First Year: Semester I

First Year: Semester II

Second Year: Semester I

Second Year: Semester II

Third Year: Semester I

Third Year: Semester II

Fourth Year: Semester II

(Students have to complete minimum 17 credits)

Sets: Elementary idea of set, set notation, set of natural numbers, rational numbers, irrational numbers and real numbers along with their geometrical representation, idea of open & closed interval, subsets, power set of a set, basic set operations and related theorems on sets and venn diagrams. Real Number system: Idea of absolute value of real number. Axioms of real number system and their application in solving algebraic equations. Equation and Inequality: Elementary idea of law of inequality, solution of equations and inequalities. Variable and Functions: Variable of a set, functions of a variable, Polynomial, graph of single polynomial functions, exponential, logarithmic, trigonometric functions and their graphs, domain and range of a function; sum, difference, product, quotient, composition and inverse of functions.

1. Seymour Lipschutz: Set Theory

2. R. David Gustafson & Peter D. Frisk: Functions and Graphs

 

Complex numbers: Definition of complex numbers and their properties. De-Moivre’s theorem (for integral and rational exponents) and its applications. Inequalities: Cauchy, Chebyshev and Jensen’s inequality. Theory of equations: Polynomials, division algorithm, fundamental theorem of algebra, multiplicity of roots, relation between roots and coefficients of algebraic equations, Descartes rule of signs.

Bernard & Child, Algebra

2. Hall & Knight, Higher Algebra

3. Rahman M.A. Algebra and Trigonometry

Shahdullah & Battacharjee, Albegra and Trigonometry

 

Vectors: Vectors and vector algebraic operations on vectors, null and unit vectors, components of vectors, scalar and vector products of two vectors, angle between two vectors, product of three and four vectors – their applications. Spherical polar and cylindrical coordinate systems – unit vectors and vector component in spherical and cylindrical systems. Vector calculus: Derivative of vectors with respect to scalars, vector operator DEL, gradient, divergence and curl – their physical significance. Outlines of line, volume and surface integrations. Green’s theorem , divergence theorem, Stokes theorem and their applications. Tensors: Definatons of tensors, fundamental metric tensor, covariant and contravariant tensors. Christoffel’s symbols, covariant differentiation of tensors..

1. Spiegel, M.R. Vector Analysis and and Introduction to Tensor Analysis

2. Jaffreys, H. and Jaffreys B: Method of Mathematical Analysis.

3. Spain, B.: Tensor Calculus

 

Matrix: Definition, elementary properties and solution of system of linear equations with the help of matrices. Differential Calculus: limit, continuity, differentiation of functions, partial differentiation, leibniitz’s theorem and its applications, maxima and minima of a function of one variable. Integral Calculus: Methods of integration, integration by parts, definite integral, area and volumes. Vector Analysis: Scalars and vectors, algebraic operations on vectors, Scalar and vector product of two vectors. derivative of vectors with respect to scalars, vector operators gradient, divergence and Curl. Out lines of line, volume and surface integration.

Ayres, F., Matrices

Kolman, B., Elementary Linear Algebra.

Thomas and Finney, Calculus and Analytic Geometry

E. W Swokowski, Calculus with Analytic Geometry

Speigel M R.: Vector analysis

 

PART-A: Differential Calculus- function , limit , continuity, differentiation , successive and partial differentiation, Rolle’s theorem , Mean value theorem, Maxima and minima. .

Integral Calculus – Integration by various methods; standard Integrals; Definite Integrals; length of curves; area bounded by plane curves; volumes and surface areas of solids of revolution .

PART–B: Co-ordinate geometry of two dimensions - Co-ordinate system, pair of straight lines; circle; tangent & normal at a point on a circle; General equation of second degree.

Co-ordinate geometry of three dimensions - distance between points; angle between two straight lines; plan through three points; angle between two planes; straight line through two points.

