User:Bomanhoo19/Mathematics

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Mathematics Topics

Map of Mathematics

Origins	Foundations	Pure Mathematics	Applied Mathematics	miscellaneous
History of Mathematics	Foundational Rules	Number Systems $\mathbb {N}$ Natural numbers $\mathbb {Z}$ Integers $\mathbb {Q}$ Rational numbers $\mathbb {R}$ Real numbers $\mathbb {C}$ Complex numbers	Mathematical Physics Mathematical Chemistry Bio Mathematics	Lists of mathematics topics Axioms Mathematical Objects
Table of mathematical symbols by introduction date	Mathematical logic	Structures Algebra Linear Agebra Number Theory Combinatorics Group Theory Order Theory	Engeneering Numerical Analysis Economics Mathematical Finance
	Set Theory	Spaces Geometry Trigonometry Fractal Geometry Toplogy Measure Theory Differential Geometry	Probability Game Theory Statistics Optimization
	Category Theory	Changes Calculus Vector Calculus Dynamical System Chaos Theory	Computer Science Machine Learning Cryptography
	Theory of Computation

Mathematics

1. Numbers

Number systems are the sets of numbers used to represent quantities and perform mathematical operations. These systems have evolved over time, and different types of number systems have been developed to meet different needs.

1. Natural Numbers: Natural numbers are the set of positive integers, including zero (0), that is, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}. Natural numbers are used for counting and ordering, and they are the foundation of all other number systems.

2. Whole Numbers: Whole numbers are the set of non-negative integers, that is, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}. Whole numbers include all natural numbers and the number zero.

3. Integers: Integers are the set of whole numbers and their negatives, that is, {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers can be used to represent quantities that have both magnitude and direction, such as temperature or altitude.

4. Rational Numbers: Rational numbers are numbers that can be expressed as the ratio of two integers, that is, a/b where a and b are integers and b is not equal to zero. Rational numbers include fractions and terminating or repeating decimals, such as 1/2, 0.75, and 0.3333….

5. Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a ratio of two integers. These numbers have decimal expansions that neither terminate nor repeat, such as √2, π, and e.

6. Real Numbers: Real numbers are the set of all rational and irrational numbers, that is, the numbers that can be represented on a number line. Real numbers include all decimal numbers, positive or negative, rational or irrational.

7. Complex Numbers: Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as i² = -1. Complex numbers can be used to represent quantities that have both real and imaginary components, such as electrical currents or magnetic fields.

8. Quaternion Numbers: Quaternion numbers are a type of hypercomplex numbers that extend the concept of complex numbers to four dimensions. Quaternion numbers are of the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary units that satisfy the relations i² = j² = k² = -1 and ij = -ji = k, jk = -kj = i, and ki = -ik = j.

9. Hyper Complex Numbers: Hyper complex numbers are a generalization of the concept of complex numbers to higher dimensions. These numbers are used in fields such as physics, engineering, and computer graphics, where quantities with more than three dimensions are common. Examples of hypercomplex numbers include split-complex numbers, dual numbers, and tessarines.

	Numbers	1,2,3...		A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words.
	Set of Numbers	Main number systems $\mathbb {N}$ Natural numbers 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ... $\mathbb {N} _{0}$ or $\mathbb {N} _{1}$ are sometimes used. $\mathbb {Z}$ Integers ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... $\mathbb {Q}$ Rational numbers a/b where a and b are integers and b is not 0 $\mathbb {R}$ Real numbers The limit of a convergent sequence of rational numbers $\mathbb {C}$ Complex numbers a + bi where a and b are real numbers and i is a formal square root of −1 Hypercomplex number A definition of a hypercomplex number is given as an element of a unital, but not necessarily associative or commutative, finite-dimensional algebra over the real numbers. $\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C}$	Set of Numbers	Number systems Complex $:\;\mathbb {C}$ Real $:\;\mathbb {R}$ Rational $:\;\mathbb {Q}$ Integer $:\;\mathbb {Z}$ Natural $:\;\mathbb {N}$ Zero: 0 One: 1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Transcendental Imaginary	.
	Arithmatic Operationss	Arithmatic Operations Addition (+) Subtraction (−) Multiplication (×) Division (÷) Exponentiation (^) nth root (√) Logarithm (log)	Basic arithmetic operators	Addition Operation. Multiplication Operation
	Properties	Properties Addition (+) Multiplication (×) Closure property $a\in S,b\in S\Rightarrow (a+b)\in S$ $a\in S,b\in S\Rightarrow (a.b)\in S$ Associative property $(a+b)+c=a+(b+c)$ $(a.b).c=a.(b.c)$ Existence of Identity $a+e=a$ $a.e=a$ Existence of Inverse $a+(-a)=e$ $a.(1/a)=e$ Commutative property $(a+b)=(b+a)$ $(a.b)=(b.a)$ Distributive Property $a.(b+c)=a.b+a.c$	Associativity Commutativity distributive property
	Group-like structure	(G,+) or (G.*) Quasi Group Closure property Semi Group Closure property Associative property Monoid Closure property Associative property Existence of Identity Group Closure property Associative property Existence of Identity Existence of Inverse Abelian Group Closure property Associative property Existence of Identity Existence of Inverse Commutative property	Algebraic structures - magma to group	Group-like structures Totality Associativity Identity Divisibility Commutativity Partial magma Unneeded Unneeded Unneeded Unneeded Unneeded Semigroupoid Unneeded Required Unneeded Unneeded Unneeded Small category Unneeded Required Required Unneeded Unneeded Groupoid Unneeded Required Required Required Unneeded Magma Required Unneeded Unneeded Unneeded Unneeded Quasigroup Required Unneeded Unneeded Required Unneeded Unital magma Required Unneeded Required Unneeded Unneeded Loop Required Unneeded Required Required Unneeded Semigroup Required Required Unneeded Unneeded Unneeded Associative quasigroup Required Required Unneeded Required Unneeded Monoid Required Required Required Unneeded Unneeded Commutative monoid Required Required Required Unneeded Required Group Required Required Required Required Unneeded Abelian group Required Required Required Required Required ^α The closure axiom, used by many sources and defined differently, is equivalent. ^β Here, divisibility refers specifically to the quasigroup axioms.
	Ring Like Structure	(G,+,*) + x Ring 1 2 3 4 5 1 2 Rng Semiring Near-ring Commutative ring Domain Integral domain Field 1 2 3 4 5 1 2 3 4 5 Division ring Lie ring	Ring-Field Theory Panda.png	rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
	Space	n-dimensional- Euclidean Space ($R^{n}$) Hilbert Space Banach Space Inner Product Space Normed Vector Space Locally Convex Space Metric Space Vector Space Topological Space Manifold	Mathematical Spaces Functions between metric spaces	Fig. 1: Overview of types of abstract spaces. An arrow indicates is also a kind of; for instance, a normed vector space is also a metric space.
	Functions	Classes/properties  Constructions Generalizations   Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective Restriction Composition λ Inverse Partial Multivalued Implicit space	Function	A function from a set X to a set Y is an assignment of an element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function. A function, its domain, and its codomain, are declared by the notation f: X→Y, and the value of a function f at an element x of X, denoted by f(x), is called the image of x under f, or the value of f applied to the argument x.
	Function Space	continuous functions endowed with the uniform norm topology continuous functions with compact support bounded functions continuous functions which vanish at infinity continuous functions that have continuous first r derivatives. smooth functions smooth functions with compact support real analytic functions , for , is the Lp space of measurable functions whose p-norm  is finite , the Schwartz space of rapidly decreasing smooth functions and its continuous dual,  tempered distributions compact support in limit topology Sobolev space of functions whose weak derivatives up to order k are in holomorphic functions linear functions piecewise linear functions continuous functions, compact open topology all functions, space of pointwise convergence Hardy space Hölder space Càdlàg functions, also known as the Skorokhod space , the space of all Lipschitz functions on  that vanish at zero.		In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.
	Functional	Duality Definite integral Inner product spaces Locality	The arc length functional has as its domain the vector space of rectifiable curves – a subspace of  – and outputs a real scalar. This is an example of a non-linear functional. The Riemann integral is a linear functional on the vector space of functions defined on [a, b] that are Riemann-integrable from a to b.

