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Sets and Propositions – Cardinality, Principal of inclusion and exclusion, Mathematical induction. Relations and Functions- Binary relations, Equivalence relations and Partitions, Partial ordered relations and Lattices, Chains and Antichains, Pigeon Hale principle.
Algebraic structures – Groups, Rings, Integral domains, Fields (Definitions, simple examples and elementary properties only). Boolean Algebras- Lattices and Algebraic structure, Duality, Distributive and Complemented Lattices. Boolean Lattices, Boolean functions and expressions.
Logic and Propositional Calculus, Propositions, Simple and compound, Basic Logical operations, Truth tables, Tautologies and contradictions, Propositional Functions, quantifiers. Discrete numeric functions and Generating functions. Recurrence relations and Recursive Algorithms – Linear Recurrence relations with constant coefficients. Homogeneous solutions. Particular solution. Total solution. Solution by the method of generating functions.
Graphs – Basic ‘terminology, Multigraphs, Weighted graphs, Paths and circuits, Shortest paths, Eulerian paths and Circuits. Travelling Salesman problem. Union, Join, Product and composition of graphs. Planar graphs and Geometric dual graphs.
Trees – Properties, Spanning tree, Binary and Rooted tree. Digraphs – Simple digraph, Asymmetric digraphs, Symmetric digraphs and complete digraphs. Digraph and Binary relations. Matrix representation of graphs and digraphs.
Series – Infinite series and Convergent series. Tests for convergence of a series- Comparison test, D’Alembert’s ratio test, Cauchy’s n-th root test, Raabe’s test, De-Morgan-Bertrand’s test, Cauchy’s condensation test, Gauss’s test, (Derivation of tests is not required). Alternating series. Absolute convergence. Taylor’s theorem. Maclaurin’s theorem. Power series expansion of a function. Power series expansion of sinx, cosx,
Exact Differential Equations: Introduction, Exact Differential Equations, Non Exact Differential Equation. Differential Equations of First Order and Higher Degree: Introduction, Differential Equation of First Order and Higher Degree, Clairaut’s Equations, Singular Solutions. Geometrical Meaning of a Differential Equation: Introduction, Geometrical meaning of a Differential Equation, Orthogonal Trajectories, Linear Differential Equations with Constant Coefficients, Ordinary Homogeneous Linear Differential Equations.
Envelopes, Maxima and Minima: of functions of two variables. Lagrange’s method of undetermined multipliers. Asymptotes. Multiple points. Curve tracing of standard curves (Cartesian and Polar curves).
Double integrals in Cartesian and Polar Coordinates, Change of order of integration. Triple integrals. Application of double and triple integrals in finding areas and volumes. Dirichlet’s integral.
Rectification: Introduction, Rectification (Length of a Curve), Different Forms of Rectification.
Polar equation of conics, polar equation of tangent, normal and asymptotes, chord of contact, auxiliary circle, dircector circle of conics. Sphere, Cone.
Cylinder, Central Conicoids – Ellipsoid, Hyperboloid of one and two sheets, tangent lines and tangent planes, Direct sphere, Normal’s.
Generating lines of hyperboloid of one sheet system of generating lines and its properties. Reduction of a general equation of second degree in three-dimensions to standard forms.
The linear programming problem. Basic solution. Some basic properties and theorems on convex sets. Fundamental theorem of L.P.P. Theory of simplex method only Duality. Fundamental theorem of duality, properties and elementary theorems on duality only
Properties of the Real Numbers: Introduction, The Real Number System, Order Structure, Bounds, Sups and Infs, the Rational Numbers Are Dense, Inductive Property of IN, The Metric Structure of R. Elementary Topology: Introduction, Compactness Arguments, Bolzano-Weierstrass Property, Cantors Intersection Property, Cousins Property, Heine-Borel Property, Compact Sets. Infinite Sum: Finite Sums, Infinite Unordered Sums, Ordered Sums: Series.
