User:Bomanhoo19/Mathematics
Video Link: https://youtu.be/OmJ-4B-mS-Y
Origins | Foundations | Pure Mathematics | Applied Mathematics | miscellaneous | ||||||||||||||||
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History of Mathematics | Foundational Rules |
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Table of mathematical symbols by introduction date | Mathematical logic |
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Set Theory |
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Category Theory |
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Theory of Computation |
Mathematics
[edit]1. Numbers
[edit]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. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Set of Numbers |
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Arithmatic Operationss |
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Properties |
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Group-like structure |
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Ring Like Structure |
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rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Space | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Functions |
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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. |
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Function Space |
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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 |
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2. Shapes & Space
[edit]Shape | |
Lists of shapes | |
List of mathematical shapes | |
List of two-dimensional geometric shapes | |
Solid geometry |
3. Operations
[edit]Operation (mathematics) | |
Set operations (mathematics) | |
Functions
Class Wise
[edit]I
[edit]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
[edit]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
[edit]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
[edit]Mathematics | ||||
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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
[edit]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
[edit]Chapter 1 Knowing Our Numbers |
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
[edit]https://byjus.com/maths/class-7-maths-index/
Chapter 1: Integers
1.2 Recall 1.3 Properties of Addition and Subtraction of Integers
1.4 Multiplication of Integers
1.5 Properties of Multiplication of Integers 1.6 Division of Integers 1.7 Properties of Division of Integers | ||||||||||||||||||||||||||||||
Chapter 2: Fractions and Decimals
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 3.6 Mode 3.7 Median
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
[edit]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
[edit]https://byjus.com/ncert-solutions-class-9-maths/
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
[edit]https://byjus.com/maths/class-10-maths-index/
Chapter 1: Real Numbers
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Chapter 2: Polynomials
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Chapter 3: Pair Of Linear Equations In Two Variables
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Chapter 4: Quadratic Equations
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Chapter 5: Arithmetic Progressions
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Chapter 6: Triangles
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Chapter 7: Coordinate Geometry
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Chapter 8: Introduction To Trigonometry
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Chapter 9: Some Applications Of Trigonometry
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Chapter 10: Circles
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Chapter 11: Constructions
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Chapter 12: Areas Related To Circles
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Chapter 13: Surface Areas And Volumes
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Chapter 14: Statistics
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Chapter 15: Probability
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XI
[edit]https://byjus.com/maths/class-11-maths-index/
Chapter 1: Sets
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Chapter 2: Relations & Functions
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Chapter 3: Trigonometric Functions
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Chapter 4: Principle of Mathematical Induction
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Chapter 5: Complex Numbers and Quadratic Equations
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Chapter 6: Linear Inequalities
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Chapter 7: Permutations and Combinations
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Chapter 8: Binomial Theorem
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Chapter 9: Sequence and Series
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Chapter 10: Straight Lines
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Chapter 11: Conic Sections
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Chapter 12: Introduction to Three–dimensional Geometry
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Chapter 13: Limits and Derivatives
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Chapter 14: Mathematical Reasoning
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Chapter 15: Statistics
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Chapter 16: Probability
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XII
[edit]https://byjus.com/maths/class-12-maths-index/
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
[edit]1. Complex Numbers
[edit]3. Progression
[edit]4. Permutation & Combination
[edit]5. Binomial Theorem
[edit]7. Determinant
[edit]3. Hyperbolic Functions, Inverse Hyperbolic Functions & Separation of Real and Imaginary Part
[edit]4. Problems Related to Heights & Distances
[edit]Geometry ( 2D & 3D)
[edit]1. Functions, Limit, Continuity & Differentiability
[edit]2. Differentiation
[edit]3. Applications of the Derivative
[edit]Integration & Differential Equations
[edit]2. Definite Integral
[edit]Vectors
[edit]Probability
[edit]1. Probability
[edit]Mechanics ( Statics & Dynamics )
[edit]Bachelors
[edit]Year I
[edit]Paper 1: Algebra, Vector Analysis and Geometry
[edit]Unit I
[edit]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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]Unit I
[edit]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
| .3.1 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
[edit]2.1 Curvature
2.1 .1 Formula for radius of Curvature
2. 1 .2 Curvature at origin
2. 1.3 Centre of Curvature
2.2.1 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
[edit]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
[edit]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
[edit]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
Year II
[edit]Paper 1: Abstract Algebra & Linear Algebra
[edit]Paper 2: Advanced Calculus & Partial Differential Equations
[edit]Year III
[edit]Paper 1: Numerical Method & Scientific Computation
[edit]Paper 2(A): Discrete Mathematics
[edit]Paper 2(B) Probability & Statistics
[edit]Paper 2(C) Integral Transform
[edit]Masters
[edit]Sem I
[edit]Paper 1: Advanced Abstract Algebra –I
[edit]Unit-1
[edit]Normal & Subnormal series of groups, Composition series,Jordan-Holder series.
