MATHEMATICS FOR IAS/IFoS

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GENERAL INFORMATION

SYLLABUS

LINEAR ALGEBRA

Introduction to Matrices 4 Topics

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Definition of Matrices

Types of Matrices

Operation on matrices

Operation on matrices-2

Rank of Matrix 3 Topics

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Rank of matrix

Echelon Form of matrix

Problems on Rank

Linear Equations 5 Topics

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Linear Equations

Homogeneous Linear Equations

Homogeneous linear equation-contd

Non-homogeneous Linear Equation

Non-homogeneous Linear Equation-contd

Revision of Matrices 5 Topics

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Equivalence of matrix

Determinant 7 Topics

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Introduction of Determinants

Properties of Determinants

Determinant Formula

Problems on Determinants

Formula for Inverse of Matrix

Kramer’s Rule

Problems on Kramer’s Rule

Vector Spaces 3 Topics

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Introduction to Vector Spaces

Subspace

Four Subspaces Associated with Matrix

Linear Equations – Vector Space Approach 5 Topics

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Homogeneous Linear equation

Homogeneous Linear Equations-2

Homogeneous Linear Equations-3

Non-homogeneous Linear Equation

Non-homogeneous Linear Equation-2

Basis and Dimensions 5 Topics

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Basis and Dimensions`

Vector Space of Matrices

Subspaces associated with matrix

Problems on Basis and Dimensions

Problems on Basis and Dimensions-contd

Change of Basis 3 Topics

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Change of Basis

Transition Matrix

Problems on Change of Basis

Orthogonality 5 Topics

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Orthogonal matrices

Orthogonal Subspaces

Projection of Vector on Line

Projectile on Inclined Plane

Gram Smidt Orhogonalisation

Introduction to Eigenvalues and Eigenvectors 3 Topics

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Introduction to Eigenvalues and Eigenvectors

Problems on Eigenvalues and Eigenvectors-1

Problems on Eigenvalues and Eigenvectors-2

Diagonalisation of Matrices and its Applications 5 Topics Sample Lesson

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Diagonalisation

Matrix Exponential

Differential Equations

Differential Equations-contd

Fibonaaci Sequence

Properties of Eigenvalues 1 Topic

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Properties of Eigenvalues

Symmetric Matrices 3 Topics

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Symmetric Matrices

Symmetric Matrices-2

Conic Sections

Cayley Hamilton Equation 1 Topic

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Cayley Hamilton Equation

Similar Matrices 1 Topic

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Similar Matrices

Introduction to Linear Transformation 3 Topics

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Introduction to Linear Transformation

Nullspace and Range

Problems on Nullspace and Range

Matrix of Linear Transformation 2 Topics

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Matrix of Linear Transformation

Change of Basis and Matrix of Linear Transformation

Introduction to Double Integrals 1 Topic

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Introduction to Double Integrals

CALCULUS

Change of Order 7 Topics Sample Lesson

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Transformation of Variables 8 Topics

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Transformation of Variables-1

Transformation of Variables-2

Transformation of Variables-3

Transformation of Variables-4

Transformation of Variables-5

Transformation of Variables-6

Transformation of Variables-7

Transformation of Variables-8

Beta and Gamma Function 2 Topics

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Beta and Gamma Function-1

Beta and Gamma Function-2

Dirichlet Theorem 2 Topics

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Dirichlet Theorem-1

Dirichlet Theorem-2

Introduction and Problems on Volume Calculation 5 Topics

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Introduction to Volumes

Problems on Volume Calculation-1

Problems on Volume Calculation-2

Problems on Volume Calculation-3

Problems on Volume Calculation-4

Calculating Volumes Using Polar Coordinates 3 Topics

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Volume Element in Spherical and Cylindrical Polar Coordinates

Volume Calculation Using Polar Coordinates

Volume Calculation Using Polar Coordinates-2

Centre of Mass 3 Topics

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Problems on Centre of Mass

Problems on Centre of Mass-2

Centre of Mass

Moment of Inertia 1 Topic

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Moment of Inertia using Volume Integral

Volume of the Region Bounded by Surface of Revolution 6 Topics

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Introduction to Volume of the Region Bounded by Surface of Revolution