1. Thomas and Finney, Calculus and Analytic Geometry

2. E. W Swokowski, Calculus with Analytic Geometry

3. H. Anton, Calculus

4. Rahman & Bhattacharjee; Co-ordinate geometry of two & three dimensions.

5. Loney, S. L.: Coordinate Geometry of Two dimensions

6. Smith, C.: The Analytical Geometry of Conic Sections

 

Sets and Algebra: Sets of real numbers, operation on sets, quadratic equations, solving linear equations and inequalities in one variable, exponents and roots, absolute value and inequalities. Linear equations and system of linear equations. The Cartesian coordinate system. Linear functions and their graphs, the slope and equation of lines, system of two linear equations in two unknowns, a traffic flow (Optional).

Matrices and Matrix Operations: Matrix and matrix product, system of linear equations, inverse of a square matrix.

Linear Programming: Linear inequalities in two variables and linear programming, introduction of graphic approach ,slack variables, the simplex method, the standard maximizing problem.

The Derivative: Introduction, limits, continuity, the derivative as the slope of a curve, the derivative as the rate of change, some differentiation formulas, the product and quotient rules, the chain rule, higher order derivatives, implicit differentiation, derivatives of exoibebtuak and integrating functions.

1.Sanley I Grossman, Applied Mathematics for the Management Life and Social Sciences

2. Finney & Thomas, Calculus and Analytic Geometry.

 

Differential calculus: Differentiation of explicit and implicit functions and parametric equations, sucfessive differentiation of various types of functions, Leibnitz’s theorem, Rolls theorem, Mean value theorem. Taylor’s theorem in finite and infinite forms. Laclaurin’s theorem in finite and infinite forms, lagrange’s form of remainder, Cauchy’s form of remainder. Expansion of function by differentiation and integration. Partial differentiation. Euler’s theorem. Tangent and normal, subtangent and subnormal in Cartesian and polar coordinates. Deteremination of maximum and minimum values of functions, point of inflexion, its applications. Evaluation of indeterminant forms by L’Hospital’s rule. Curvature, radius of curvature, centre of curvature and chord of curvature. Evolute and involute. Asymptotes, Envelopes, curve tracing.

Matrices: Definition of matrix. Different types of matrices. Algebra of matrices. Adjoint and inverse of a matrix. Rank and elementary transformations of matrices. Normal and canonical forms. Solution of linear equations. Quadratic forms. Matrix polynomials. Caley Hamilton theorem and eigenvectors.

1.Thomas and Finney: Calculus and Analytic Geometry

2. Swokowski, E. W.: Calculus with Analytic Geometry

3. Mohammad and Bhattacharjee: Defferential Calculus

4. Spiegel, M. R.: Vector Analysis

5.Bell, R. J. T.: An Elementary Treatise on Coordinate Geometry of Three Dimension.

6.Rahman and Bhattacharjee: A Text Book on Coordinate Geometry

 

Trigonometry: Complex numbers and functions, De Moiver’s theorem and it’s application, summation of finite trigonometric series, hyperbolic functions. Matrices: Type of matrices, null and unit matrices, algebraic operations in matrices, Determinant of square matrices, Matrix equivalence, adjoint and inverse of a matrix, orthogonal and unitary of a matrices, Linear equations, vector spaces, Linear transformations, similar matrices, Charectiristic roots and vectors, diagonalizations of matrices. Complex Variable: Complex numbers and their properties, functions of a complex variable, limit, continuity, analytic functions, Cauchy – Reimann equations, Cauchy’s theorems, Simple contour integrations.

Sarder and Others: Higher Trigonometry

Ayres, F: Matrices

A G Hamilton: Linear Algebra

A Rahman: College Linear Algebra

M L Khanna: Linear Algebra

I S Sokolnikoff & R M Redheffer: Mathematics for Physics & Modern Engineering

KK Kodaira: Introduction to Complex analysis

H Jaffreys & B Jaffreys: Methods of Mathematical Physics

 

Trigonometry: Complex numbers and functions, De Moiver’s theorem and it’s application, summation of finite trigonometric series, hyperbolic functions. Vector algebra: Scalars and vectors, algebraic operations on vectors, null and unit vectors, components of vectors, scalar and vector products of two vectors, angle between two vectors, product of three & four vectors – their applications, spherical polar and cylindrical coordinate systems - unit vectors and vector components in these systems. Vector Calculus: Derivative of vectors with respect to scalar, vector operators DEL, Gradient, Divergence and Curl – their physical significance, outlines of line, volume and surface integration. Geometry: Locus of a point, equations for straight line, circle, parabola, ellipse and hyperbola, pairs of straight lines, equations for plane and straight line in space, sphere, cylinder, cone & ellipsoid.