Unit-2
[edit]Solvable & Nilpotent groups.
Unit-3
[edit]Extension fields. Roots of polynomials, Algebraic and transcendental extensions. Splitting fields. Separable and inseparable extension.
Unit-4
[edit]Perfect fields, Finite fields, Algebraically closed fields.
Unit-5
[edit]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
[edit]Unit-I
[edit]Definition and existence of Riemann- Stieltjes integral and its properties, Integration and differentiation.
Unit-II
[edit]Integration of vector- valued functions, Rectifiable curves. Rearrangements of terms of a series. Riemann’s theorem.
Unit-III
[edit]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
[edit]Functions of several variables, linear transformations, Derivatives in an open subset of R n , Chain rule, Partial derivatives, Differentiation, Inverse function theorem.
Unit-V
[edit]Derivatives of higher order, Power series, uniqueness theorem for power series, Abel's and Tauber's theorems, Implicit function theorem,
Unit – I
[edit]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
[edit]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
[edit]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
[edit]First and Second Countable spaces. Lindelof’s theorems. Separable spaces. Second Countability and Separability. [G.F.,Simmons]
Unit- V
[edit]Path-connectedness, connected spaces. Connectedness on Real line. Components, Locally connected spaces. [J.R. Munkers]
Paper 4: Complex Analysis-I
[edit]Unit-I
[edit]Complex integration, Cauchy – Goursat theorem, Cauchy integral formula, Higher order derivatives
Unit-II
[edit]Morera’s theorem. Cauchy’s inequality. Liouville’s theorem. The fundamental theorem of algebra. Taylor’s theorem.
Unit-III
[edit]The maximum modulus principle. Schwartz lemma. Laurent series. Isolated singularities. Meromorphic functions, The argument principle. Rouche’s theorem. Inverse function theorem.
Unit – IV
[edit]Residues. Cauchy’s residue theorem. Evaluation of integrals. Branches of many valued functions with special reference to argz,log z, z a .
Unit – V
[edit]Bilinear transformations, their properties and classification. Definitions and examples of conformal mappings.
Paper 5:
[edit](a) Advanced Discrete Mathematics-I
[edit]Unit-I
[edit]Semigroups and monoids, subsemigroups and submonoids, Homomorphism of semigroups and monoids, Congruence relationand Quotient semigroups, Direct products, Basic Homomorphism
Theorem.
Unit-II
[edit]Lattices- Lattices as partially ordered sets, their properties, Lattices as Algebraic systems, sublattices, Bounded lattices, Distributive Lattices, Complemented lattices.
Unit-III
[edit]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
[edit]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
[edit]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
[edit]Unit-I
[edit]Linear differential equation of second order, ordinary
simultaneous differential equations [As given in Sharma and
Gupta].