Problems on Volume of the Region Bounded by Surface of Revolution-1

Problems on Volume of the Region Bounded by Surface of Revolution-2

Problems on Volume of the Region Bounded by Surface of Revolution-3

Problems on Volume of the Region Bounded by Surface of Revolution-4

Problems on Volume of the Region Bounded by Surface of Revolution-5

Surface Area 4 Topics

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Introduction to Surface Area

Problems on Surface Area-2

Problems on Surface Area-1

Problems on Surface Area-3

Introduction to Limits 4 Topics

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Definition of Limits

Definition of Limits -contd

Problems on Limit

Important Results

Methods of Finding Limits 10 Topics

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Finding Limits using Factorization

Finding Limits When x tends to infinity

Trigonometric Limits-1

Trigonometric limits-2

Trigonometric Limits-3

L Hospital Rule-1

L Hospital Rule-2

L Hospital Rule-3

Logarithmic and Exponential limits

Finding Limits using Expansion

Continuity 4 Topics

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Definition of Continuity

Problems on Continuity

Problems on Continuity – contd

Types of Discontinuity

Differentiability 4 Topics

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Introduction to Differentiability

Differentiability using first principle

Problems on Differentiability

Monotonicity 5 Topics

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Increasing and Decreasing Function

Monotonicity

Problems on Monotonicity

Problems on Monotonicity-2

Problems on Monotonicity-3

Critical Points 3 Topics

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Critical and Stationary Points

Point of Inflexion

Problems on Point of Inflexion

Maxima and Minima 3 Topics

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Local maxima and minima

Problems on local maxima and minima

Global maxima and minima

Rolle Theorem 2 Topics

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Rolle Theorem

Problems on Rolle Theorem

Lagrange Mean Value Theorem 3 Topics Sample Lesson

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Introduction to Lagrange Mean Value Theorem

Problems on Lagrange Mean Value Theorem

Problems on Lagrange Mean Value Theorem-2

Taylors Theorem 2 Topics

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Introduction to Taylor’s Theorem

Problems on Taylor’s Theorem

Function of Two Variables 7 Topics

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Introduction to Function of Two Variables

limits of Function of Two Variables

Continuity of Function of Two Variables

Diffrentiability of Function of Two Variables

Partial Derivatives

Partial Derivatives of Higher Order

Homogeneous Function

Maxima and Minima of Function of Several Variables 2 Topics

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Maxima and Minima of Function of two Variables

Maxima and Minima of Function of Several Variables

ORDINARY DIFFERENTIAL EQUATION

Introduction to Differential Equations 2 Topics

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Order and Degree

Formation of Differential Equation

Differential Equation of First order and First Degree 9 Topics

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Separation of Variables

Homogeneous Equation

Equation Reducible to Homogeneous Form

Exact Differential Equation

Integrating Factor

Integrating Factor-contd

Linear Differential Equation of first order

Equation reducible to Linear Form

Product of Exponential and other Function

Orthogonal Trajectory 2 Topics Sample Lesson

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Introduction to Orthogonal Trajectory

Problems on Orthogonal Trajectory

Differential Equation with Constant Coefficient 6 Topics

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Introduction to Differential Equation with Constant Coefficient

Complementary Function

Particular Integral

Exponential Function

Trigonometric Function

Polynomial Function

Homogeneous Linear Differential Equation 4 Topics

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Introduction to Homogeneous Linear Differential Equation

Problems on Cauchy Euler Equation-1

Problems on Euler Cauchy Equation-2

Equation reducible to cauchy Euler form

Differential Equation of Second Order 6 Topics

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Introduction to Differential Equation of Second Order

Problems on Differential Equation of Second Order

Problems on Differential equation of second order-2

Variation of Parameters

Problems on Variation of parameters

Problems on Differential Equation of Second Order

DYNAMICS AND STATICS

Gradient 2 Topics

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Directional Derivative

Properties of Gradient

Tangent Plane and Normal 2 Topics

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Tangent Plane and Normal

Problems on Tangent Plane and Normal

Problems on Gradient 1 Topic

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Problems on Gradient

Divergence and Curl 4 Topics

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Divergence and Curl

Problems on Divergence and Curl-1

Problems on Divergence and Curl-2

Problems on Divergence and Curl-3

Line Integral 4 Topics Sample Lesson

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Introduction to Line Integral

