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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 are Functions of x, y, z 26; Solution of Partial Differential Equations 34; Lagrange's Method of Getting the General Solution in the Form f(u,v) = 0 35; General Solution of Lagrange's Equation 35; Some Special Types of Equations which can be Solved Easily by Methods other than the General Method 53; Standard Form I 53; Standard Form II 58; Standard Form III 64; Standard Form IV or Clairaut's Form 70; Charpit's Method 72; Compatible Differential Equations of First Order 85. UNIT-II 3. Linear Partial Differential Equations with Constant Coefficients 90-142 Introduction 90; Solution of Linear Partial Differential Equation 90; Complementary Solutions 90; When Auxiliary Equation has Two Equal Roots 92; Integration 99; Particular Integral (P.I.) 100; Short-cut Method 106; Particular Case, when F(a,b) = 0 112; General Method for Finding the Particular Integral 118; Non-homogeneous Linear Differential Equations 121; Particular Integrals (P.I.) 123; Partial Differential Equations Reducible to Equations with Constant Coefficients 136. LAPLACE TRANSFORMS UNIT-III 4. LAPLACE TRANSFORM 143-196 Integral Transform 143; Laplace Transform 143; Properties of Laplace Transform 147; Laplace Transform of Discontinuous Functions 162; Existence Theorem of Laplace Transforms 166; Laplace Transform of Derivatives of F(t) 168; Differentiation of Laplace Transforms 169; Integration of Laplace Transforms 170; Initial Value Theorem 184; Final Value Theorem 185; Laplace Transform of Integrals 185; Evaluation of Integrals with the help of Laplace Transform 188; Periodic Function 194. 5. THE INVERSE LAPLACE TRANSFORMS 197-250 Inverse Laplace Transform 197; Properties of Inverse Laplace Transform 198; Methods of Finding Inverse Laplace Transforms by Using Partial Fractions 214; Convolution 238; Convolution Theorem or Convolution Property 238. UNIT-IV 6. APPLICATIONS OF LAPLACE TRANSFORMS 251-272 Solution of Linear Differential Equations with Constant Coefficients 251; Procedure for Application of Laplace Transform 251. 7. SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 273-276 Definition 273; Theorem 273.

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.

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 Finite Group 25; Regular Permutation Group 26; Cayley’s Theorem for Infinite Group 26. 3. Counting Principle 33-44 Conjugate Elements and Conjugacy Relation 33; Conjugate Classes 33; Conjugate Subgroups 33; Conjugate Class of a Subgroup 34; Self Conjugate Elements 34; Normalizer or Centralizer of an Element 34; Normalizer of a Subgroup of a Group 34; Self-Conjugate Subgroups 34; Counting Principle 34. UNIT-II 4. Introduction to Rings and Subrings 45-72 Introduction 45; Ring 45; Examples of a Ring 46; Properties of a Ring 46; Types of Rings 46; Some Properties of a Ring 61; Integral Multiples of the Elements of a Ring 63; Some Special Kinds of Ring 63; Cancellation Laws in a Ring 65; Invertible Elements in a Ring with Unity 66; Division Rings or Skew Field 67; Quotient Ring or Factor Ring or Ring of Residue Classes 67; Subrings 69; Smallest Subring 72. 5. Integral Domain 73-84 Integral Domain 73; Sub-domain 74; Ordered Integral Domain 75; Inequalities 76; Well-ordered 76; Field 76; Some Theorems 76; The Characteristic of a Ring 78. 6. Ideals 85-100 Ideal 85; Theorem 85; Improper and Proper Ideals 86;Unit and Zero Ideals 86; Some Theorems 89; Smallest Ideal Containing a given Subset of a Ring 91; Principal Ideal 91; Principal Ideal Ring (or Principal Ideal Domain) 91; Prime Ideal 91; Maximal Ideal 92; Minimal Ideal 93; Sum of Two Ideals 93; Theorems 93; Product of Two Ideals 94; Important Theorems 94. DIFFERENTIAL EQUATIONS & FOURIER SERIES UNIT-III 1. Series Solutions of Differential Equations : Power Series Method 1-33 Power Series Method 1; Analytic or Regular or Holomorphic Function 1; Singular Point of the Differential Equation 1; Power Series 2; General Method for Solving a Differential Equation by Power Series Method 2; Frobenius Method 9; When Two Roots of Indicial Equation are Unequal and Differ by a Quantity not an Integer 10; Roots of the Indicial Equation Unequal and Differing by an Integer 17; When Roots of Indicial Equation are Equal 23; Series Solution Near an Ordinary Point (Power Series Method) 28. 2. Legendre’s Equation, Legendre’s Polynomial, Generating Function, Recurrence Formulae and Orthogonal Legendre’s Polynomials 34-78 Legendre’s Equation 34; Solution of Legendre’s Equation 34; Legendre’s Functions and its Properties 36; Legendre’s Functions 36; Legendre’s Function of the First Kind 37; Legendre’s Function of the Second Kind 37; Another Form of Legendre’s Polynomial Pn(x) 37; General Solution of Legendre’s Equation 39; Associated Legendre’s Functions 39; Generating Function for Legendre’s Polynomial 40; Orthogonal Properties of Legendre’s Polynomials 49; Recurrence Formulae 51; Beltrami’s Result 53; Christoffel’s Expansion Formula 54; Christoffel’s Summation Formula 55; Rodrigue’s Formula Pn(x) 65; Laplace’s Integral for Pn(x) 67; Some Bounds on Pn(x) 68. 3. Bessel’s Equation, Recurrence Formula 79-106 Bessel’s Equation 79; Solution of Bessel’s Equation 79, Bessel’s Functions 81; Bessel’s Function of the First Kind of Order n (or Index n) 81; Bessel’s Function of the Second Kind of Order n (or Neumann's Function) 82; General Solution of Bessel’s Equation 82; Integration of Bessel’s Equation for n = 0 and Bessel’s Functions of Zeroeth Order 82; Linear Dependence of Bessel Functions Jn(x) and J-n(x) 84; Recurrence Relations for Jn(x) 84; Elementary Functions 90. UNIT-IV 4. Fourier Series 107-134 Introduction 107; Periodic Function 107; Even and Odd Functions 107; Fourier Series for Even and Odd Functions 108; Euler’s Formulae 109; Orthogonal Functions 109; Important Definite Integrals 110; To Determine the Fourier Coefficients a0, an and bn 110; Dirichlet Conditions 122; Fourier Series for Discontinuous Functions 122; Change of Interval 127. 5. Half Range Fourier Sine and Cosine Series 135-152 Half Range Series : Fourier Sine and Cosine Series 135; Parseval’s Theorem 143; Complex Form of Fourier Series 146.