Sets of Real Numbers: Introduction, Points, Sets, Elementary Topology. Contraction Maps: Introduction, Applications of Contraction Maps (I), Applications of Contraction Maps (II), Compactness, Continuous Functions on Compact Sets, Total Boundedness, Compact Sets in C [a, b]. The Integral: Introduction, Cauchy’s First Method, Scope of Cauchy’s First Method, Properties of the Integral, Cauchy’s Second Method, Cauchy’s Second Method (Continued), The Riemann Integral, The Improper Riemann Integral.
Differentiation: Introduction, Defined the Derivative, Mean Value Theorem, Monotonicity, Dini Derivates, Convexity. Sequences: Introduction, Sequences, Divergence, Convergence, Sub sequences. Sequences and Series of Functions: Introduction, Point-wise Limits, Uniform Limits, Uniform Convergence and Continuity.
Continuous Functions: Introduction, Limits (å-ä Definition), Limits (Sequential Definition), Limits (Mapping Definition), One-Sided Limits, Infinite Limits, Properties of Limits. More on Continuous Functions and Sets: Introduction, the Baire Category Theorem, Cantor Sets, An Arithmetic Construction of K, The Cantor Function, Borel Sets. Metric Spaces: Introduction, Metric Spaces, Additional Examples, Function Spaces, Convergence, Functions.
The LP Spaces: Introduction, The Basic Inequalities, The lp and Lp Spaces (1 d” p < “), The Spaces l” and L”, Separability, The Spaces l2 and L2, Continuous Linear Functional, The Lp Spaces (0 < p < 1). The Euclidean Spaces: Introduction, The Algebraic Structure of Rn , The Metric Structure of Rn , Elementary Topology of Rn , Sequences in Rn , Coordinate-Wise Convergence, Functions and Mappings, Limits of Functions from Rn ’! Rm , Coordinate-Wise Convergence. Differentiation on Euclidean Spaces: Introduction, Partial and Directional Derivatives, Integrals Depending on a Parameter, Differentiable Functions.
Equations of First Order and First Degree: Introduction, Homogeneous Equations, Non- homogeneous Equations of First Degree in x and y, Exact Differential Equations, Integrating Factors (I.F). Equations of First Order but not of First Degree: Introduction, Equations which can be factorised into Factors of First Degree, Equations which cannot be factorised into Factors of First Degree, Equations Solvable for y, Equations Solvable for x, Equations in which either x or y is absent, Equations Homogeneous in x and y, Equations of First Degree in x and y.
Linear Equations with Constant Coefficients: Introduction, Symbolic Operator, Method of finding C. F, Methods of finding Particular Integral, To Find Particular Integral when X= eax where ‘a’ is 1 m constant, To Find Particular Integral when X=cos ax or sin ax, To Find the value of f D x where m is a positive integer.
Homogeneous Linear Equations with Variable Coefficients: Introduction, Method of Solution, To Find Complementary Function, Symbolic Notation in 5ØÉÞ, To find Particular Integral, Particular 1 m f x case to find, Equations Reducible to Homogeneous Linear Equations. Exact Differential Equations and Equations of Particular Forms: Introduction, Condition for the Exactness of the Linear Differential Equation, Solution of Non-linear Equations which are Exact, Equations of the form y (n) =f(x), Equations of the form y(2) =f(y), Equations that do not contain y directly, Equations that do not Contain x Directly, Equations in which y Appears in only Two Derivatives whose Orders Differ by Two, Homogeneous Equations. Linear Equations of Second Order: Introduction, Method of solving Equation when an integral included in the C.F. is known, Method of Solving Equation by Changing the Dependent Variable, Method of Solving Equation by Changing the Independent Variable, Solution by Factorization of the Operator, Method of Variation of Parameters, Method of Undetermined Coefficients.
Simultaneous Differential Equations: Introduction, Simultaneous Equations with Constant Coefficients, Simultaneous Equations with Variable Coefficients, Method of Solution of Equations in Symmetrical Form, Method of Introduction of a New Variable.
