,UNICEF, WHO, ILO, Declarations for women and children,

Universal Declaration of Human Rights.

Human Rights in India – Fundamental rights and Indian Constitution, Rights for

children and women, Scheduled Castes, Scheduled Tribes, Other Backward Castes and

Minorities

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Types of numbers, HCF & LCM of integers, Fractions, Simplifications (VBODMAS

rule), Squares and square roots, Ratio and proportion, Percentage, Profit & loss

Quadratic equations (Solution of quadratic equations with real roots only), Permutations

and combinations – simple applications, Trigonometry- introduction, values of trigonometric

ratios of 0⁰, 30⁰, 45⁰, 60⁰ & 90⁰, Heights and distances

and combinations – simple applications

ratios of 0⁰, 30⁰, 45⁰, 60⁰ & 90⁰, Heights and distances

Simple interest, Compound interest, Time and work, Work and wages, Time and distance,

Exponential series and logarithmic series

Elementary mensuration – Area and perimeter of polygons, Elementary Algebra, monomial,

binomial, polynomial (linear, quadratic & cubic), simple factorization of quadratic and cubic

polynomials, Differential Calculus – Differentiation – Standard results (derivatives), Product rule, Quotient rule and function of function rule (without proof) and simple problems

binomial, polynomial (linear, quadratic & cubic), simple factorization of quadratic and cubic

polynomials

Binary operations, Isomorphic binary structures, Groups-definition

and examples, elementary properties of groups, finite groups and group tables, subgroups,

cyclic subgroups, cyclic groups, elementary properties of cyclic groups.

Groups of permutations, Cayley’s theorem,

orbits, cycles and the alternating groups, cosets and the theorem of Lagrange, direct

products.

Homomorphisms, properties of homomorphisms, factor

groups, The Fundamental Homomorphism theorem, normal subgroups and inner

automorphisms, simple groups.

automorphisms

Definitions and basic properties, homomorphisms and isomorphisms,

Integral domains- divisors of zero and cancellation, integral domains, the characteristic of a

ring. Ideals and factor rings. Homomorphisms and factor rings.

A review of algebra of matrices is followed by some applications of matrices, analytic

geometry, systems of linear equations and difference equations. Systems of linear equations:

elementary matrices, the process of Gaussian elimination, Hermite or reduced row-echelon

matrices. Linear combinations of rows (columns), linear independence of columns, row

equivalent matrices, rank of a matrix, column rank, normal form, consistent systems of

equations.

Invertible matrices, left and right inverse of a matrix, orthogonal matrix, vector spaces,

subspaces, linear combination of vectors, spanning set, linear independence and basis.

Linear mappings: Linear transformations, Kernel and range, Rank and Nullity, Linear

isomorphism. Matrix connection: Ordered basis, Matrix of f relative to a fixed ordered basis,

Transition matrix from a basis to another, Nilpotent and index of nilpotency.

Eigenvalues and eigenvectors: Characteristic equation, Algebraic multiplicities, Eigen space,

Geometric multiplicities, Eigenvector, diagonalisation, Tri-diagonal matrix.

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Continuous Functions – Sequential Criterion for Continuity, Combinations of Continuous Functions, Composition of Continuous Functions, Continuous Functions on Intervals, Boundedness Theorem, Maximum-Minimum Theorem, Location of Roots Theorem, Bolzano’s Intermediate Value Theorem, Preservation of Intervals Theorem. Uniform continuity, Non uniform Continuity Criteria, Uniform Continuity Theorem, Lipschitz Functions. Monotone and Inverse Functions, Continuous Inverse Theorem, The nth Root Function

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L’ Hospital’s Rules.

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differentiation formulas, Cauchy-Riemann equation, sufficient condition for differentiability,

analytic functions, examples, harmonic functions. Elementary functions, the Exponential

function, logarithmic function, complex exponents, trigonometric functions, hyperbolic

functions, inverse trigonometric and hyperbolic functions.

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examples, upper bounds for moduli of contour integrals, antiderivates , Cauchy-Goursat

theorem (without proof ), simply and multiply connected domains, Cauchy’s integral formula,

an extension of Cauchy’s integral formula, Liouville’s theorem and fundamental theorem of

algebra, maximum modulus principle.

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Convergence of sequences and series, Taylor’s series, proof of Taylor’s theorem, examples,

Laurent’s series (without proof), examples.

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Laurent’s series

Isolated singular points, residues, Cauchy’s residue theorem, three types of isolated singular

points, residues at poles, examples. Applications of residues, evaluation of improper integrals,

examples

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points. Applications of residues, evaluation of improper integrals,

An introduction to graph. Definition of a Graph, More definitions, Vertex Degrees, Sub graphs, Paths and cycles, the matrix representation of graphs.