2. Shapes & Space

Area

Basic Shapes

	Shape
	Lists of shapes
	List of mathematical shapes
	List of two-dimensional geometric shapes
	Solid geometry

3. Operations

Operators

	Operation (mathematics)
	Set operations (mathematics)

Functions

Class Wise

I

Chapter 1 - Shape and Space

Chapter 2 - Numbers from One to Nine

Chapter 3 - Addition

Chapter 4 - Subtraction

Chapter 5 - Numbers from Ten to Twenty

Chapter 6 - Time

Chapter 7 - Measurement

Chapter 8 - Numbers from Twenty-one to Fifty

Chapter 9 - Data Handling

Chapter 10 - Patterns

Chapter 11 - Numbers

Chapter 12 - Money

Chapter 13 - How Many

II

Chapter 1: What is Long, What is Round?

Chapter 2: Counting in Groups

Chapter 3: How Much Can You Carry?

Chapter 4: Counting in Tens

Chapter 5: Patterns

Chapter 6: Footprints

Chapter 7: Jugs and Mugs

Chapter 8: Tens and Ones

Chapter 9: My Funday

Chapter 10: Add Our Points

Chapter 11: Lines and Lines

Chapter 12: Give and Take

Chapter 13: The Longest Step

Chapter 14: Birds Come, Birds Go

Chapter 15: How Many Ponytails

III

Chapter 1 Where to Look from?
Chapter 2 Fun with Numbers
Chapter 3 Give and Take
Chapter 4 Long and Short
Chapter 5 Shapes and Designs
Chapter 6 Fun with Give and Take
Chapter 7 Time Goes on
Chapter 8 Who Is Heavier?
Chapter 9 How Many Times?
Chapter 10 Play with Patterns
Chapter 11 Jugs and Mugs
Chapter 12 Can We Share?
Chapter 13 Smart Charts
Chapter 14 Rupees and Paise

IV

	Mathematics
1.	Building with Bricks
2.	Long and Short
3.	A Trip to Bhopal
4.	Tick-Tick-Tick
5.	The Way The World Looks
6.	The Junk Seller
7.	Jugs and Mugs
8.	Carts and Wheels
9.	Halves and Quarters
10.	Play with Patterns
11.	Tables and Shares
12.	How Heavy? How Light?
13.	Fields and Fences
14.	Smart Charts

V

Lesson 1 The Fish Tale

Lesson 2 Shapes and Angles

Lesson 3 How Many Squares?

Lesson 4 Parts and Wholes

Lesson 5 Does it Look the Same?

Lesson 6 Be My Multiple,I’ll be Your Factor

Lesson 7 Can You See the Pattern?

Lesson 8 Mapping Your Way

Lesson 9 Boxes and Sketches

Lesson 10 Tenths and Hundredths

Lesson 11 Area and its Boundary

Lesson 12 Smart Charts

Lesson 13 Ways to Multiply and Divide

Lesson 14 How Big? How Heavy?

