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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 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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