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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.
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.
ANALYSIS
Unit-1 1. METRIC SPACE 1-42 Metric and Metric Space 1; Quasi-Metric Space 5; Pseudo-Metric Space 5; Distance between Point and Set 6; Distance between Two Sets 6; Diameter of a Set 7; Some Important Inequalities 7; Product 11; Finite Product in General 12; Product of the Metric Spaces 13; Open Sphere 18; Open Disk (in Real Plane) 18; Open Disk (in Complex Plane) 18; Neighbourhood of a Point 18; Limit Point of a Set 18; Derived Set 19; Interior Point 19; Open Set 19; Closed Sphere 21; Closed Disk (in Real Plane) 21; Closed Disk (in Complex Plane) 21; Open and Closed Balls in RK 21; Convexity in RK 21; Closed Set 22; Closure of a Set 26; Interior of a Set 29; Exterior of a Set 29; Boundary Poi...
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...
A Manual for Dryland Afforestation and Management
Community-oriented conservation of natural resources and promotion and protection of trees in drylands are examples to deal with climatic adversities. This book provides knowledge on climatic, ecological, social and economic condition of dry areas and lay out approaches and strategies to restore degraded lands. There are 15 chapters and first five deals with physiography of Rajasthan, drylands ecology, problems of land degradation, its economic evaluation and the approaches and strategies of restoration and rehabilitation. Next two chapters describe the problems of sand drift, salinity, water logging and effluent inflicted areas and strategies to control them. Chapters 8-10 deal with seed pr...
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...
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 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.
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...
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...