Total Differential Equations: Introduction, Condition of Integrability, Method of Obtaining the Primitive, Solution by Inspection, Non-Integrable Single Differential Equations, Equations Containing More Than Three Variables, Equations Containing More Than Three Variables of Method of Solution.
Partial Differential Equations of First Order: Introduction, Classification of Integrals, Singular Integral, Geometrical Interpretation of three Types of Integrals, Singular Integral from the Partial Differential Equation Directly, Derivation of Partial Differential Equations by the Elimination of Arbitrary Functions, Solution of Partial Differential Equations.
Linear Partial Differential Equations With Constant Co-efficient: Introduction, to find Complementary Function, Particular Integral.
Partial Diff Equations of Order Two with Variable Co-efficient: Introduction, Laplace’s Transformation, Non-Linear Partial Differential Equations of order two, Monge’s Method of integrating Rr+Ss+Tt=V.
Legendre’s Equation and Simple Properties of Pn (x): Introduction, Solution of Legendre’s Equation, Legendre Polynomial, Rodrigue’s Formula, Recurrence Formulae, Laplace’s First Integral for Pn(x), Laplace’s Second Integral for Pn(x).
Bessel’s Equation and Bessel Function: Introduction, Solution of Bessel’s Equation, Recurrence Formula for Jn(x).
Laplace Transform and its Application to Differential Equations: Introduction, Laplace Transform of some Elementary Functions, Properties of Laplace Transforms, Laplace Transform of Derivatives, Laplace Transform of Integrals, Properties of Inverse Laplace Transforms, Applications to Differential Equations.
Fourier Transform and Its Application to Partial Differential Equations: Introduction, Derivative of Fourier Transform, Fourier Sine and Cosine-Transforms, Finite Fourier Transform, Application to Partial Differential Equations.
Approximations and Errors in Computation: Introduction, Accuracy of Numbers, Error in the Approximation of a Function, Error in a Series Approximation, Order of Approximation, Propagation of Error.
Finite Differences: Introduction, Finite Differences, Differences of a Polynomial, Factorial Notation, Effect of an Error on a Difference Table, Relations between the Operators, Application to Summation of Series.
Interpolation: Introduction, Newton’s Forward Interpolation Formula, Newton’s Backward Interpolation Formula.
Central Differences Interpolation Formula: Introduction, Central Difference Interpolation Formulae, Gauss’s Forward Interpolation Formula, Gauss’s Backward Interpolation Formula, Stirling’s Formula, Bessel’s Formula, Laplace-Everett’s Formula, Choice of an Interpolation Formula.
Interpolation for Unequal Intervals: Introduction, Lagrange’s Interpolation Formula, Divided Differences, Newton’s Divided Difference Formula, Hermite’s Interpolation Formula, Spline Interpolation, Lagrange’s Method, Iterative Method.
Inverse Interpolation: Introduction, Lagrange’s Method, Summation of a Series, Cubic-Spline Interpolation Formulas, Bivariate Interpolation, Least Square Approximation.
Solution of Algebraic And Transcendental Equations: Introduction, Basic Properties of Equations, Transformation of Equations, Bisection (Or Bolzano) Method, Method of False Position or Regula-FalsiMethod, Newton-Raphson Method.
Solutions of Simultaneous Linear Equations: Introduction, Direct Methods of Solution, Comparison of Various Methods.
Numerical Solution of Differential Equations: Introduction, Formulae for Derivatives, Numerical Integration, Newton-Cotes Quadrature Formula, Euler-Maclaurin Formula.
Numerical Solution of Linear Differential Equations: Introduction, Picard’s Method, Taylor’s Series Method, Euler’s Method, Runge’s Method, Runge-Kutta Method.