Sarder and Others: Higher Trigonometry

Speigel M R.: Vector analysis

Smith C.: An elementary treatise on coordinate geometry of three dimension

Rahman & Bhattacharjee: A Text Book on coordinate geometry

Harun Ar Rashid: A Text Book on coordinate geometry

 

Matrices: Types of matrices, null and unit matrices, algebraic operations on matrices, determinant of a square matrix, matrix equivalence, adjoint and inverse of a matrix, orthogonal and unitary matrices, linear equations, vector spaces, linear transformations, similarity, characteristic roots and vectors, diagonalization of matrices.

Vector Analysis: Scalars and vectors; operation on vectors, Null and unit vectors; components of a vectors, scalar and vector products of two, three and four vectors, their applications; vector components in spherical and cylindrical systems, derivative of vectors, vector operators, Del, Gradient, Divergence and Curl, their physical significance, vector integration, line, surface and volume integrals, Greens’, Gauss’ and Stokes’ theorem and their applications.

Geometry: Review of equation of a straight line, circle, parabola, ellipse and hyperbola. Pair of straight lines, General equation of the second degree. Three-dimensional coordinates. Equations for a plane, sphere, cylinder, cone, ellipsoid and paraboloid.

1. Ayres, F., Matrices

2. Kolman, B., Elementary Linear Algebra.

3. Speigel M R.: Vector analysis

4. Smith C.: An elementary treatise on coordinate geometry of three dimension

5. Rahman & Bhattacharjee: A Text Book on coordinate geometry

6. Harun Ar Rashid: A Text Book on coordinate geometry

 

Trigonometry: Complex numbers and functions, De Moivre’s theorem and its applications, summation of finite trigonometric series, hyperbolic functions. Vector Algebra: Scalars and vectors, algebraic operations on vectors, null and unit vectors, components of vectors, scalar and vector products of two vectors, angle between two vectors, product of three and four vectors with their applications; spherical, polar and cylindrical coordinate systems, unit vectors and vector components in these systems, derivative of vectors with respect to scalars, vector operators - DEL, gradient, divergence, curl & their physical significance.

1. Spiegel, M. R.: Vector Analysis and an Introduction to Tensor Analysis

2. Das & Mukharjee : Higher trigonometry

 

Group-A: Differential Calculus. Functions of real variables and their graphs. Limit, Continuty and derivative. Physical meaning of derivative of a function, higher derivatives. Leibnitz’s theorem , Rolle’s theorem , Mean Value theorem ,Taylor’s theorem ,Taylor’s and Maclaurin’s series without proof. Maximum and Minimum of a function , functions of two and three variables , partial and total derivatives, concavity and convexity of a function.

Group-B: Integral Calculus. Physical meaning of integration of a function , evaluation of indefinite integral, definition of Reimann integral, fundamental theorem of integral calculus and its application to definite integral double and triple integration , application of integration in finding lengths areas and volumes.

1. Dass &Mukherjeee , Integral Calculu

2. Dass &Mukherjeee , Integral Calculus

3. Thomas and Finney: Calculus and Analytic Geometry

4. Swokowski, E. W.: Calculus with Analytic Geometry

5. Tierney, Calculus with Analytic Geometry

 

Differential Calculus: Functionof a real variable and their Graphs, limit, continuity and derivatives, Physical meaning of derivative of a function, successive derivative, Leibnitz’s theorem, Rolle’s theorem, Mean value & Taylors theorem (statement only), Taylor’e & Maclaurins series and Expansion of function, Maximum & minimum Values of functions, functions of two and three variables. Partial and total derivative. Integral calculus: Physical meaning of integration, integration as a inverse process of differentiation, definite integral as the limit of a sum and as an area, Definition of Reimann integral, fundamental theorem of integral calculus and its application to definit integrals, reduction formula, improper integrals, double integration, evaluation of areas and volumes by integration. Differential equations: Definition and solution of ordinary differential equations, first order ordinary differential equations, second order ordinary linear differential equations with constant coefficients, initial value problems.

