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DIFFERENTIAL EQUATION AND ANALYSIS
UNIT I Exact Differential equations. Linear differential Equation. Equation reducible to linear form. First order and higher degree equations solvable for x, y and p. Clairaut's differential equations. Orthogonal trajectories. UNIT II Linear differential equation with constant coefficient. Operator method to find the particular integral. Linear differential equation of second order. Method of Variation of parameter. UNIT III Sequences. Theorem on limit of sequences. Bounded and Monotonic Sequences. Cauchy Sequences. Cauchy's convergence criterion. UNIT IV Series of non-negative terms. Comparison test, Cauchy's integral test, Ratio test. Alternating Series, Leibnitz's Theorem, Absolute and conditional convergence. Series of arbitrary terms.
ADVANCED CALCULUS & GROUP THEORY
Unit I Group : Definition of Group with example and properties, Sub-group, Cosets, Normal Subgroup. Unit II Permutation groups, product of permutations, even and odd permutation. Cyclic group. Group homorphism and isomorphism. Fundamental theorem of homomorphism. Unit III Limit and continuity of function of two variables. Partial differentiation. Chain rule, Differential. Unit IV Jacobins, Homogeneous function, and Euler’s theorem, Maxima & Minima and Saddle point of function of two variables, Lagrange’s multiplier method.
PHYSICAL CHEMISTRY
Unit-I : Thermodynamics -I (A) Recapitulation of thermodynamic terms : System, surrounding types of system (closed, open & isolated), Thermodynamic, variables, intensive & extensive properties, thermodynamic processes (isothermal, adiabatic, isobaric, cyclic, reversible & irreversible) State function & path functions, properties of state functions (exact differential, cyclic rule), integrating factor, concept of heat & work. [3L] (B) Statements of first law of thermodynamics : Definition of internal energy & enthalpy, heat capacity at constant volume & at constant pressure, Joule-Thomson experiment, Joule Thomson coefficient & Inversion temperature, calculations of W,Q,ΔE & ΔH for expansio...
ALGEBRA & TRIGONOMETRY
Unit I De-Moivre’s theorem and its applications, Square root of complex number. Inverse circular and hyperbolic functions. Logarithm of complex quantity. Summation of series. C+iS methods based on binomial, Geometric, Exponential, sin x and cos x. Unit II Definition of rank of a matrix. Theorems on consistency of a system of linear equations. Application of matrices to a system of linear (homogeneous and non-homogeneous equations). Eigen values, Eigen vectors and characteristic equation of a matrix. Caley Hamilton’s theorem Unit III Relation between roots and coefficients of a general polynomial equation in one variable, Transformation of equations. Descarte’s rule of signs. Solution of cubic equations (Cardon’s method). Unit IV Divisibility, Definition and elementary properties. Division Algorithm, G.C.D. and L.C.M. of two integers, Basic properties of G.C.D., Euclidean algorithm. Primes. Euclid’s theorem. Unique factorization theorem.
LINEAR ALGEBRA
Unit-1 1. Analytic Functions, Cauchy-Riemann Equations, Harmonic Functions 1-40 Complex Number System 1; Complex Numbers as Ordered Pairs 1; The Polar Form 1; Function of a Complex Variable 2; Single Valued Function(or Uniform Function) 2; Multiple-Valued Function(or Many-Valued Function) 3; Limit of a Function 3; Theorems on Limits 3; Continuity 3; Fundamental Operations as Applied to Continuous Function 4; Continuity in Terms of Real and Imaginary Parts of f(z) 4; Uniform Continuity 4; Differentiability of a Complex Function 5; Geometric Interpretation of the Derivative 5; Partial Derivative 6; Analytic Function 6; The Necessary Conditions for f(z) to be Analytic [(Cauchy-Riemann Equations...