VI

Chapter 1 Knowing Our Numbers Bodmas

Chapter 2 Whole Numbers

Chapter 3 Playing with Numbers

Chapter 4 Basic Geometrical Ideas

Chapter 5 Understanding Elementary Shapes

Chapter 6 Integers

Chapter 7 Fractions

Chapter 8 Decimals

Chapter 9 Data Handling

Chapter 10 Mensuration

Chapter 11 Algebra

Chapter 12 Ratio and Proportion

Chapter 13 Symmetry

Chapter 14 Practical Geometry

VII

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Chapter 1: Integers 1.2 Recall 1.3 Properties of Addition and Subtraction of Integers Type of Numbers Operation Result Example Positive + Positive Add Positive (+) 10 + 15 = 25 Negative + Negative Add Negative (-) (-10) + (-15) = -25 Positive + Negative* Subtract Positive (+) (-10) + 15 =5 Negative + Positive* Subtract Negative (-) 10 + (-15)= -5 1.4 Multiplication of Integers (+) × (+) = + Plus x Plus = Plus (+) x (-) = – Plus x Minus = Minus (-) × (+) = – Minus x Plus = Minus (-) × (-) = + Minus x Minus = Plus 1.5 Properties of Multiplication of Integers 1.6 Division of Integers 1.7 Properties of Division of Integers

Chapter 2: Fractions and Decimals Hundreds Tens Ones Decimal Point (.) Tenths Hundredths Thousandths Ten-thousandths 2.1 Introduction 2.2 How well have you learnt about fractions? 2.3 Multiplication of Fractions 2.4 Division of Fractions 2.5 How well have you learn about Decimal Numbers 2.6 Multiplication of Decimal Numbers 2.7 Division of Decimal Numbers

Chapter 3: Data Handling 3.1 Introduction to Data Handling 3.2 Collecting Data 3.3 Organisation of Data 3.4 Representative Values 3.5 Arithmetic Mean ${\bar {x}}={\frac {1}{n}}\left(\sum _{i=1}^{n}{x_{i}}\right)={\frac {x_{1}+x_{2}+\dots +x_{n}}{n}}$ 3.6 Mode 3.7 Median Type Description Example Result Arithmetic mean Sum of values of a data set divided by number of values (1+2+2+3+4+7+9) / 7 4 Median Middle value separating the greater and lesser halves of a data set 1, 2, 2, 3, 4, 7, 9 3 Mode Most frequent value in a data set 1, 2, 2, 3, 4, 7, 9 2 3.8 Use of Bar Graphs with a Different Purpose 3.9 Chance and Probability

Chapter 4: Simple Equations 4.1 A Mind-reading Game 4.2 Setting up an Equation 4.3 Review of What we know 4.4 What Equation is? 4.5 More Equations 4.6 From Solution to Equation 4.7 Applications of Simple Equations to Practical Situations

Chapter 5: Lines and Angles 5.1 Introduction to Lines and Angles 5.2 Related Angles 5.3 Pairs of Lines 5.4 Checking for Parallel Lines

Chapter 6: The Triangle and its Properties 6.1 Introduction to Triangles 6.2 Medians of a Triangle 6.3 Altitudes of a Triangle 6.4 Exterior Angle of a Triangle and its Property 6.5 Angle Sum Property of a Triangle 6.6 Two Special Triangles: Equilateral and Isosceles 6.7 Sum of the Lengths of Two Sides of a Triangle 6.8 Right-angled triangles and Pythagoras Property

Chapter 7: Congruence of Triangles 7.1 Introduction 7.2 Congruence of Plane Figures 7.3 Congruence among Line Segments 7.4 Congruence of Angles 7.5 Congruence of Triangles 7.6 Criteria for Congruence of Triangles 7.7 Congruence among Right-angled triangles

Chapter 8: Comparing Quantities 8.1 Introduction 8.2 Equivalent Ratios 8.3 Percentage – Another way of Comparing Quantities 8.4 Use of Percentage 8.5 Prices Related to an Item or Buying and Selling 8.6 Charge Given on Borrowed Money or Simple Interest

Chapter 9: Rational Numbers 9.1 Introduction 9.2 Need for Rational Numbers 9.3 What are Rational Numbers? 9.4 Positive and Negative Rational Numbers 9.5 Rational Numbers on a Number Line 9.6 Rational Numbers in Standard Form 9.7 Comparison of Rational Numbers 9.8 Rational Numbers Between Two Rational Numbers 9.9 Operations on Rational Numbers

Chapter 10: Practical Geometry 10.1 Introduction 10.2 Construction of a Line Parallel to a Given line, through a Point not on the Line 10.3 Construction of Triangles 10.4 Constructing a Triangle When The Lengths of Its Three Sides Are Known (SSS Criterion) 10.5 Constructing a Triangle When The Lengths of Its Two Sides and the Measure of the Angle Between them are known. (SAS Criterion) 10.6 Constructing a Triangle when the Measures of Two of its Angles and The Length of the Side included Between Them is Given (ASA Criterion) 10.7 Constructing a Right-angled Triangle when the Length of one leg and its hypotenuse are given (RHS Criterion)

Chapter 11: Perimeter and Area 11.1 Introduction to Area and Perimeter 11.2 Squares and Rectangles 11.3 Area of a Parallelogram 11.4 Area of a Triangle 11.5 Circles 11.6 Conversion of Units 11.7 Applications

Chapter 12: Algebraic Expressions 12.1 Introduction 12.2 How are Expressions Formed? 12.3 Terms of an Expression 12.4 Like and Unlike Terms 12.5 Monomials, Binomials, Trinomials and Polynomials 12.6 Addition and Subtraction of Algebraic Expressions 12.7 Finding the Value of an Expression 12.8 Using Algebraic Expressions – Formulas and Rules

Chapter 13: Exponents and Powers 13.1 Introduction 13.2 Exponents 13.3 Laws of Exponents 13.4 Miscellaneous Examples Using the Laws of Exponents 13.5 Decimal Number System 13.6 Expressing Large Numbers in the Standard Form

Chapter 14: Symmetry 14.1 Introduction 14.2 Lines of Symmetry for Regular polygons 14.3 Rotational Symmetry 14.4 Line Symmetry and Rotational Symmetry

Chapter 15: Visualizing Solid Shapes 15.1 Introduction: Plane Figures and Solid Shapes 15.2 Faces, Edges and Vertices 15.3 Nets for Building 3-D Shapes 15.4 Drawing Solids on a Flat Surface 15.5 Viewing Different Sections of a Solid