Unit-II
[edit]Total differential equations, Picard Iteration Methods,
Existence and uniqueness theorem [As given in Sharma and
Gupta].
Unit-III
[edit]Systems of first order equations, Existence and Uniqueness
theorem. [As given in Deo, Lakshmikantham and
Raghvendra].
Unit-IV
[edit]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
[edit]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)
[edit]Unit-I
[edit]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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]Unit-I
[edit]Transcendental and Polynomial Equations Bisection Method,
Iteration methods based on First & Second degree equation Rate of
convergence.
Unit-II
[edit]General iteration methods, System of Non-linear equations, Method
for complex roots, Polynomial equation, Choice of an iterative method
and implementation.
Unit-III
[edit]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
[edit]Interpolation – Introduction, Lagrange and Newton interpolation,
Finite difference operators, Interpolating Polynomials using Finite
Differences, Hermite interpolation.
Unit-V
[edit]Piecewise and spline interpolation, Bivariate interpolation
approximation least squares approximation. Uniform approximation,
rational approximation. Choice of the method.
Sem II
[edit]Paper 1: Advanced Abstract Algebra –II
[edit]Paper 2: Lebesque Measure and Integration-II
[edit]Paper 3: Topology-II
[edit]Paper 4: Complex Analysis-II
[edit]Paper 5:
[edit](a) Advanced Discrete Mathematics-II
[edit](b) Differential and integral Equations-II
[edit](c) Programming in “C” (Theory and Practical)
[edit](d) Advanced Numerical Analysis -II
[edit]Sem III
[edit]Paper 1: Functional Analysis-I
[edit]Unit-I
[edit]Normed linear spaces. Banach Spaces and examples.
Properties of normed linear spaces, Basic properties of finite
dimensional normed linear spaces.
Unit-II
[edit]Normed linear subspace, equivalent norms, Riesz lemma and
compactness. Qutient space of normed linear spaces and its
Unit-III
[edit]Linear operator, Bounded linear operator and continuous
Unit-IV
[edit]Linear functional, bounded linear functional, Dual spaces
with examples.
Unit-V
[edit]Hilbert space, orthogonal complements, orthonormal sets
and sequences. Representation of functional on Hilbert
spaces.
Paper 2: Partial differential Equations-I
[edit]Unit-I
[edit]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
[edit]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
[edit]function, separation of variables method, Solution of
diffusion equation in cylindrical coordinates, solution of
diffusion equation in spherical coordinates.
Unit-IV
[edit]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
[edit]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
[edit]Unit-I
[edit]Retractions and fixed point. Brouwer’s fixed point for Disc.
(Art-55).
Unit-II
[edit]Deformation retracts and homotopy type (Art58)
Unit-III Fundamental group of ‘s n ’ and fig 8 and torus (Art 59 &
Art 60)
Unit-IV
[edit]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.
[edit](Art 63) and Imbedding graphs in the plane, Theta
space (Art 64)
Paper 4: Advanced Graph Theory –I
[edit]Unit-I
[edit]Revision of graph theoretic preliminaries. Isomorphism of
graphs, subgraphs.
Unit-II
[edit]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
[edit]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
[edit]Cut-sets, Properties of a cut-set, Fundamental circuits and
cut-sets, connectivity and separability.
Unit-V
[edit]Planar graphs, Kuratowski’s two graphs, Different
Representations of a planer graph, Detection of Planarity,
Geometric Dual, Combinational Dual.