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 (C-R Equations)] 6; The Sufficient Condition for f(z) to be Analytic 8; Polar Form of Cauchy-Riemann Equations 9; Derivative of w in Polar Form 11; Functions of a Function 12; Derivative of a Function of a Function 12; Inverse Function 12; Laplace Equation 13; Harmonic Function 13; Theorem 13; Conjugate Harmonic Functions 14; Theorem 14; Determination of the Conjugate Function 14; To Construct a Function f(z) when One Conjugate Function is Given 15; Orthogonal System 16; Theorem 16. 2. Mobius Transformation, Cross Ratio 41-68 Elementary Functions 41; Mapping or Transformation 43; Definition of Mapping 43; Mobius Transformation or Bilinear Transformation or Fractional Transformation 43; Inverse Transformation 44; Critical Points and Critical Mapping 44; Resultant or Product of Two Mobius Transformations (Group Property) 45; Some Theorems 46; Fixed Points (or Invariant Points) of Mobius Transformation 47; Theorem 48; Cross Ratio 48; Some Theorems 49; The Circle 54; Inverse Points with Respect to a Circle 55; To find the Relation between the Inverse Points with Respect to the Circle 55; Nature of Transformations (Elliptic, Hyperbolic and Parabolic Transformations) 56; Some Theorems 58. 3. Vector Space 69-123 Vector Space 69; Various Notations 70; General Properties (Elementary Properties) of Vector Spaces 70; Vector Subspace 77; Union and Intersection of Subspaces 83; Sum of Subspaces 85; Some Definitions 91; Basis of a Vector Space 91; Dimension of a Vector Space 92; Finite Dimensional Vector Space 102; Some Theorems on Finite Dimensional Spaces 102; Quotient Space 115. 4. Linear Transformation 124-158 Definition 124; Purpose 124; Image 124; Existence and Uniqueness 124; Types of Linear Transformation 127; Determining whether a Mapping is Linear Transformation or Not 127; Isomorphism of Vector Spaces 133; Theorems on Isomorphism 134; Kernel of Linear Transformation T or Kernel of a Homomorphism T 142; Theorem 142; Range of a Linear Transformation 143; Theorem 143; Lemma 144; Sylvester Law of Nullity [Rank-Nullity Theorem] 144; Fundamental Theorem of Vector Space Homomorphism 146. 5. Inner Product Spaces 159-200 Inner Product 159; Usual or Standard Inner Product 159; Inner Product Space 162; Theorems 162; Some Important Terms about Vectors 168; Norm or Length of a Vector a in an Inner Product Space 168; Theorems 169; Orthogonality 170; Orthogonal Complement 171; Orthogonal Basis and Orthonormal Basis 172; Gram-Schmidt Orthogonalization Process 172; Theorems on Orthogonal/ Orthonormal Bases 173; Cauchy-Schwarz’s Inequality or Schwarz’s Inequality 186; Bessel’s Inequality 188; Normed Vector Space or Normal Vector Space 194; Distance in an Inner Product Space 195.

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.

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.

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.

