IIT JAM MATHS is jointly conducted by IITs and IISc for admission into their two year M.Sc maths,Integrated M.Sc PhD,M.Sc-PhD dual degree and joint M.Sc PhD degree courses.Off late IIT JAM MATHS score has benchmark for admission into other colleges and universities like IISERs,NITs etc.TRAJECTORY EDUCATION has developed a comprehensive integrated classroom course for training students for IIT JAM and other M.Sc entrance exams like ISI,TIFR,CMI,BHU,DU etc.

SYLLABUS

##### Sequences and Series of real numbers: Sequences and series of real numbers. Convergent and divergent sequences, bounded and monotone sequences, Convergence criteria for sequences of real numbers, Cauchy sequences, absolute and conditional convergence; Tests of convergence for series of positive terms – comparison test, ratio test, root test, Leibnitz test for convergence of alternating series.

Functions of one variable: limit, continuity, differentiation, Rolle’s Theorem, Mean value theorem. Taylor’s theorem. Maxima and minima.

Functions of two real variable: limit, continuity, partial derivatives, differentiability, maxima and minima. Method of Lagrange multipliers, Homogeneous functions including Euler’s theorem.

Integral Calculus: Integration as the inverse process of differentiation, definite integrals and their properties, Fundamental theorem of integral calculus. Double and triple integrals, change of order of integration. Calculating surface areas and volumes using double integrals and applications. Calculating volumes using triple integrals and applications.

Differential Equations: Ordinary differential equations of the first order of the form y’=f(x,y). Bernoulli’s equation, exact differential equations, integrating factor, Orthogonal trajectories, Homogeneous differential equations-separable solutions, Linear differential equations of second and higher order with constant coefficients, method of variation of parameters. Cauchy- Euler equation.

Vector Calculus: Scalar and vector fields, gradient, divergence, curl and Laplacian. Scalar line integrals and vector line integrals, scalar surface integrals and vector surface integrals, Green’s, Stokes and Gauss theorems and their applications.

Group Theory: Groups, subgroups, Abelian groups, non-abelian groups, cyclic groups, permutation groups; Normal subgroups, Lagrange’s Theorem for finite groups, group homomorphisms and basic concepts of quotient groups (only group theory).

Linear Algebra: Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations, matrix representation with respect to an ordered basis, Range space and null space, rank-nullity theorem; Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions. Eigenvalues and eigenvectors. Cayley-Hamilton theorem. Symmetric, skewsymmetric, hermitian, skew-hermitian, orthogonal and unitary matrices.

Real Analysis: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets; completeness of R, Power series (of real variable) including Taylor’s and Maclaurin’s, domain of convergence, term-wise differentiation and integration of power series.