Numerical Solution of Ordinary Differential Equations: Introduction, Classification of Second Order Equations, Finite Difference Approximations to Partial Derivatives, Elliptic Equations, Solution of Laplace Equation, Solution of Poisson’s Equation, Solution of Elliptic Equations by Relaxation Method, Parabolic Equations.
Vector: Introduction, Addition of Vectors, Rectangular Resolution of a Vector, Unit Vector, Position Vector of a Point, Ratio Formula, Vector Product or Cross Product, Moment of a Force, Angular Velocity.
Vector Calculus: Introduction, Vector Differentiation, Gradient, Divergence and Curl, More Identities Involving, Vector Integration.
Vector Theorem: Introduction, Theorems of Gauss, Green’s Theorems, Stokes’s Theorems, Verification of Stocks and Gauss Theorem.
Ordinary Differential Equations: Introduction, Concept and Formation of a Differential Equation, Order and Degree of a Differential Equation.
Theory of Equations : Polynomials in one variable and the division algorithm. Relations between the roots and the coefficients. Transformation of equations. Descartes rule of signs. Solution of cubic and biquadratic (quartic) equations.
Matrix Addition and Multiplication : Diagonal, permutation, triangular, and symmetric matrices. Rectangular matrices and column vectors. Non-singular transformations. Inverse of a Matrix. Rank-nullity theorem. Equivalence of row and column ranks. Elementary matrices and elementary operations. Equivalence and canonical form. Determinants. Eigenvalues, eigenvectors, and the characteristic equation of a matrix. Cayley-Hamilton theorem and its use in finding the inverse of a matrix. Matrix Theory and Linear Algebra :
Matrix Theory and Linear Algebra in R”. Systems of linear equations, Gauss elimination, and consistency. Subspaces of R”, linear dependence, and dimension. Matrices, elementary row operations, row-equivalence, and row space.
Linear Equations as Matrix Equations : Systems of linear equations as matrix equations, and the invariance of its solution set under row-equivalence. Row-reduced matrices, row-reduced echelon matrices, row-rank, and using these as tests for linear dependence. The dimension of the solution space of a system of independent homogeneous linear equations.Linear transformations and matrix representation.
Modern Algebra : Commutative rings, integral domains, and their elementary properties. Ordered integral domain: The integers and the well-ordering property of positive elements. Finite induction. Divisibility, the division algorithm, primes, GCDs, and the Euclidean algorithm.
The Fundamental Theorem of Arithmetic : Congruence modulo n and residue classes. The rings Z„ and their properties. Units in Zn, and Zp for prime p. Subrings and ideals. Characteristic of a ring. Fields.Sets, relations, and mappings. Bijective, injective, and surjective maps. Composition and restriction of maps. Direct and inverse images and their properties. Finite, infinite, countable, uncountable sets, and cardinality.
Equivalence Relations and Partitions : Ordering relations. Definition of a group, with examples and simple properties. Groups of transformations. Subgroups. Generation of groups and cyclic groups. Various subgroups of GL2(R). Coset decomposition. Lagrange’s theorem and its consequences. Fermat’s and Euler’s theorems. Permutation groups. Even and odd permutations. The alternating groups An.
Isomorphism and Homomorphism : Normal subgroups. Quotient groups.First homomorphism theorem. Cayley’s theorem.Trigonometry. De-Moivre’s theorem and applications. Direct and inverse, circular and hyperbolic, functions. Logarithm of a complex quantity. Expansion of trigonometric functions.
Linear Algebra : Vector spaces over a field, subspaces. Sum and direct sum of subspaces. Linear span. Linear dependence and independence. Basis. Finite dimensional spaces. Existence theorem for bases in the finite dimensional case. Invariance of the number of vectors in a basis, dimension. Existence of complementary subspace of any subspace of a finite-dimensional vector space. Dimensions of sums of subspaces. Quotient space and its dimension.