VIII

Chapter 1: Rational Numbers

Chapter 2: Linear Equation in One Variable

Chapter 3: Understanding Quadrilaterals

Chapter 4: Practical Geometry

Chapter 5: Data Handling

Chapter 6: Squares and Square Roots

Chapter 7: Cubes and Cube Roots

Chapter 8: Comparing Quantities

Chapter 9: Algebraic Expressions and Identities

Chapter 10: Visualising Solid Shapes

Chapter 11: Mensuration

Chapter 12: Exponents and Powers

Chapter 13: Direct and Inverse Proportions

Chapter 14: Factorisation

Chapter 15: Introduction to Graphs

Chapter 16: Playing with Numbers

IX

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Chapter 1: Number Systems

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equations in Two Variables

Chapter 5: Euclid’s Geometry

Chapter 6: Lines and Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms and Triangles

Chapter 10: Circles

Chapter 11: Constructions

Chapter 12. Heron’s Formula

Chapter 13: Surface Areas and Volumes

Chapter 14: Statistics

Chapter 15: Probability

X

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Chapter 1: Real Numbers 1.1 Introduction to Real Numbers 1.2 Euclid’s Division Lemma 1.3 The Fundamental Theorem of Arithmetic 1.4 Revisiting Irrational Numbers 1.5 Revisiting Rational Numbers and Their Decimal Expansions 1.6 Summary

Chapter 2: Polynomials 2.1 Introduction Polynomials 2.2 Geometrical Meaning of the Zeroes of a Polynomial 2.3 Relationship between Zeroes and Coefficients of a Polynomial 2.4 Division Algorithm for Polynomials 2.5 Summary

Chapter 3: Pair Of Linear Equations In Two Variables 3.1 Introduction to Pair Of Linear Equations In Two Variables 3.2 Pair of Linear Equations in Two Variables 3.3 Graphical Method of Solution of a Pair of Linear Equations 3.4 Algebraic Methods of Solving a Pair of Linear Equations 3.5 Equations Reducible to a Pair of Linear Equations in Two Variables 3.6 Summary

Chapter 4: Quadratic Equations 4.1 Introduction to Quadratic Equations 4.2 Quadratic Equations 4.3 Solution of a Quadratic Equation by Factorisation 4.4 Solution of a Quadratic Equation by Completing the Square 4.5 Nature of Roots 4.6 Summary

Chapter 5: Arithmetic Progressions 5.1 Introduction Arithmetic Progressions 5.2 nth Term of an AP 5.3 Sum of First n Terms of an AP 5.4 Summary

Chapter 6: Triangles 6.1 Triangles Introduction 6.2 Similar Figures 6.3 Similarity of Triangles 6.4 Criteria for Similarity of Triangles 6.5 Areas of Similar Triangles 6.6 Pythagoras Theorem 6.7 Summary

Chapter 7: Coordinate Geometry 7.1 Coordinate Geometry Introduction 7.2 Distance Formula 7.3 Section Formula 7.4 Area of a Triangle 7.5 Summary

Chapter 8: Introduction To Trigonometry 8.1 Introduction To Trigonometry 8.2 Trigonometric Ratios 8.3 Trigonometric Ratios of Some Specific Angles 8.4 Trigonometric Ratios of Complementary Angles 8.5 Trigonometric Identities 8.6 Summary

Chapter 9: Some Applications Of Trigonometry 9.1 Introduction to Some Applications Of Trigonometry 9.2 Heights and Distances 9.3 Summary

Chapter 10: Circles 10.1 Circles Introduction 10.2 Tangent to a Circle 10.3 Number of Tangents from a Point on a Circle 10.4 Summary

Chapter 11: Constructions 11.1 Introduction to Constructions 11.2 Division of a Line Segment 11.3 Construction of Tangents to a Circle 11.4 Summary

Chapter 12: Areas Related To Circles 12.1 Areas Related To Circles Introduction 12.2 Perimeter and Area of a Circle-A Review 12.3 Areas of Sector and Segment of a Circle 12.4 Areas of Combination of Plane Figures 12.5 Summary

Chapter 13: Surface Areas And Volumes 13.1 Surface Areas And Volumes Introduction 13.2 Surface Area of a Combination of Solids 13.3 Volume of a Combination of Solids 13.4 Conversion of Solid from One Shape to Another 13.5 Frustum of a Cone 13.6 Summary

Chapter 14: Statistics 14.1 Statistics Introduction 14.2 Mean of Grouped Data 14.3 Mode of Grouped Data 14.4 Median of Grouped Data 14.5 Graphical Representation of Cumulative Frequency Distribution 14.6 Summary

Chapter 15: Probability 15.1 Probability Introduction 15.2 Probability-A Theoretical Approach 15.3 Summary

XI

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Chapter 1: Sets Sets and Their Representations Empty Set Finite and Infinite Sets Equal Sets Subsets Subsets of a set of real numbers especially intervals (with notations) Power Set Universal Set Venn Diagrams Union and Intersection of Sets Difference of Sets Complement of a Set Properties of Complement

Chapter 2: Relations & Functions Ordered pairs and Cartesian product of sets Number of elements in the cartesian product of two finite sets Cartesian product of the sets of real (up to R × R) Definition of − Relation Pictorial diagrams Domain, Co-domain and Range of a relation Function as a special kind of relation from one set to another Pictorial representation of a function, domain, co-domain and range of a function Real valued functions, domain and range of these functions − Constant Identity Polynomial Rational Modulus Signum Exponential Logarithmic Greatest integer functions (with their graphs) Sum, difference, product and quotients of functions

Chapter 3: Trigonometric Functions Introduction to Trigonometric Functions Positive and negative angles Measuring angles in radians and in degrees and conversion of one into other Definition of trigonometric functions with the help of unit circle Truth of the sin2x + cos2x = 1, for all x Signs of trigonometric functions Domain and range of trigonometric functions Graphs of Trigonometric Functions Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a

Chapter 4: Principle of Mathematical Induction Introduction to Principle of Mathematical Induction Process of the proof by induction − Motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers The principle of mathematical induction and simple applications

Chapter 5: Complex Numbers and Quadratic Equations Need for complex numbers, especially √1, to be motivated by inability to solve some of the quadratic equations Algebraic properties of complex numbers Argand plane and polar representation of complex numbers Statement of Fundamental Theorem of Algebra Solution of quadratic equations in the complex number system Square root of a complex number