VECTOR CALCULUS, GEOMETRY AND DIFFERENCE EQUATION
UNIT I Vector triple product. Product of four Vectors. Vector differentiation. Gradient, divergence and curl. Solenoidal and irrotational vector field. UNIT II Double integration. Properties of double integration. Iterated integral. Change of order of Integration. Transformation of double integral in polar form. UNIT III Spheres, Plane section of a sphere. Intersection of two sphere. Sphere through a given circle cone. Equation of cone with Vertex at origin. Right circular cone. Right circular cylinder. UNIT IV Formation of difference equation. Order of difference equation. Linear difference equation. Homogeneous linear equation with constant coefficient. Non homogeneous linear equation. Particular integrals.
INORGANIC CHEMISTRY
1. IONIC SOLIDS 1-15 Types of Solids 1; Space Lattice, Lattice Point and Unit Cell of a Crystal 1; Ionic Crystal Structures 2; Structure of Sodium Chloride (Nacl) 3, Structure of Cesium Chloride (CsCl) 3; Limitations of Radius Ratio Rule 6; Lattice Energy 6; Factors Affecting Lattice Energy 7; Born- Haber Cycle 7; Solvation Energy 10; Definition of Solvation Energy 11; Factors Affecting Solvation and Solvation Energy 11; Polarization, Polarizing Power and Polarizability 12; Fajan's Rules 12. 2. METALLIC BONDING 16-23 Metallic Bonding 16; Factors Favoring the Formation of Metallic Bond 16; Electron Sea Theory 16; Metallic Properties 17; Thermal Conductivity 17; Electrical Conductivity 17; Mal...
DIFFERENTIAL EQUATIONS & LAPLACE TRANSFORMS
UNIT-I 1. Total Differential Equation (Pfaffian Differential Equations) 1-18 Introduction 1; Methods for Solving the Equation Pdx+Qdy+Rdz=0 1. 2. Partial Differential Equations of the First Order, Lagrange's Equations, Charpit's General Method 19-89 Introduction 19; Partial Differential Equations 19; Order of Partial Differential Equations 19; Degree of the Partial Differential Equations 19; Linear Partial Differential Equations 20; Formation of a Partial Differential Equations 20; Formation of a Partial Differential Equation by Elimination of Arbitrary Constants 20; Formation of Partial Differential Equation by Elimination of Arbitrary Function f from the Equation f(u, v) = 0, where u, v ar...
ABSTRACT ALGEBRA, DIFFERENTIAL EQUATION & FOURIER SERIES
ABSTRACT ALGEBRA UNIT-I 1. Group Automorphism, Inner Automorphism, Group of Automorphisms 1-22 Introduction 1; Homomorphism of Group 1; Types of Homomorphism 1; Kernel of a Homomorphism 3; Some Theorems (Properties of Group Homomorphism) 3; Isomorphism of Groups 3; Fundamental Theorem of Homomorphism of Groups 3; More Properties of Group Homomorphism 4; Automorphism of a Group 4; Inner Automorphism 8; Theorem 4; Definition of Inner Automorphism 8; Centre of a Group 9; Group of Automorphisms 12; Group of Automorphisms of a Cyclic Group 14. 2. Cayley's Theorem 23-32 Permutation Groups and Transformations 23; Equality of Two Permutations 24; Identity Permutations 24; Cayley’s Theorem for Fini...
DIFFERENTIAL & INTEGRAL CALCULUS
Unit I Limit and Continuity (e and d definition). Types of Discontinuities. Theorems on Limit and Continuity. Differentiability of Functions. Successive Differentiation. Leibnitz's Theorem. Unit II Mean Value Theorem. Rolle's Theorem. Cauchy's Generalised Mean Value Theorem. Lagranges Mean value Theorem. Taylors Theorem with Lagranges & Cauchy's form of remainder. Maclaurin's Series & Taylor's Series of sin x, cos x, ex, log(1+x), (1+x)m. Unit III Improper integrals, Gamma function, Properties of Gamma function. Beta function. Properties of Beta function. Indeterminate forms L. Hospitals Rule. Unit IV Double Integration. Properties of Double Integration. Iterated Integral. Change of order Integration. Transformation of Double Integral in Polar Form.