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 Points 31; Subspace of a Metric Space 32; Relative Open Set 32; Convergence of a Sequence in a Metric Space 33; Cauchy Sequence 33; Bounded Set and Bounded Sequence 34; Complete Metric Space 36; Completeness 37; Nested Sequence 38; Contraction Mapping 39; Contraction Principle or Banach Fixed Point Theorem 40. 2. COMPACTNESS 43-57 Cover 43; Subcover 43; Finite Subcover 43; Open Cover 43; Compact Set and Compact Space 43; Some Theorems 44; Bolzano-Weierstrass Property 46; Sequential Compactness 46; Theorems 47; Heine-Borel Theorem 49; e-Net 50; Totally Bounded 50; Some Theorems 50; Lebesgue Number 52; Lebesgue Covering Lemma 52; Theorem 52; Theorem 53; Finite Intersection Property 53; Some Theorems 53. Unit-2 3. RIEMANN INTEGRAL 58-91 Introduction 58; Definition 58; Upper and Lower Riemann Sums 59; Some Important Theorems 59; Upper and Lower Riemann Integrals 62; Darboux Theorem 63; Riemann Integral 64; Oscillatory Sum 64; Integrability of Continuous Function 75; Integrability of Monotonic Function 76; Properties of Riemann Integral 76; Continuity and Differentiability of Integral Function 82; Second Fundamental Theorem 83; Mean Value Theorems 84. Unit-3 4. COMPLEX INTEGRATION 92-143 Complex Integration 92; Some Definitions 92; Rieman's Definition of Integration or Line Integral or Definite Integral or Complex Line Integral 96; Relation between Real and Complex Line Integrals 97; Some Properties of Line Integrals 97; Evaluation of the Integrals with the Help of the Direct Definition 97; Complex Integral as the Sum of Two Real Line Integrals 99; An Upper Bound for a Complex Integral 112; Cauchy's Fundamental Theorem or Cauchy's Original Theorem or Cauchy's Integral Theorem 113; Cauchy-Goursat Theorem or Cauchy's Integral Theorem (Revised Form) 114; Corollary 117; Cross-Cut or Cut 117; A More General Form of Cauchy's Integral Theorem 117; Extension of Chauch's Theorem Multi-Connected Region 118; Cauchy's Integral Formula 118; Extension of Cauchy's Integral Formula to Multiply Connected Regions 120; Cauchy's Integral Formula for the Derivative of an Analytic Function 120; Analytic Character of Higher Order Derivatives of an Analytic Function 121; Corollary 122; Cauchy's Inequality Theorem 123; Integral Functions or Entire Function 123; Converse of Cauchy's Theorem or Morera's Theorem 123; Indefinite Integrals or Primitives 124; Theorem 124; Fundamental Theorem of Integral Calculus for Complex Functions 125; Liouville's Theorem 125; Maximum Modulus Theorem or Maximum Modulus Principle 126; Minimum Modulus Principle or Minimum Modulus Theorem 127. 5. SINGULARITY 144-169 The Zeroes of an Analytic Function 144; Zeroes are Isolated 144; Singularities of an Analytic Function 145; Different Types of Singularities 145; Meromorphic Functions 149; Theorem 149; Theorem 150; Theorem 150; Entire Function or Integral Function 150; Theorem 150; Theorem (Due to Riemann) 151; Theorem (Weierstrass Theorem) 151; Theorem 152; The Point at Infinity 153; Limit Point of Zeroes 153; Limit Point of Poles 154; Identity Theorem 154; Theorem 154; Theorem 154; Theorem 155; Theorem 155; Theorem 156; Theorem 157; Detection of Singularity 157; Rouche's Theorem 158; Fundamental Theorem of Algebra 160. 6. RESIDUE THEOREM 170-274 Definition of the Residue at a Pole 170; Residue of f(z) at a Simple Pole z = a 170; Theorem 171; Residue of f(z) at a Pole of Order m 171; Rule of Finding the Residue of f(z) at a Pole z = a of any Order 172; Theorem 172; Definition of Residue at Infinity 173; Theorem 174; Theorem 175; Cauchy's Theorem of Residues or Cauchy's Residue Theorem 175; Theorem 176; Liouville's Theorem 177; Evaluation of Real Definite Integrals by Contour Integration 195; Theorem 195; Theorem 196; Jordan's Inequality 196; Jordan's Lemma 197; Integration Round the Unit Circle 197; Evaluation of the Integral ò-¥+¥ f(x) dx 218; Poles on the Real Axis 247; Evaluation of Integrals whose Integrands Involve many Valued Functions 257; Integration along Contour other than Circle or Semi-circle 262. Unit-4 7. FOURIER TRANSFORMS 275-283 Periodic Function 275; Even and Odd Functions 275; Dirichlet Conditions 275; Fourier Series 276; Fourier Integral Theorem 276; Fourier Sine and Cosine Integrals 276; Complex Form of Fourier Integral 277; Finite Fourier Transform of f (x) or Finite Fourier Sine/Cosine Transform of f (x) 277; Determination of f (x) 282. 8. INFINITE FOURIER TRANSFORMS 284-297 Definition 284; Properties 284; Inverse Fourier Transform 291; Fourier Transform or Complex Fourier Transform 294. 9. PROPERTIES OF FOURIER TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS 298-304 Theorems 298; Fourier Transform of Partial Derivatives 300.