Algebra of Linear Transformations : Matrices and linear transformations, change of basis and similarity. Algebra of linear transformations. The rank-nullity theorem. Change of basis. Dual space. Bidual space and natural isomorphism. Adjoints of linear transformations. Eigenvalues and eigenvectors. Determinants, characteristic and minimal polynomials,
Cayley-Hamilton Theorem : Cayley-Hamilton Theorem. Annihilators. Diagonalization and triangularization of operators. Invariant subspaces and decomposition of operators. Canonical forms. Inner product spaces. Cauchy-Schwartz inequality. Orthogonal vectors and orthogonal complements. Orthonormal sets and bases. Bessel’s inequality. GramSchmidt orthogonalization method. Hermitian, Self-Adjoint, Unitary, and Orthogonal transformation for complex and real spaces. Bilinear and Quadratic forms. The Spectral Theorem. The structure of orthogonal transformations in real Euclidean spaces. Applications to linear differential equations with constant coefficients.
Advanced Group Theory : Advanced Group Theory. Group automorphisms, inner automorphisms. Automorphism groups and their computations. Conjugacy relation. Normalizer. Counting principle and the class equation of a finite group. Center of a group. Free abelian groups. Structure theorem of finitely generated abelian groups.
Ring Theory : Rings and ring homomorphisms. Ideals and quotient rings. Prime and maximal ideals. The quotient field of an integral domain. Euclidean rings. Polynomial rings. Polynomials over Q and Eisenstein’s criterion. Polynomial rings over arbitrary commutative rings. UFDs. If A is a UFD, then so is A[x\, x2,…, xn]
Complex Functions: The complex number system, Polar form of complex numbers, Square roots, Stereographic projection. Möbius transforms, Polynomials, rational functions and power series
Analytic Functions: Conformal mappings and analyticity, Analyticity of power series; elementary functions, Conformal mappings by elementary functions
Integration: Complex integration, Goursat’s theorem, Local properties of analytic functions, A general form of Cauchy’s integral theorem, Analyticity on the Riemann sphere
Singularities: Singular points, Laurent expansions and the residue theorem, Residue calculus, The argument principle
Harmonic functions: Fundamental properties Dirichlet’s problem
Entire functions: Sequences of analytic functions, Inûnite products, Canonical products, Partial fractions. Hadamard’s theorem
The Riemann mapping theorem The Gamma function: Complex Numbers and the Complex Plane, Complex Numbers and Their Properties, Complex Plane, Polar Form of Complex Numbers, Powers and Roots, Sets of Points in the Complex Plane, Applications Complex Functions and Mappings : Complex Functions,
Complex Functions as Mappings, Linear Mappings, Special Power Functions, The Power Function zn , The Power Function z1/n, Reciprocal Function, Limits and Continuity, Limits, Continuity, Applications
Analytic Functions : Diûerentiability and Analyticity, Cauchy-Riemann Equations, HarmonicFunctions, Applications,
Elementary Functions : Exponential and Logarithmic Functions, Complex Exponential Function, Complex Logarithmic Function, Complex Powers, Trigonometric and Hyperbolic Functions, Complex Trigonometric Functions, Complex Hyperbolic Functions, Inverse Trigonometric and Hyperbolic Functions, Applications
Integration in the Complex Plane : Real Integrals, Complex Integrals, Cauchy-Goursat Theorem, Independence of Path, Cauchy’s Integral Formulas and Their Consequences, Cauchy’s. Two Integral Formulas, Some Consequences of the Integral Formulas, Applications
Laurent Series : Zeros and Poles, Residues and Residue Theorem, Some Consequences of the Residue Theorem, Evaluation of Real Trigonometric Integrals, Evaluation of Real Improper Integrals, Integration along a Branch Cut, The Argument Principle and Rouch´e’s Theorem, Summing Inûnite Series, Applications
Conformal Mappings: Conformal Mapping, Linear Fractional Transformations, Schwarz-Christoûel Transformations, Poisson Integral Formulas, Applications, Boundary-Value Problems, Fluid Flow
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