### Course Curriculum

Section 1: INTEGRAL CALCULUS | |||

DOUBLE INTEGRAL | |||

Introduction to Double Integrals | 00:12:00 | ||

Change of Order | 01:50:00 | ||

Transformation of Variables | 01:01:00 | ||

BETA AND GAMMA FUNCTION | |||

Beta and Gamma Function | 00:33:00 | ||

Dirichlet Theorem | 00:21:00 | ||

VOLUME | |||

Introduction and Problems on Volume Calculation | 01:04:00 | ||

Calculating Volumes Using Polar Coordinates | 00:55:00 | ||

Centre of Mass | 00:43:00 | ||

Moment of Inertia | 00:08:00 | ||

Volume of the Region Bounded by Surface of Revolution | 00:57:00 | ||

SURFACE AREA | |||

Surface Area | 00:40:00 | ||

Section 2: VECTOR CALCULUS | |||

GRADIENT, DIVERGENCE AND CURL | |||

Gradient | 01:12:00 | ||

Tangent Plane and Normal | 00:18:00 | ||

Problems on Gradient | 00:07:00 | ||

Divergence and Curl | 00:41:00 | ||

LINE INTEGRAL | |||

Line Integral | 00:56:00 | ||

GREENS THEOREM | |||

Greens Theorem | 01:06:00 | ||

SURFACE INTEGRAL | |||

Surface Integral | 01:19:00 | ||

GAUSS DIVERGENCE THEOREM | |||

Gauss Divergence Theorem | 01:36:00 | ||

STOKES THEOREM | |||

Stokes Theorem | 00:56:00 | ||

CONSERVATIVE VECTOR FIELD | |||

Conservative Vector Field | 00:23:00 | ||

Section 3: DIFFERENTIAL CALCULUS | |||

Part A: FUNCTION OF ONE VARIABLE | |||

LIMITS | |||

Introduction to Limits | 00:42:00 | ||

Methods of Finding Limits | 01:50:00 | ||

CONTINUITY | |||

Continuity | 00:45:00 | ||

DIFFERENTIABILITY | |||

Differentiability | 00:41:00 | ||

APPLICATION OF DERIVATIVES | |||

Monotonicity | 01:07:00 | ||

Critical Points | 00:54:00 | ||

Maxima and Minima | 00:45:00 | ||

MEAN VALUE THEOREM | |||

Rolle Theorem | 00:25:00 | ||

Lagrange Mean Value Theorem | 00:33:00 | ||

Taylors Theorem | 00:33:00 | ||

Part B: FUNCTION OF TWO VARIABLES | |||

Function of Two Variables | 01:32:00 | ||

Maxima and Minima of Function of Several Variables | 01:02:00 | ||

Section 4: DIFFERENTIAL EQUATION | |||

INTRODUCTION | |||

Introduction to Differential Equations | 00:25:00 | ||

DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE | |||

Differential Equation of First order and First Degree | 02:01:00 | ||

ORTHOGONAL TRAJECTORY | |||

Orthogonal Trajectory | 00:27:00 | ||

DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS | |||

Differential Equation with Constant Coefficient | 01:19:00 | ||

CAUCHY EULER EQUATIONS | |||

Homogeneous Linear Differential Equation | 00:49:00 | ||

DIFFERENTIAL EQUATION OF SECOND ORDER | |||

Differential Equation of Second Order | 01:24:00 | ||

Section 5: LINEAR ALGEBRA | |||

INTRODUCTION | |||

Introduction to Matrices | 02:17:00 | ||

Rank of Matrix | 00:54:00 | ||

Linear Equations | 02:04:00 | ||

Determinants | 00:00:00 | ||

VECTOR SPACE AND LINEAR EQUATIONS | |||

Revision of Matrices | 02:03:00 | ||

Vector Spaces | 01:05:00 | ||

Linear Equations – Vector Space Approach | 01:43:00 | ||

Basis and Dimensions | 02:35:00 | ||

Change of Basis | 01:34:00 | ||

ORTHOGONALITY | |||

Orthogonality | 01:52:00 | ||

EIGENVALUES AND EIGENVECTORS | |||

Introduction to Eigenvalues and Eigenvectors | 00:36:00 | ||

Diagonalisation of Matrices and its Applications | 00:59:00 | ||

Properties of Eigenvalues | 00:18:00 | ||

Symmetric Matrices | 01:09:00 | ||

Cayley Hamilton Equation | 00:08:00 | ||

Similar Matrices | 00:16:00 | ||

LINEAR TRANSFORMATION | |||

Introduction to Linear Transformation | 01:02:00 | ||

Matrix of Linear Transformation | 00:50:00 | ||

Section 6: GROUP THEORY | |||

Defining Groups | 00:33:00 | ||

INTRODUCTION TO GROUP | |||

Modulo Arithmetic | 00:42:00 | ||

Cayley Table | 00:12:00 | ||

Unit Group | 00:12:00 | ||

Group of Matrices | 00:22:00 | ||

Dihedral Groups | 01:31:00 | ||

Subgroup | 00:47:00 | ||

Problems on Groups | 01:39:00 | ||

CYCLIC GROUP | |||

Cyclic Groups | 01:30:00 | ||

PERMUTATION GROUP | |||

Permutation Group | 01:57:00 | ||

ISOMORPHISM OF GROUPS | |||

Isomorphism of Groups | 02:17:00 | ||

EXTRNAL DIRECT PRODUCT | |||

External Direct Product | 00:51:00 | ||

HOMOMORPHISM OF GROUP | |||

Homomorphism of Groups | 04:12:00 | ||

Section 7: REAL ANALYSIS | |||

SET THEORY | |||

Set Theory | 00:00:00 | ||

Relation and Functions | 00:47:00 | ||

Countability of Sets | 00:32:00 | ||

REAL NUMBERS | |||

Real Numbers | 01:32:00 | ||

TOPOLOGY ON REAL LINE | |||

Metric Space | 01:09:00 | ||

Open Sets | 01:10:00 | ||

Closed Sets | 00:18:00 | ||

Limit Points | 01:00:00 | ||

Compactness and Connectedness | 00:30:00 | ||

REAL SEQUENCES | |||

Introduction to Real Sequences | FREE | 01:03:00 | |

Monotonic Sequence | 00:37:00 | ||

Cauchy Theorems on Limits | 00:19:00 | ||

Limit Points | 01:00:00 | ||

Subsequence | 00:11:00 | ||

Cauchy Sequence | 00:16:00 | ||

INFINITE SERIES | |||

Introduction to Infinite Series | 00:52:00 | ||

Test of Convergence of Infinite Series | 01:34:00 | ||

Alternating Series | 00:12:00 | ||

POWER SERIES | |||

Power Series | 00:32:00 |

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