DOUBLE INTEGRAL

Introduction to Double Integrals

1 Topic

Introduction to Double Integrals

Change of Order

7 Topics

Transformation of Variables

8 Topics

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

Beta and Gamma Function

2 Topics

Beta and Gamma Function-1

Beta and Gamma Function-2

Dirichlet Theorem

2 Topics

Dirichlet Theorem-1

Dirichlet Theorem-2

VOLUME

Introduction and Problems on Volume Calculation

5 Topics

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

Volume Element in Spherical and Cylindrical Polar Coordinates

Volume Calculation Using Polar Coordinates

Volume Calculation Using Polar Coordinates-2

Centre of Mass

3 Topics

Centre of Mass

Problems on Centre of Mass

Problems on Centre of Mass-2

Moment of Inertia

1 Topic

Moment of Inertia using Volume Integral

Volume of the Region Bounded by Surface of Revolution

6 Topics

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

Surface Area

4 Topics

Introduction to Surface Area

Problems on Surface Area-1

Problems on Surface Area-2

Problems on Surface Area-3

LIMITS

Introduction to Limits

4 Topics

Definition of Limits

Definition of Limits -contd

Problems on Limit

Important Results

Methods of Finding Limits

10 Topics

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

Continuity

4 Topics

Definition of Continuity

Problems on Continuity

Problems on Continuity – contd

Types of Discontinuity

DIFFERENTIABILITY

Differentiability

4 Topics

Introduction to Differentiability

Problems on Differentiability

Differentiability using first principle

Problems on Differentiability

APPLICATION OF DERIVATIVES

Monotonicity

5 Topics

Increasing and Decreasing Function

Monotonicity

Problems on Monotonicity

Problems on Monotonicity-2

Problems on Monotonicity-3

Critical Points

3 Topics

Critical and Stationary Points

Point of Inflexion

Problems on Point of Inflexion

Maxima and Minima

3 Topics

Local maxima and minima

Problems on local maxima and minima

Global maxima and minima

MEAN VALUE THEOREM

Rolle Theorem

2 Topics

Rolle Theorem

Problems on Rolle Theorem

Lagrange Mean Value Theorem

3 Topics

Introduction to Lagrange Mean Value Theorem

Problems on Lagrange Mean Value Theorem

Problems on Lagrange Mean Value Theorem-2

Taylors Theorem

2 Topics

Introduction to Taylor’s Theorem

Problems on Taylor’s Theorem

Part B: FUNCTION OF TWO VARIABLES

Function of Two Variables

7 Topics

Introduction to Function of Two Variables

limits of Function of Two Variables

Continuity of Function of Two Variables

Partial Derivatives

Diffrentiability of Function of Two Variables

Partial Derivatives of Higher Order

Homogeneous Function

Maxima and Minima of Function of Several Variables

2 Topics

Maxima and Minima of Function of two Variables

Maxima and Minima of Function of Several Variables

INTRODUCTION

Introduction to Differential Equations

2 Topics

Order and Degree

Formation of Differential Equation

DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE

Differential Equation of First order and First Degree

8 Topics

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

ORTHOGONAL TRAJECTORY

Orthogonal Trajectory

2 Topics

Introduction to Orthogonal Trajectory

Problems on Orthogonal Trajectory

DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

Differential Equation with Constant Coefficient

7 Topics

Introduction to Differential Equation with Constant Coefficient

Complementary Function

Particular Integral

Exponential Function