Chapter 6: Linear Inequalities Introduction to Linear inequalities Algebraic solutions of linear inequalities in one variable and their representation on the number line Graphical solution of linear inequalities in two variables Graphical solution of system of linear inequalities in two variables

Chapter 7: Permutations and Combinations Introduction to Permutations and Combinations Fundamental principle of counting Factorial n (n!) Permutations and combinations Derivation of formulae and their connections Simple applications

Chapter 8: Binomial Theorem History Statement and proof of the binomial theorem for positive integral indices Pascal’s triangle General and middle term in binomial expansion Simple applications

Chapter 9: Sequence and Series Sequence and Series Arithmetic Progression (A.P.) Arithmetic Mean (A.M.) Geometric Progression (G.P.) Arithmetic and Geometric series infinite G.P. and its sum Geometric mean (G.M.) Relation between A.M. and G.M.

Chapter 10: Straight Lines Introduction to Straight Lines Brief recall of two dimensional geometries from earlier classes Shifting of origin Slope of a line and angle between two lines Various forms of equations of a line − Parallel to axis Point-slope form Slope-intercept form Two-point form Intercept form Normal form General equation of a line Equation of family of lines passing through the point of intersection of two lines Distance of a point from a line

Chapter 11: Conic Sections Sections of a cone − Circles Ellipse Parabola Hyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of − Parabola Ellipse Hyperbola Standard equation of a circle

Chapter 12: Introduction to Three–dimensional Geometry Three–dimensional Geometry Coordinate axes and coordinate planes in three dimensions Coordinates of a point Distance between two points Section Formula

Chapter 13: Limits and Derivatives Limits Derivatives Limits of the trigonometric functions Algebra of the derivative of the function

Chapter 14: Mathematical Reasoning Introduction to Mathematical Reasonin Mathematically acceptable statements Connecting words/ phrases Validating the statements involving the connecting words difference between contradiction, converse and contrapositive

Chapter 15: Statistics Introduction to Statistics Measures of dispersion − Range Mean deviation Variance Standard deviation of ungrouped/grouped data Analysis of frequency distributions with equal means but different variances.

Chapter 16: Probability Probability Introduction Random experiments − Outcomes Sample spaces (set representation) Types of Events Occurrence of events, ‘not’, ‘and’ and ‘or’ events Exhaustive events Mutually exclusive events Axiomatic (set theoretic) probability Connections with the theories of earlier classes Probability of − An event Probability of ‘not’, ‘and’ and ‘or’ events

XII

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Chapter 1: Relations & Functions

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity and Differentiability

Chapter 6: Applications of Derivatives

Chapter 7: Integrals

Chapter 8: Application of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

JEE

Bachelors

Paper 1: Algebra,  Vector Analysis and Geometry

Unit I

l. I Historical  background:

l.l.l  Development of Indian Mathematics:

Later  Classical Period  (500  -1250)

1.1.2 A brief biography  of Varahamihira and Aryabhana

1.2 Rank of a Matrix

|  .l  Echelon  and Normal form of a matrix

1.4 Characteristic equations  ola matrix

1.4.1  Eigen-values

| .4.2 Eieen-vectors

Unit II

2.1 Cayley Hamilton  theorem

2.2 Application  of Cayley  llamilton theorem to find the inverse ol  a

matrix.

2.3 Application  of matrix to solve  a system of linear  equations

2.4 Theorems  on consislency  and inconsistency  ofa system of linear

eouataons

2.5 Solving linear equations  up to three unknowns

Unit III

3. I Scalar and Vector products  ofthree  and four vectors

3.2 Reciprocal vectors

3.3 Vector differentiation

3.3. I Rules of differentiation

.  3.3.2 Derivatives of Triple Products

3.4 Gradient,  Divergence  and Curl

3.5 Directional  derivatives

3.6 Vector  ldentities

3.7 Vector  Eouations

Unit IV

4.  I Vector  Integration

4.2 Gauss  theorem  (without proof) and problems  based on it

4.3 Green theorem  (without proof) and problems based on it

4.4 Stoke theorem  (without proof) and problems based on it

Unit V

5.1 General  equation  ofsecond  degree

5.2 Tracing of conics

5.3 System of conics

5.4 Cone

5.4. I Equation  ofcone  with given  base

5.4.2 Cenerators  of cone

5.4.3 Condition  for three mutually  perpendicular  generators

5.4.4 Right circu  lar cone

5.5 Cylinder

5.5. I Equation  of cylinder  and its propenies

5.5.2  Right Circular  Cylinder

1.5.3

Enveloping C5 linder

Paper 2: Calculus  and Differential Equations

Unit I

L  l  Historical  background:

l.l  .  I Develooment of Indian Mathematics:

Ancient  and Early Classical Period (till 500 CE)