Problems on Line Integral-1

Problems on Line Integral-2

Problems on Line Integral-3

Greens Theorem 6 Topics

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Introduction to Green’s Theorem

Problems on Green’s Theorem-1

Problems on Green’s Theorem-2

Problems on Green’s Theorem-3

Problems on Green’s Theorem-4

Problems on Green’s Theorem-5

Gauss Divergence Theorem 8 Topics

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Introduction to Gauss Divergence Theorem

Problems on Gauss Divergence Theorem-1

Problems on Gauss Divergence Theorem-2

Problems on Gauss Divergence Theorem-3

Problems on Gauss Divergence Theorem-4

Problems on Gauss Divergence Theorem-5

Problems on Gauss Divergence Theorem-6

Problems on Gauss Divergence Theorem-7

Stokes Theorem 5 Topics

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Introduction to Stoke’s Theorem

Problems on Stoke’s Theorem-1

Problems on Stoke’s Theorem-2

Problems on Stoke’s Theorem-3

Problems on Stoke’s Theorem-4

Conservative Vector Field 3 Topics

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Introduction to Conservative Vector Field

Problems on Conservative Vector Field-1

Problems on Conservative Vector Field-2

ALGEBRA

Defining Groups 2 Topics

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Algebraic Structures

Definition of Group

Modulo Arithmetic 2 Topics

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Introduction to Modulo Arithmetic

Applications of Modulo Arithematic

Cayley Table 1 Topic

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Cayley Table

Unit Group 1 Topic

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Unit Group

Group of Matrices 1 Topic

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Group of Matrices

Dihedral Groups 1 Topic Sample Lesson

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Dihedral Groups

Subgroup 2 Topics

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Subgroup and its tests

Examples of Subgroups

Problems on Groups 1 Topic

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Problems on Groups

Cyclic Groups 3 Topics

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Introduction to Cyclic Group

Important Theorems on Cyclic Groups

Problems on Cyclic Groups

Permutation Group 5 Topics

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Introduction to Permutation Group

Properties of Permutation Groups

Properties of Permutation Group-contd

Problems on Permutation Groups

Even and Odd permutations

Isomorphism of Groups 8 Topics

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Introduction to Isomorphism

Examples of Isomorphism

Cayley Theorem

Properties of Isomorphism

Automorphism

Examples of Automorphism

Inner automorphism

Problems on Isomorphism

External Direct Product 3 Topics

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Introduction to External Direct Product

Problems on External Direct Product

Problems on External Direct Product-contd

Homomorphism of Groups 8 Topics

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Introduction to Homomorphism

Examples of Homomorphism

Equivalence Relation on Set

Cosets and Lagrange Theorem

Quotient Groups and First Isomorphic Theorem

Properties of Homomorphism

Problems On Homomorphism

Problems on Homomorphism-contd

REAL ANALYSIS

Set Theory 4 Topics

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Relation and Functions 2 Topics

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Cartesian Product and Relation

Functions

Countability of Sets 1 Topic

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Countability of Sets

Real Numbers 4 Topics

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Algebraic Properties of Real Numbers

Order Properties of Real Numbers

Completeness of R

Application of Supremum Properties

Metric Space 4 Topics Sample Lesson

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Metric Space

Function Space

Important Definitions

Problems on Metric Space

Open Sets 4 Topics

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Neighbourhood and Open Sets

Properties of Open Sets

Examples of open sets

Problems on Interior of Sets

Closed Sets 1 Topic

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Closed Sets

Limit Points 3 Topics

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Limit Points of Sets

Problems on Limit Points

Theorems on Limit Points

Compactness and Connectedness 2 Topics

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Compact Sets

Connectedness

Introduction to Real Sequences 4 Topics

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Introduction to Real Sequences

Limit of Sequence

Problems on Limits

Algebra of Limits

Monotonic Sequence 2 Topics

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Introduction to Monotonic Sequence

Problems on Monotonic Sequence

Cauchy Theorems on Limits 1 Topic

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Cauchy Theorems on Limits

Limit Points 3 Topics

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Limit Points of Sequence

Limit Superior and Limit Inferior

Properties of Limsup and Liminf

Subsequence 1 Topic

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Subsequence

Cauchy Sequence 1 Topic

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Cauchy Sequence

Introduction to Infinite Series 2 Topics

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Introduction to Infinite Series

