IIT JAM MATHS

DOUBLE INTEGRAL
Introduction to Double Integrals
1 Topic
Introduction to Double Integrals
Change of Order
7 Topics
Change of Order-1
Change of Order-2
Change of Order-3
Change of Order-4
Change of Order-5
Change of Order-6
Change of Order-7
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
GRADIENT, DIVERGENCE AND CURL
Gradient
2 Topics
Directional Derivative
Properties of Gradient
Tangent Plane and Normal
2 Topics
Tangent Plane and Normal
Problems on Tangent Plane and Normal
Problems on Gradient
1 Topic
Problems on Gradient
Divergence and Curl
4 Topics
Divergence and Curl
Problems on Divergence and Curl-1
Problems on Divergence and Curl-2
Problems on Divergence and Curl-3
LINE INTEGRAL
Line Integral
4 Topics
Introduction to Line Integral
Problems on Line Integral-1
Problems on Line Integral-2
Problems on Line Integral-3
GREENS THEOREM
Greens Theorem
6 Topics
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
SURFACE INTEGRAL
Surface Integral
6 Topics
Introduction to Surface Integral
Problems on Surface Integral-1
Problems on Surface Integral-2
Problems on Surface Integral-3
Problems on Surface Integral-4
Problems on Surface Integral-5
GAUSS DIVERGENCE THEOREM
Gauss Divergence Theorem
8 Topics
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
Stokes Theorem
5 Topics
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
Conservative Vector Field
3 Topics
Introduction to Conservative Vector Field
Problems on Conservative Vector Field-1
Problems on Conservative Vector Field-2
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
Revision of Matrices
6 Topics
Matrices- Revision
Rank of matrix
Problems on Rank
Elementary Matrix
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
Symmetric Matrices
3 Topics
Symmetric Matrices
Symmetric Matrices-2
Conic Sections
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
Defining Groups
2 Topics
Algebraic Structures
Definition of Group
Modulo Arithmetic
2 Topics
Introduction to Modulo Arithmetic
Applications of Modulo Arithematic
Cayley Table
1 Topic
Cayley Table
Unit Group
1 Topic
Unit Group
Group of Matrices
1 Topic
Group of Matrices
Dihedral Groups
1 Topic
Dihedral Groups
Subgroup
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
Set Theory
4 Topics
Introduction to Sets
Subsets
Family of Sets
Operation on Sets
Relation and Functions
2 Topics
Cartesian Product and Relation
Functions
Countability of Sets
1 Topic
Countability of Sets
REAL NUMBERS
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
Closed Sets
1 Topic
Closed Sets
Limit Points
3 Topics
Limit Points of Sets
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
Limit of Sequence
Problems on Limits
Algebra of Limits
Monotonic Sequence
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
Subsequence
1 Topic
Subsequence
Cauchy Sequence
1 Topic
Cauchy Sequence
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 Root Test
Raabe test
Logarithmic Test
Gauss Test
Cauchy Integral and Cauchy Condensation Test
Alternating Series
1 Topic
Alternating Series
POWER SERIES
Power Series
1 Topic
Power Series
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Change of Order-5

IIT JAM MATHS Change of Order Change of Order-5
video

Example 1 :

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

Solution :

The region of integration is bounded by

C_{1} : y = x^{2}

and C_{2} : y^{2} = x

The region of integration is shown in figure

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

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

= \int_{0}^{1} \left ( \dfrac{x^{3}}{3} + xy^{2} \right )_{y^{2}}^{\sqrt{y}} dy

= \int_{0}^{1} \dfrac{1}{3} (y^{3/2} + y^{6}) + (y^{5/2} - y^{4})dy

= \left [ \int_{0}^{1} \dfrac{2y^{5/2}}{15} - \frac{1 y^{7}}{21} + \dfrac{2}{7} y^{7/2} - \dfrac{1}{5} y^{5} \right ]_{0}^{1}

= \dfrac{6}{35}
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