Trees. Definitions and Simple properties, Bridges, Spanning trees. Cut vertices and Connectivity. Euler’s Tours, the Chinese postman problem. Hamiltonian graphs & the travelling salesman problem.

Metric Spaces – Definition and Examples, Open sets, Closed Sets, Cantor set.

Convergence, Completeness, Continuous Mapping (Baire’s Theorem included).

method and the simplex method)

method and the simplex method

General Mathematical Model of LPP, Guidelines on linear Programming model formulation

and examples of LP Model formulation, Introduction to graphical method, Definitions, Graphical

solution methods of LP Problems, Special cases in linear Programming, Introduction to simplex

method, Standard form of an LPP, Simplex algorithm (Maximization case), Simplex algorithm

(Minimization case), The Big M Method, Some complications and their resolution, Types of

Linear Programming solutions

and examples of LP Model formulation

solution methods of LP Problems, Special cases in linear Programming

method, Standard form of an LPP, Simplex algorithm (Maximization case)

(Minimization case), The Big M Method, Some complications and their resolution, Types of

Linear Programming solutions

Introduction, Formulation of Dual LPP, Standard results on duality, Advantages of

Duality, Theorems of duality with proof

Introduction, Mathematical model of Transportation Problem, The Transportation Algorithm,

Methods for finding Initial solution, Test for optimality, Variations in Transportation Problem,

Maximization Transportation problem, Introduction and mathematical models of Assignment

problem, Solution methods of Assignment problem, Variations of the assignment problem

Methods for finding Initial solution, Test for optimality

problem, Solution methods of Assignment problem

Introduction, Two-person zero sum games, Pure strategic (Minimax and Maximin principles),

Games with saddle point, Mixed strategies, Games without saddle point, The rules of dominance, Solution methods: Games without saddle point (Arithmetic method, Matrix method,

Graphical method and Linear programming method)

Graphical method and Linear programming method)

Introduction, Basic concepts. Sub graphs. Degrees of vertices. Paths and Connectedness, Automorphism of a simple graph, line graphs, Operations on graphs, Graph Products. Directed Graphs: Introduction, basic concepts and tournaments.

Connectivity : Introduction, Vertex cuts and edge cuts, connectivity and edge connectivity, blocks, Cyclical edge Connectivity of a graph. Trees; Introduction, Definition, characterization and simple properties, centres and cancroids, counting the number of spanning trees, Cayley’s formula. Applications

Eulerian and Hamiltonian Graphs: Introduction, Eulereian Hamiltonian Graphs, Hamiltonian around’ the world’ game graphs, Graph Colorings: Introduction, Vertex Colorings, Applications of Graph

Coloring, Critical Graphs, Brooks’ Theorem.

Coloring.

Planarity: Introduction, Planar and Non planar Graphs, Euler Formula and Its Consequences, K₅ and K_3,3 are Nonplanar Graphs, Dual of a Plane Graph, The Four-Color Theorem and the Heawood Five-Color Theorem . Spectral Properties of Graphs: Introduction, The Spectrum of a Graph, Spectrum of the Complete Graph Kn, Spectrum of the Cycle C_n.

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spaces and subspaces over a field and their

properties, including the basis structure of vector space.

transformations

and matrices of linear

transformations and change of basis, including

kernel, range and isomorphism

eigenvectors, eigenvalues and eigenspaces, as well

as the geometric and the algebraic multiplicities of

an eigenvalue and apply the basic diagonalization

Result.

triangulations, Direct sum decompositions and

Invariant direct sums .

Vector spaces, subspaces, basis and dimension, Co-ordinates, summary of row-equivalence, Computations concerning subspaces

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Linear transformations, the algebra of linear transformations, isomorphism,

representation of transformations by matrices, linear functional, double dual,

transpose of a linear transformation.

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transpose of a linear transformation

Determinants: Commutative Rings, Determinant functions, Permutation and

uniqueness of determinants, Additional properties of determinants.

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uniqueness of determinants, Additional properties of determinants.

Introduction to elementary canonical forms, characteristic values, annihilatory

Polynomials, invariant sub spaces, Direct sum Decomposition

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Definition of a topological space – Examples of topological spaces-Bases and subbases – subspaces.

Closed sets and Closures – Neighbourhoods, Interior and Accumulation points – Continuity and Related Concepts – Making functions continuous , Quotient spaces

Smallness conditions on a space, Connectedness.