Convergence of Infinite Series

Test of Convergence of Infinite Series 7 Topics

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Comparision Test

d Alembert Ratio Test

Cauchy Integral and Cauchy Condensation Test

Alternating Series 1 Topic

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Alternating Series

Power Series 1 Topic

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Power Series

NUMERICAL ANALYSIS AND COMPUTER PROGRAMMING

Bisection Method 4 Topics

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Bisection Method

Introduction to Bisection Method

Problems on Bisection Method

Problems on Bisection Method-contd

Regula Falsi Method 2 Topics

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Regula Falsi Method

Problems on Regula Falsi Method

Newton Raphson Method 3 Topics

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Newton Raphson Method

Problems on Newton Raphson Method

Applications of Newton Raphson Method

Order of Convergence 2 Topics

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Order of Convergence

Order of Convergence-contd

Solution of Linear Equations using Numerical Methods 2 Topics

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Numerical Solution to Linear Equations

Gauss Siedal Method

Interpolation 5 Topics

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Introduction to Interpolation

Differences

Interpolation Formula

Problems on Interpolation

Problems on Interpolation-contd

Error in Interpolation Formula 2 Topics

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Error in Interpolation Formula

Problems on Error in Interpolation Formula

Methods of Numerical Integration 3 Topics

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Methods of Numerical Integration

Numerical Integration-contd

Simpson 3/8 Rule

Error in Methods of Numerical Integration 2 Topics

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Error in Trapezoidal Method

Error in Simpson 1/3 Rule

Problems on Numerical Integration 1 Topic

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Problems on Numerical Integration

Euler and Runga Kutta Method 5 Topics

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Euler’s Method

Problems on Euler’s Method

Runga Kutta Method

Problems on Runga Kutta Method

Problems on Runga Kutta Method-contd

MECHANICS AND FLUID DYNAMICS

Generalized Notation 9 Topics

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Degree of Freedom

Generalized Coordinates

Constraints

Generalised Notation

Generalised Displacement

Generalised Velocity and Acceleration

Kinetic Energy

Generalised Momentum

Generalised Force

DAlembert Principle and Lagrange Equation 8 Topics

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Introduction to Lagrange’s Formulation

D’Alembert Principle

Lagrangian Equation of Motion

Charge in Electromagnetic Field

Dissipative Systems

Wedge and Mass Systems

Problems on Rotation

Problems on Non-Holonomic Systems

Problems on Lagrangian 4 Topics

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Atwood Machine

Problems on Lagrangian-1

Problems on Lagrangian-2

Problems on Lagrangian-3

Lagrange Equation from Variational Principle 6 Topics

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Variational Principle

Euler equation of Motion

Minimum Surface of Revolution

Brachistochrone Problems

Hamilton Principle

Non-holonomic Systems

Symmetry and Conservation Laws 4 Topics

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Symmetry and Conservation Laws

Translational Symmetry and Conservation of Linear Momentum

Rotational Symmetry and Conservation of Angular Momentum

Symmetry and Conservation of Energy

Hamiltonian Formulation 2 Topics

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Hamiltonian Formulation

Hamilton’s Canonical Equations

Problems on Hamiltonian 2 Topics

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Problems on Hamiltonian-1

Problems on Hamiltonian-2

Small Oscillation in One Dimension 5 Topics

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Simple Harmonic Oscillation

Examples of Small Oscillation

Dependence of Time Period on Amplitude

Problems on Small Oscillation-1

Problems on Postulates-1

Coupled Oscillation 3 Topics

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Coupled Oscillation

Normal Modes

Problems on Normal Modes

Degree of Freedom and Euler Angles 3 Topics

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Degree of Freedom

Generalised Coordinates associated with Rigid Body

Euler Angles

Rotation about Fixed Axis 4 Topics

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Torque

Moment of Inertia

Moment of Inertia of Famous Rigid Bodies

Rotation about Fixed axis

Translation + Rotation 2 Topics

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Translation + Rotation

Rolling

Angular Momentum 4 Topics

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Angular Momentum

Problems on Angular Momentum-1

Problems on Angular Momentum-2

Problems on Angular Momentum-3

Euler Equations and Its Applications 3 Topics

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Euler Equation of motion

Free Rotation of Rigid Body

Gyroscope Effect