User:Bomanhoo19/Mathematics

Video Link: https://youtu.be/OmJ-4B-mS-Y

Mathematics Topics

= Map of Mathematics =

= 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. {| class="wikitable" ! ! ! ! ! ! !Numbers !Set of Numbers $$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$ !Arithmatic Operationss !Properties !Group-like structure {| class="wikitable" ! ! colspan="5" |
 * 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.
 * 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.
 * NumberSetinC.svgCayley Q8 quaternion multiplication graph.svg
 * Basic arithmetic operators.svg
 * Addition Operation.svgMultiplication Operation.svg
 * Addition Operation.svgMultiplication Operation.svg
 * Associativity of binary operations (without question marks).svgCommutativity of binary operations (without question mark).svgIllustration of distributive property with rectangles.svg
 * Associativity of binary operations (without question marks).svgCommutativity of binary operations (without question mark).svgIllustration of distributive property with rectangles.svg

(G,+) or (G.*)
!Ring Like Structure {| class="wikitable" ! ! colspan="10" |
 * 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.svg
 * rowspan="3" |
 * 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.svg
 * rowspan="3" |
 * Algebraic structures - magma to group.svg
 * rowspan="3" |
 * rowspan="3" |

(G,+,*)
! ! colspan="5" |+ ! colspan="5" |x
 * Ring
 * 1
 * 2
 * 3
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 * 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
 * }
 * Near-ring
 * Commutative ring
 * Domain
 * Integral domain
 * Field
 * 1
 * 2
 * 3
 * 4
 * 5
 * 1
 * 2
 * 3
 * 4
 * 5
 * Division ring
 * Lie ring
 * }
 * Commutative ring
 * Domain
 * Integral domain
 * Field
 * 1
 * 2
 * 3
 * 4
 * 5
 * 1
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 * 3
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 * 5
 * Division ring
 * Lie ring
 * }
 * Integral domain
 * Field
 * 1
 * 2
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 * 4
 * 5
 * 1
 * 2
 * 3
 * 4
 * 5
 * Division ring
 * Lie ring
 * }
 * Field
 * 1
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 * 5
 * 1
 * 2
 * 3
 * 4
 * 5
 * Division ring
 * Lie ring
 * }
 * 2
 * 3
 * 4
 * 5
 * Division ring
 * Lie ring
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 * Lie ring
 * }
 * Lie ring
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 * Lie ring
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 * Lie ring
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 * Lie ring
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 * Lie ring
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!Space !Functions 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 !Functional
 * 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
 * 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.svgFunctions between metric spaces.svg
 * Mathematical implication diagram-alt-large-print.svg
 * Mathematical implication diagram-alt-large-print.svg
 * Function machine2.svg
 * 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 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.
 * rowspan="2" |
 * 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.
 * 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.
 * 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.
 * Duality
 * Definite integral
 * Inner product spaces
 * Locality
 * Arclength.svg 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.]]Integral as region under curve.svg is a linear functional on the vector space of functions defined on [a, b] that are Riemann-integrable from a to b.]]
 * }
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 * }

3. Operations


Functions

= Class Wise =

VII
https://byjus.com/maths/class-7-maths-index/

IX
https://byjus.com/ncert-solutions-class-9-maths/

X
https://byjus.com/maths/class-10-maths-index/

XI
https://byjus.com/maths/class-11-maths-index/

XII
https://byjus.com/maths/class-12-maths-index/

= JEE =

2. Dynamics
= Bachelors =

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

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

Paper 2(C) Integral Transform
= Masters =

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.

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,

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]

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.

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.

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]

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.

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.

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.

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.

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)

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.

(j) Spherical Trigonometry and Astronomy-II
= 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

UNIT – 2== ==

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.

7. Test For Linear Hypothesis, Analysis Of Variance, Linear Regression
== 8. Multivariant Normal Distribution, Inference For Parameters, Partial & Multiple Correlation Coefficients ==

48. Classical Mechanics (II)
= Research Level / Ph.D (Mathematics) =

Major Research Fields
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