Trigonometric Function

Polynomial Function

Product of Exponential and other Function

CAUCHY EULER EQUATIONS

Homogeneous Linear Differential Equation

4 Topics

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

Differential Equation of Second Order

6 Topics

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

Section 5: LINEAR ALGEBRA

Introduction to Matrices

7 Topics

Definition of Matrices

Types of Matrices

Operation on matrices

Operation on matrices-2

Transpose of matrix

Symmetric and Skew-symmetric matrices

Orthogonal matrices

Rank of Matrix

3 Topics

Rank of Matrix

Echelon Form of matrix

Problems on Rank

Linear Equations

6 Topics

Linear Equations

Homogeneous Linear equation

Homogeneous linear equation-contd

Non-homogeneous Linear Equation

Non-homogeneous Linear Equation-contd

Finding Inverse using elementary transformation

VECTOR SPACE AND LINEAR EQUATIONS

6 Topics

Equivalence of matrix

Normal form of Matrix

Vector Spaces

3 Topics

Introduction to Vector Spaces

Subspace

Subspaces associated with matrix

Linear Equations – Vector Space Approach

5 Topics

Homogeneous Linear Equations

Homogeneous Linear Equations-2

Homogeneous Linear Equations-3

Non-homogeneous Linear Equation

Non-homogeneous Linear Equation-2

Basis and Dimensions

5 Topics

Basis and Dimensions`

Vector Space of Matrices

Four Subspaces Associated with Matrix

Problems on Basis and Dimensions

Problems on Basis and Dimensions-contd

Change of Basis

3 Topics

Change of Basis

Transition Matrix

Problems on Change of Basis

ORTHOGONALITY

Orthogonality

6 Topics

Orthogonal Vectors

Orthogonal Subspaces

Projection of Vector on Line

Projection of Vector on Plane

Gram Smidt Orhogonalisation

Problems on orthogonality

EIGENVALUES AND EIGENVECTORS

Introduction to Eigenvalues and Eigenvectors

3 Topics

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

Diagonalisation

Matrix Exponential

Differential Equations

Differential Equations-contd

Fibonaaci Sequence

Properties of Eigenvalues

1 Topic

Properties of Eigenvalues

3 Topics

Cayley Hamilton Equation

1 Topic

Cayley Hamilton Equation

Similar Matrices

1 Topic

Similar Matrices

LINEAR TRANSFORMATION

Introduction to Linear Transformation

3 Topics

Introduction to Linear Transformation

Nullspace and Range

Problems on Nullspace and Range

Matrix of Linear Transformation

2 Topics

Matrix of Linear Transformation

Change of Basis and Matrix of Linear Transformation

Section 6: GROUP THEORY

2 Topics

2 Topics

Introduction to Modulo Arithmetic

Applications of Modulo Arithematic

1 Topic

1 Topic

1 Topic

1 Topic

2 Topics

Examples of Subgroups

Subgroup and its tests

Problems on Groups

1 Topic

Problems on Groups

CYCLIC GROUP

Cyclic Groups

3 Topics

Introduction to Cyclic Group

Problems on Cyclic Groups

Important Theorems on Cyclic Groups

PERMUTATION GROUP

Permutation Group

5 Topics

Introduction to Permutation Group

Properties of Permutation Groups

Problems on Permutation Groups

Properties of Permutation Group-contd

Even and Odd permutations

ISOMORPHISM OF GROUPS

Isomorphism of Groups

8 Topics

Introduction to Isomorphism

Examples of Isomorphism

Cayley Theorem

Properties of Isomorphism

Automorphism

Examples of Automorphism

Inner automorphism

Problems on Isomorphism

EXTRNAL DIRECT PRODUCT

External Direct Product

3 Topics

Introduction to External Direct Product

Problems on External Direct Product

Problems on External Direct Product-contd

HOMOMORPHISM OF GROUP