I . | .2 A brief biography of Bhaskaracharya

(with special  reference to Lilavati) and Madhava

1.2 Successive differentiation

l.2.l Leibnitz theorem

I .2.2 Maclaurin's series  exoansion

I .2.3 Taylor's series expansion

| .3 Partial Differentiation

| .3.1 Partial  derivatives of higher  order

1.3.2 Euler's  theorem  on homogeneous functions

1.4 Asymptotes

I .4.1  Asymptotes of algebraic  curves

1.4.2 Condition for Existence of Asymptotes.

1.4.3  Parallel  Asymptotes

1.4.4 Asymptotes of polar  curves

Unit II

2.1 Curvature

2.1 .1 Formula for radius  of Curvature

2. 1 .2 Curvature at origin

2. 1.3 Centre  of Curvature

2.2 Concavity  and Convexity

2.2.1  Concavity  and Convexity  of curves

2.2.2 Point of inflexion

2.2.3 Singular point

2.2.4 Multiple points

2.3 Tracing of curves

2.3.1 Curves represented by Cartesian equation

2.3.2 Curves reDresented  by Polar equation

Unit III

3.1 Integration  of transcendental  functions

3.2 Introduction  to Double and Triple Integral

3.3 Reduction formulae

3.4 Quadrature

3.4.1 For Cartesian  coordinates

3.4.2 For Polar cooidinates

3.5 Rectification

3.5. I For Cartesian  coordinates

3.5.2 For Polar coordinates

Unit IV

4.1 Linear differential  equations

4.1.1 Linear equation

4. 1.2 Equations  reducible to the linear form

4.1 .3 Change of variables

4.2 Exact differential  equations

4.3 First order and higher degree  differential equations

4.3. I Equations  solvable lor x, y and p

4.3.2 Equations  homogenous  in x and y

4.3.3  Clairaut's  equation

4.3.4 Singular solutions

4.3.5  Ceometrical  meaning  of differential equations

4.3.6  Orthogonal  trajectories

Unit V

Linear differential equation with constant  coefficients

Homogeneous  linear ordinary  differential equations

Linear differential equations  of second  order

Trarisforrnation  of  equations  by changing the

indeoendent  variable

Method of variation  of parameters

Masters

Paper 1: Advanced Abstract Algebra –I

Unit-1   

Normal  &  Subnormal  series  of  groups,  Composition  series,Jordan-Holder series.

Unit-2 

Solvable & Nilpotent groups.

Unit-3 

Extension  fields.  Roots  of  polynomials,  Algebraic  and transcendental extensions. Splitting  fields.  Separable  and inseparable extension.

Unit-4

Perfect fields, Finite fields, Algebraically closed fields.

Unit-5 

Automorphism  of  extension,  Galois  extension.  Fundamental theorem  of  Galois  theory  Solution  of  polynomial  equations  by radicals, Insolubility of general equation of degree 5 by radicals.

Paper 2: Real Analysis  

Unit-I 

Definition and existence of Riemann- Stieltjes integral and its properties, Integration and differentiation.   

Unit-II 

Integration  of  vector-  valued  functions,  Rectifiable  curves. Rearrangements of terms of a series. Riemann’s theorem.

Unit-III 

Sequences  and  series  of  functions,  Point  wise  and  uniform convergence,  Cauchy  criterion  for  uniform  convergence, Weierstrass M-test, uniform convergence and continuity, uniform convergence  and  Riemann-Stieltjes  integration,  uniform convergence and differentiation.  

Unit–IV 

Functions  of  several  variables,  linear  transformations, Derivatives  in  an  open  subset  of  R n ,  Chain  rule,  Partial derivatives, Differentiation, Inverse function theorem.

Unit-V

Derivatives of higher order, Power series, uniqueness theorem for power  series,  Abel's  and  Tauber's  theorems,  Implicit  function theorem,   

Paper 3: Topology-I

Unit – I 

Countable  and  uncountable  sets.  Infinite  sets  and  the  Axiom  of Choice.  Cardinal  numbers  and  its  arithmetic.  Schroeder- Bernstein  theorem,  statements  of  Cantor's  theorem  and  the Continuum  hypothesis.  Zorn's  lemma.  well-  ordering  theorem. [G.F. Simmons and K.D. Joshi]  

Unit- II 

Definition  and  examples  of  topological  spaces.  Closed  sets.Closure.  Dense  subsets.  Neighbourhoods,  interior  exterior  and boundary. Accumulation points and derived sets. Bases and sub- bases, Subspaces and relative topology. [G.F. Simmons]  

Unit-III 

Alternate methods of defining a topology in terms of Kuratowski Closure  Operator  and  Neighbourhood  Systems.  Continuous functions and omeomorphism. [G.F. Simmons, K.D. Joshi, J.R. Munkers]

Unit- IV 

First  and  Second  Countable  spaces.  Lindelof’s  theorems. Separable  spaces.  Second  Countability  and  Separability.  [G.F.,Simmons]   

Unit- V 

Path-connectedness,  connected  spaces.  Connectedness  on  Real line. Components, Locally connected spaces. [J.R. Munkers]

Paper 4: Complex Analysis-I  

Unit-I 

Complex integration, Cauchy – Goursat theorem, Cauchy integral formula, Higher order derivatives

Unit-II 

Morera’s  theorem.  Cauchy’s  inequality.  Liouville’s  theorem.  The fundamental theorem of algebra. Taylor’s theorem.

Unit-III 

The  maximum  modulus  principle.  Schwartz  lemma.  Laurent series.  Isolated  singularities.  Meromorphic  functions,  The argument principle. Rouche’s theorem. Inverse function theorem.

Unit – IV 

Residues.  Cauchy’s  residue  theorem.  Evaluation  of  integrals. Branches  of  many  valued  functions  with  special  reference  to argz,log z, z a .

Unit – V 

Bilinear  transformations,  their  properties  and  classification. Definitions and examples of conformal mappings.   

(a) Advanced Discrete Mathematics-I

Unit-I 

Semigroups  and  monoids,  subsemigroups  and  submonoids, Homomorphism of semigroups and monoids, Congruence relationand Quotient semigroups, Direct products, Basic Homomorphism

Theorem.

Unit-II 

 Lattices-  Lattices  as  partially  ordered  sets,  their  properties, Lattices  as  Algebraic  systems,  sublattices,  Bounded  lattices, Distributive Lattices, Complemented lattices.

Unit-III 

Boolean  Algebra-  Boolean  Algebras  as  lattices,  various  Boolean identities.  Joint  irreducible  elements,  minterms,  maxterms, minterm  Boolean  forms,  canonical  forms,  minimization  of Boolean  functions.  Applications  of  Boolean  Algebra  to  switching theory (Using AND, OR, & NOT gates) the Karnaugh method.  

Unit-IV 

Graph  Theory-  Defintion  and  types  of  graphs.  Paths  &  circuits. Connected  graphs.  Euler  graphs,  weighted  graphs  (undirected) Dijkstra’s Algorithm. Trees, Properties of trees, Rooted & Binary trees, spanning trees, minimal spanning tree.

Unit-V 

Complete  Bipartite  graphs,  Cut-sets,  properties  of  cut  sets, Fundamental Cut-sets & circuits, Connectivity and Separability, Planar  graphs,  Kuratowski’s  two  graphs,  Euler’s  formula  for planar graphs.  