Spaces with special properties: – Local connectedness and Paths

Separation axioms:- Hierarchy of separation axioms

Direct products and finitely generated abelian groups, Fundamental theorem, Applications, Factor groups, Fundamental homomorphism theorem, Normal subgroups and inner automorphisms, Group action on a set, Isotropy subgroups, Applications of G- sets to counting

Isomorphism theorems, Sylow theorems, Applications of the Sylow theory

Fermat’s and Euler Theorems, The field of quotients of an integral domain, Rings of polynomials, Factorisation of polynomials over a field

Non commutative examples, Homeomorphisms and factor rings, Prime and Maximal Ideals

Introduction, properties of monotonic functions, functions of bounded variation, total variation, additive property of total variation, total variation on  as a function of , functions of bounded variation expressed as the difference of increasing functions, continuous functions of bounded variation, curves and paths, rectifiable path and arc length, additive and continuity properties of arc length, equivalence of paths, change of parameter.

Definition and existence of the integral, properties of the integral, integration and differentiation, integration of vector valued functions.

Discussion of main problem, Uniform convergence, Uniform convergence and Continuity, Uniform convergence and Integration, Uniform convergence and Differentiation.

Equicontinuous families of functions, the Stone – Weierstrass theorem, Power series, the exponential and logarithmic functions, the trigonometric functions, the algebraic completeness of complex field.

Introduction to extension fields, Algebraic extensions, Geometric Constructions, Finite fields

Unique factorization domains, Euclidean domains, Gaussian integers and multiplicative norms

Automorphism of fields, The isomorphism extension theorem , Splitting fields

Separable extensions, Galois Theory, Illustrations of Galois Theory, Cyclotomic Extensions

Compactness and Separation axioms , The Urysohn Characterisation of normality –Tietze Characterisation of normality.

Cartesian products of families of sets – The product topology -Productive properties

Evaluation functions into products – Embedding lemma and Tychonoff Embedding – The Urysohn Metrisation Theorem – Variation of compactness  .

Definition and convergence of nets, Homotopy of paths

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calculating residues

theorem .

The spherical representation of complex numbers , Riemann Sphere,

Stereographic projection, Distance between the stereographic projections

Elementary Theory of power series,Abel’s Theorem on convergence of the

power series, Hadamard’s formula, Abel’s limit Theorem

Arcs and closed curves, Analytic functions in regions, Conformal mappings,

Length and area ,Linear transformations , The cross ratio, Symmetry, Oriented

circles, Families of circles.

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Length and area ,Linear transformations , The cross ratio, Symmetry, Oriented

circles, Families of circles.

Fundamental theorems on complex integration: line integrals, rectifiable arcs,

line integrals as functions of arcs, Cauchy’s theorem for a rectangle, Cauchy’s

theorem in a disk,

Cauchy’s integral formula: the index of a point with respect to a cloud curve,

the integral formula.

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Higher derivatives. Differentiation under the sign of integration, Morera’s

Theorem, Liouville’s Theorem, Fundamental Theorem, Cauchy’s estimate

Local properties of analytical functions: removable singularities, Taylor’s

theorem, zeroes and poles,Weirstrass Theorem on essential singularity, the

local mapping, the maximum principle.Schwarz lemma.

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Theorem, Liouville’s Theorem, Fundamental Theorem, Cauchy’s estimate.

theorem, zeroes and poles,Weirstrass Theorem on essential singularity, the

local mapping, the maximum principle.Schwarz lemma

The general form of Cauchy’s theorem: chains and cycles, simple

connectivity, homology, general statement of Cauchy’s theorem, proof of

Cauchy’s theorem, locally exact differentiation, multiply connected regions

Calculus of Residues: the residue theorem, the argument principle, evaluation

of definite integrals.

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Cauchy’s theorem, locally exact differentiation, multiply connected regions .

programs.

handling.

Calculations and variables, creating strings, lists are more powerful than strings, tuples,  If statements, if-then-else statements, if and elif statements, combining conditions, the difference between strings and numbers, using for loops, while we are talking about looping, using functions, parts of a function, using modules. The functions abs, float, int, len, max, min, range, sum, complex numbers.

Defining Symbols and Symbolic Operations, Working with Expressions, Solving Equations and Plotting Using SymPy, problems on factor finder, summing a series and solving single variable inequalities.

Finding the limit of functions, finding the derivative of functions, higher-order derivatives and finding the maxima and minima and finding the integrals of functions are to be done. In the section programming challenges, the following problems – verify the continuity of a function at a point, area between two curves and finding the length of a curve.

Interpolation and Curve Fitting – Polynomial Interpolation – Lagrange’s Method, Newton’s Method and Limitations of Polynomial Interpolation, Roots of Equations – Method of Bisection and Newton-Raphson Method.

Gauss Elimination Method, Doolittle’s Decomposition Method only from LU Decomposition Methods

Numerical Integration, Newton-Cotes Formulas, Trapezoidal rule, Simpson’s rule and Simpson’s 3/8 rule.

Introduction, Lebesgue outer measure, The  algebra of Lebesgue measurable sets, Outer and inner approximation of Lebesgue measurable sets, Countable additivity, continuity and Borel-Cantelli Lemma – Non measurable sets – The Canter set and Canter Lebesgue function