Homomorphism of Groups

8 Topics

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

SET THEORY

4 Topics

Relation and Functions

2 Topics

Cartesian Product and Relation

1 Topic

REAL NUMBERS

4 Topics

Algebraic Properties of Real Numbers

Order Properties of Real Numbers

Completeness of R

Application of Supremum Properties

TOPOLOGY ON REAL LINE

Metric Space

4 Topics

Metric Space

Function Space

Important Definitions

Problems on Metric Space

Open Sets

4 Topics

Neighbourhood and Open Sets

Properties of Open Sets

Examples of open sets

Problems on Interior of Sets

1 Topic

3 Topics

Problems on Limit Points

Theorems on Limit Points

Compactness and Connectedness

2 Topics

Compact Sets

Connectedness

REAL SEQUENCES

Introduction to Real Sequences

4 Topics

Introduction to Real Sequences

2 Topics

Introduction to Monotonic Sequence

Problems on Monotonic Sequence

Cauchy Theorems on Limits

1 Topic

Cauchy Theorems on Limits

Limit Points

3 Topics

Limit Points of Sequence

Limit Superior and Limit Inferior

Properties of Limsup and Liminf

1 Topic

1 Topic

INFINITE SERIES

Introduction to Infinite Series

2 Topics

Introduction to Infinite Series

Convergence of Infinite Series

Test of Convergence of Infinite Series

7 Topics

Comparision Test

d Alembert Ratio Test

Cauchy Integral and Cauchy Condensation Test

1 Topic

POWER SERIES

1 Topic

Next Topic

Change of Order-6

IIT JAM MATHS Change of Order Change of Order-6

Example 1 :

Evaluate $\int \int \cos (x + y) dx dy$ over the region enclosed by $x = 0, y = \pi$ and $y = x$ .

Solution :

The region of integration is shown in the figure

$I = \int \int \cos (x + y)dx dy$

$= \int_{0}^{\pi} \int_{0}^{y} \cos(x + y) dx dy$

$= \int_{0}^{\pi} \left [ \sin (x + y) \right ]_{0}^{y} dy$

$= \int_{0}^{\pi} \left ( \sin 2y - \sin y \right ) dy$

$= - \dfrac{1}{2} \left . \cos 2y + \cos y \right |_{0}^{\pi}$

$= -2$

Example 2 :

Evaluate the integral $\int \int xe^{x^{2} - y^{2}} dx dy$ over the region bounded by the lines y = x, y = x – 1, y = 0 and y = 1.

Solution :

The region of integration is shown in figure. It is easier to evaluate the integral first w.r.t. x and then w.r.t. y.

$I = \int_{0}^{1} \int_{y}^{y + 1} xe^{x^{2} - y^{2}} dx dy$

$= \dfrac{1}{2} \int_{0}^{1} e^{-y^{2}} \left [ e^{x^{2}} \right ]_{y}^{y + 1} dy$

$= \dfrac{1}{2} \int_{0}^{1} \left (e^{2y + 1} - 1 \right ) dy$

$= \dfrac{1}{2} \left [ \dfrac{e}{2} e^{2y} - y \right ]_{0}^{1}$

$= \dfrac{1}{4} [e^{3} - e - 2]$

Example 3 :

Evaluate $\int \int (x^{2} + y^{2}) dx dy$ over the region bounded by xy = 1, y = 0, y = x and x = 2.

Solution :

The region of integration is as shown in figure. It has to be partitioned by line x = 1, before integration first w.r.t. y and then w.r.t x,

$I = \int \int (x^{2} + y^{2})dy dx$

$= \int \int (x^{2} + y^{2}) dy dx + \int \int (x^{2} + y^{2})dy dx = I_{1} + I_{2}$

$I_{1} = \int_{0}^{1} \int_{0}^{x} (x^{2} + y^{2})dy dx$

$= \int_{0}^{1} \left [ x^{2} y + \dfrac{y^{3}}{3} \right ]_{0}^{x} dx$

$= \int_{0}^{1} \dfrac{4}{3} x^{3} dx$

$= \dfrac{4}{3} \left [ \dfrac{x^{4}}{4} \right ]_{0}^{1} = \dfrac{1}{3}$

$I_{2} = \int_{1}^{2} \int_{0}^{1/x} (x^{2} + y^{2}) dy dx$

$= \int_{1}^{2} x^{2} y + \left . \dfrac{y^{3}}{3} \right |_{0}^{1/x} dx$

$= \int_{1}^{2} \left ( x + \dfrac{1}{3x^{3}} \right ) dx$

$= \left [ \dfrac{x^{2}}{2} - \dfrac{1}{6x^{{2}}} \right ]_{1}^{2}$

$= \dfrac{39}{24}$

So, $I = I_{1} + I_{2}$

$= \dfrac{47}{24}$

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