(b) Differential and integral Equations-I  

Unit-I 

Linear  differential  equation  of  second  order,  ordinary

simultaneous differential equations [As given in Sharma and

Gupta].

Unit-II 

Total  differential  equations,  Picard  Iteration  Methods,

Existence and uniqueness theorem [As given in Sharma and

Gupta].

Unit-III

  Systems  of  first  order  equations,  Existence  and  Uniqueness

theorem.  [As  given  in  Deo,  Lakshmikantham  and

Raghvendra].

Unit-IV 

Solution  of  non  homogeneous  voltera  integral  equation  of

second  kind  by  method  of  successive  substitution  and  also

method of successive approximation. Determination of some

resolvent kernels. Voltera integral equation of first kind. [As

given in Shanti Swarup].

Unit-V 

Solution of the Fredholm integral equation by the method of

successive  substitution  and  also  the  successive

approximation, Iterated Kernels and reciprocal functions. [As

given in Shanti Swarup]

(c) Fundamentals of computers (Theory and Practical)  

Unit-I 

Characteristics  of  Computers,  Block  Diagram  of  Computer,

Generation  of  Computers,  Classification  of  Computers,  Memory

and  Types  of  Memory,  Hardware  &  Software,  System  Software,

Application  software.  Compiler,  Interpreter,  Programming

Languages,  Types  of  Programming  Languages  (Machine

Languages,  Assembly  Languages,  High  Level  Languages).

Algorithm and Flowchart. Number system.

Unit-II 

Introduction  to  MS-DOS  History  and  version  of  DOS,  internal

and  external  DOS  command,  creating  and  executing  batch  file,

booting  process,  Disk,  Drive  Name,  FAT,  File  and  Directory

Structure and Naming Rules, Booting Process, DOS System Files,

DOS  Commands;  Internal-  DIR,  MD,  RD,  COPY,  COPY  CON,

DEL,  REN  VOL,  DATE,  TIME,  CLS,  PATH,  TYPE,  VER  etc.

External CHKDSK, XCOPY, PRINT, DISKCOPY, DOSKEY, TREE,

MOVE, LABEL, FORMAT.

Unit-III 

Introduction for windows System, WINDOWS XP : Introduction to

Windows  XP  and  its  Features.  Hardware  Requirements  of

Windows.  Windows  Concepts,  Windows  Structure,  Desktop,

Taskbar, Start Menu, My Pictures, My Music- Restoring a deleted

file, Emptying the Recycle Bin. Managing Files, Folders and Disk-

Navigating  between  Folders,  Manipulating  Files  and  Folders,

Creating New Folder, Searching Files and Folders.

Unit-IV 

MS Word : Introduction to MS Office, Introduction to MS Word,

Features  &  area  of  use.  Working  with  MS  Word,  Menus  &

Commands,  Toolbars  &  Buttons,  Shortcut  Menus,  Wizards  &

Templates, Creating a new Document, Different Page Views and

Layouts, Applying various Text Enhancements.  

Unit-V 

MS Excel : Introduction and area of use, working with MS Excel,

Toolbars, Menus and Keyboard Shortcuts, Concepts of Workbook

& worksheets, Using different features with Data, Cell and Texts,

Inserting, Removing & Resizing of Columns & Rows.

   MS  PowerPoint  :  Introduction  &  area  of  use,  Working  with  MS

PowerPoint,  Creating  a  New  Presentation,  working  with

presentation,  Using  Wizards  :  Slides  &  its  different  views,

Inserting, Deleting and Copying  of Slides.

(d) Advanced Numerical Analysis -I

Unit-I 

Transcendental  and  Polynomial  Equations  Bisection  Method,

Iteration  methods based on First & Second degree equation  Rate of

convergence.  

Unit-II 

 General  iteration  methods,  System  of  Non-linear  equations,  Method

for complex roots, Polynomial equation, Choice of an iterative method

and implementation.

Unit-III 

System  of  linear  algebraic  equations  and  Eigen  value  problems,

Direct  method,  Iteration  methods,  Eigen  values  and  Eigen  Vectors,

Bounds  on  Eigen  values,    Jacobi  Givens  Household’s  symmetric

matrices. Rutishauser method for arbitrary matrices, Power method,

inverse power methods.  

Unit-IV 

Interpolation  –  Introduction,  Lagrange  and  Newton  interpolation,

Finite  difference  operators,  Interpolating  Polynomials  using  Finite

Differences, Hermite interpolation.

Unit-V 

Piecewise  and  spline  interpolation,  Bivariate  interpolation

approximation  least  squares  approximation.  Uniform  approximation,

rational approximation. Choice of the method.

Paper 1: Functional Analysis-I  

Unit-I 

Normed  linear  spaces.  Banach  Spaces  and  examples.

Properties of normed linear spaces,  Basic properties of finite

dimensional normed linear spaces.

Unit-II 

Normed linear subspace, equivalent norms, Riesz lemma and

compactness. Qutient space of normed linear spaces and its

completeness.

Unit-III 

Linear  operator,  Bounded  linear  operator  and  continuous

operators.

Unit-IV   

Linear  functional,  bounded  linear  functional,  Dual  spaces

with examples.  

Unit-V 

Hilbert  space,  orthogonal  complements,  orthonormal  sets

and  sequences.  Representation  of  functional  on  Hilbert

spaces.

Paper 2: Partial differential Equations-I  

Unit-I 

Derivation  of  Laplace  equation,  derivation  of  passions

equation,  boundary  value  problems  (BVPs),  properties  of

harmonic function: the spherical mean, mean value theorem

for  harmonic  function.  Maximum-minimum  principle  and

consequences.

Unit-II 

Separation  of  variables,  solution  of  Laplace  equation  in

cylindrical  coordinates,  solution  of  Laplace  equation  in

spherical  coordinates,  parabolic  differential  equation

occurrence of the diffusion equation, boundary conditions.  