Paper 5:
[edit](a) Advanced special function-I
[edit](b) Theory of linear operators-I
[edit](d) Fuzzy sets and their applications-I
[edit](e) Operations research –I
[edit](g) Integral Transform –I
[edit](h) Advanced Programming in ‘C’ Theory & Practical-I
[edit](i) Integration Theory-I
[edit](j) Spherical Trigonometry and Astronomy-I
[edit]Sem IV
[edit]Paper 1: Functional Analysis-II
[edit]Paper 2: Partial differential Equations-II
[edit]Paper 3: Algebraic topology-II
[edit]Paper 4: Advanced Graph Theory –II
[edit]Paper 5:
[edit](a) Advanced special function-II
[edit](b) Theory of linear operators-II
[edit](c) Mechanics –II
[edit](d) Fuzzy sets and their applications-II
[edit](e) Operations research –II
[edit](f) Wavelets-II
[edit](g) Integral Transform –II
[edit](i) Integration Theory-II
[edit](j) Spherical Trigonometry and Astronomy-II
[edit]Research Entrance(CSIR NET)
[edit]UNIT – 1
[edit]Analysis:
[edit]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,
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:
[edit]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
==UNIT – 2==
[edit]Complex Analysis:
[edit]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:
[edit]Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle,
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:
[edit]basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.
UNIT – 3
[edit]Ordinary Differential Equations (ODEs):
[edit]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 :
[edit]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:
[edit]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:
[edit]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:
[edit]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.
UNIT – 4
[edit]Descriptive statistics, exploratory data analysis
[edit]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.
Notes
[edit]Volume-1
[edit]1. General Probability Theory Expectation & Moments
[edit]2. Marginal & Conditional Distribution, WLLN, CLT
[edit]3. Markov Analysis
[edit]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.
7. Test For Linear Hypothesis, Analysis Of Variance, Linear Regression
[edit]8. Multivariant Normal Distribution, Inference For Parameters, Partial & Multiple Correlation Coefficients
[edit]Volume-2
[edit]10. Sequence and Series and uniform convergence, Fourier series, Power series
[edit]12. Reimann integral, functions of bunded variation, Lebesgue Measure.
[edit]14. Metric Space
[edit]15. Vector algebra, Green, Gauss & Stoke's Theorem
[edit]Volume-3
[edit]19. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms
[edit]21. Quadratic Forms, Classification of Quadratic Forms
[edit]22. Linear programming problem, inventory models
[edit]23. Steady-state solutions of Markovian queuing models
[edit]24. Transportation problems/ assignment problems, Laplace Transform, Fourier transform
[edit]Volume-4
[edit]25.Complex Numbers
[edit]26. Analytic and Harmonic function
[edit]27. Contour Integral, Cauchy theorem, Cauchy integral formula
[edit]28. Taylor's series, Laurent series, calculus of residues
[edit]29. Liouville's Theorem, Maximum Modulus principle Schwarz Lemma
[edit]30. Conformal mapping, Mobius transformations
[edit]31. Ordinary differential equation
[edit]32. ODE's, Variation of parameters, Sturm Liouill problem, Green's function
[edit]Volume-5
[edit]33. PDE of 1st order Lagrange's Charpit method and Cauchy problem
[edit]34. Higher order PDE's BVP
[edit]35. Permutation, Combinations, Pigeon hole principle, inclusion-exclusion principle, Derangement
[edit]36. Set Relation and fundamental theorems of algebra Chinese remainder theorem
[edit]37. Groups subgroups normal, quotient groups and their homomorphism
[edit]38. Cyclic groups, permutation groups, sylow theorems, Group action
[edit]39. Rings, Ideal, Euclidean Domain, PID
[edit]40. Polynomial ring, field, filed extension
[edit]Volume-6
[edit]41. Numerical Analysis ( Iterative method for solving algebraic equations and system of linear equations)
[edit]42. Interpolation Numerical Integration Differentiation Solution of Differential Equation
[edit]43. Variation of Parameters
[edit]44. Variation method of boundary value problems
[edit]45. Integral Equations
[edit]46. Integral Equation for separable Kernel
[edit]47. Classical Mechanics (I)
[edit]48. Classical Mechanics (II)
[edit]Research Level / Ph.D (Mathematics)
[edit]Research Methodology
[edit]Major Research Fields
[edit]Y1====== Y2====== Y3======