Unit-III 

Elementary  solution  of  diffusion  equation,  Dirac  delta

function,  separation  of  variables  method,  Solution  of

diffusion  equation  in  cylindrical  coordinates,  solution  of

diffusion equation in spherical coordinates.  

Unit-IV

  Maximum  and  minimum  principle  and  consequence,

Hyperbolic  Differential  equation  :  Occurrence  of  the  Wave

Equation,  Derivation  of  One  Dimensional  Wave  Equation,

Solution  of  One  dimensional  Wave  Equation  by  Canonical

Reduction,  The  Initial  Value  Problem  :  D’  Alembert’s

solution.  

Unit-V 

Vibrating  string-variables  Separable  solution,  Forced

Vibrations-  solution  of  nonhomogeneous  equation,

boundary  and  initial  value  problems  for  two  dimentional wave equation-method of Eigen function, periodic solution of

one-dimensional  wave  equation  in  cylindrical  coordinates,

periodic  solution  of  one-dimensional  wave  equation  in

spherical polar coordinates.  

Paper 3: Algebraic topology-I

Unit-I 

Retractions and fixed point. Brouwer’s fixed point for Disc.

(Art-55).

Unit-II 

Deformation retracts and homotopy type (Art58)

Unit-III  Fundamental group of ‘s n ’ and fig 8 and torus (Art 59 &

Art 60)  

Unit-IV 

Jordan  separation  theory,  nul  homotopy  lemma,

homotopy extension lemma. Borsuk lemma. Invariance

of domain Art. 61 and 62.  

Unit-V 

The Jordan curve theorem. A non separation theorem.

(Art 63) and Imbedding graphs in the plane, Theta

space (Art 64)

Paper 4: Advanced Graph Theory –I

Unit-I 

Revision  of  graph  theoretic  preliminaries.  Isomorphism  of

graphs, subgraphs.

Unit-II 

Walks, Paths and circuits, Connected graphs, Disconnected

graphs  and  components,  Euler  Graphs,  Operations  on

Graphs,  Hamiltonian  paths  and  circuits,  The  traveling

salesman problem.

Unit-III 

Trees,  Properties  of  trees,  Distance  and  centers  in  a  tree,

Rooted  and  Binary  trees,  Spanning  trees,  Fundamental

circuits, spanning trees in a weighted graph.  

Unit-IV 

Cut-sets, Properties of a cut-set, Fundamental circuits and

cut-sets, connectivity and separability.  

Unit-V 

Planar  graphs,  Kuratowski’s  two  graphs,  Different

Representations  of  a  planer  graph,  Detection  of  Planarity,

Geometric Dual, Combinational Dual.  

Research Entrance(CSIR NET)

Analysis:

Elementary set theory, finite, countable and uncountable sets, Real number system as a

complete ordered field, Archimedean property, supremum, infimum.

Sequences and series, convergence, limsup, liminf.

Bolzano Weierstrass theorem, Heine Borel theorem.

Continuity, uniform continuity, differentiability, mean value theorem.

Sequences and series of functions, uniform convergence.

Riemann sums and Riemann integral, Improper Integrals.

Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure,

Lebesgue integral.

Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.

Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.

Linear Algebra:

Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations.

Algebra of matrices, rank and determinant of matrices, linear equations.

Eigenvalues and eigenvectors, Cayley-Hamilton theorem.

Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms,

triangular forms, Jordan forms.

Inner product spaces, orthonormal basis.

Quadratic forms, reduction and classification of quadratic forms

Complex Analysis:

Algebra of complex numbers, the complex plane, polynomials, power series,

transcendental functions such as exponential, trigonometric and hyperbolic functions.

Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem.

Taylor series, Laurent series, calculus of residues.

Conformal mappings, Mobius transformations.

Algebra:

Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle,

derangements.

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem,

Euler’s Ø- function, primitive roots.

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation

groups, Cayley’s theorem, class equations, Sylow theorems.

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal

domain, Euclidean domain.

Polynomial rings and irreducibility criteria.

Fields, finite fields, field extensions, Galois Theory.

Topology:

basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.

Ordinary Differential Equations (ODEs):

Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs.

General theory of homogenous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function.

Partial Differential Equations (PDEs):

Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.

Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations.

Numerical Analysis :

Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.

Calculus of Variations:

Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema.

Variational methods for boundary value problems in ordinary and partial differential equations.

Linear Integral Equations:

Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with

separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.

Classical Mechanics:

Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s principle and principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical equations for the motion of a rigid body about an axis, theory of small oscillations.

Descriptive statistics, exploratory data analysis

Sample space, discrete probability, independent events, Bayes theorem. Random variables and distribution functions (univariate and multivariate); expectation and moments. Independent random variables, marginal and conditional distributions. Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. case).

Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution, Poisson and birth-and-death processes.

Standard discrete and continuous univariate distributions. sampling distributions, standard errors and asymptotic distributions, distribution of order statistics and range. Methods of estimation, properties of estimators, confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, likelihood ratio tests. Analysis of discrete data and chi-square test of goodness of fit. Large sample tests.

Simple nonparametric tests for one and two sample problems, rank correlation and test for independence.

Elementary Bayesian inference.

Gauss-Markov models, estimability of parameters, best linear unbiased estimators, confidence intervals,

tests for linear hypotheses. Analysis of variance and covariance. Fixed, random and mixed effects models.

Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression.

Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation. Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ratio and regression methods.

Completely randomized designs, randomized block designs and Latin-square designs. Connectedness and orthogonality of block designs, BIBD. 2 K  factorial experiments: confounding and construction.

Hazard function and failure rates, censoring and life testing, series and parallel systems.

Linear programming problem, simplex methods, duality. Elementary queuing and inventory models.

Steady-state solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C,

M/M/C with limited waiting space, M/G/1.

3. Markov Analysis

Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step

transition probabilities, stationary distribution, Poisson and birth-and-death processes.

13. Function of several variable, metric spaces, normed linear spaces.

Mathematical Spaces

Research Level / Ph.D (Mathematics)

Major Research Fields

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