Thomas & finney: Calculus and analytic geometry

Swokowski E W: Calculus with analytic geometry

Mohammed & Bhattacharjee: Differential calculus

Das & Mucharjee: Differential calculus

Mohammed & Bhattacharjee: Integral calculus

Das & Mucharjee: Integral calculus

Ayres, F : Differential calculus

 

Differential Calculus: Functionof a real variable and their Graphs, limit, continuity and derivatives, Physical meaning of derivative of a function, successive derivative, Leibnitz’s theorem, Rolle’s theorem, Mean value & Taylors theorem (statement only), Taylor’e & Maclaurins series and Expansion of function, Maximum & minimum Values of functions, functions of two and three variables. Partial and total derivative. Integral calculus: Physical meaning of integration, integration as a inverse process of differentiation, definite integral as the limit of a sum and as an area, Definition of Reimann integral, fundamental theorem of integral calculus and its application to definit integrals, reduction formula, improper integrals, double integration, evaluation of areas and volumes by integration. Differential equations: Definition and solution of ordinary differential equations, first order ordinary differential equations, second order ordinary linear differential equations with constant coefficients,solutions by the method of undetermined coefficient and variation of parameter, initial value problems.

Thomas & finney: Calculus and analytic geometry

Swokowski E W: Calculus with analytic geometry

Mohammed & Bhattacharjee: Differential calculus

Das & Mucharjee: Differential calculus

Mohammed & Bhattacharjee: Integral calculus

Das & Mucharjee: Integral calculus

Ayres, F : Differential calculus

 

Differential Calculus: Funtions of a real variables and their plots, limit, continuity and derivatives,

Physical meaning of derivative of a function, Leibnitz Theorem, Rolles Theorem,,Mean value theorem and Taylors theorem (statements only).Taylors and Maclaurins series and expansion of functions,

Maximum and minimum values of function, Functions of two or three variables, Partial and total derivatives .

Integral Calculus: Physical meaning of a integration of a function , Integration as an inverse process of differentiation, different techniques of Integration’s, definite integrate as the limit of a sum and as an area, definition of Riemann integrals, Fundamental theorem of integral calculus and its application to definite integral, Improper integral, Reduction formula , improper integrals, Double integration, Evaluation of area and volume by integration.

Differential Equations: Definition and solution of Ordinary Differential Equation, First order Ordinary Differential Equation , second order Ordinary Linear Differential Equation with constant coefficients. Initial value problems

1. Das & Mukherjee; differential Calculus

2. Das & Mukherjee, integral Calculus

3. J. Edwards; differential Calculus

4. J. Edwards; integral calculus

5. R.A. Sardar; differential Calculus

6. S. L. Ross; Differential equations

 

Differential Calculus: Physical meaning of derivative of a function, Higher derivatives, Leibnitz Theorem, Rolles Theorem,

,Mean value theorem ,Taylors thorem,Taylors and Maclaurins series , Maximum and minimum values of function, Functions of two or three variables, Partial and total derivatives , Taylors series for multivariable functions, Convexity of a function.

Integral calculus: Physical meaning of a integration of a function , Evaluation of a indefinite integral, definition of Riemann integrals, Fundamental theorem of integral calculus and its application to definite integral, Improper integral, Double integration Evaluation of area, volume and revolution by integration.

1. Das & Mukherjee; differential Calculus

2. Das & Mukherjee, integral Calculus

3. M.R. Spiegel; Advanced Calculus

4. J. Edwards; differential Calculus

5. J. Edwards; integral calculus

6. R.A. Sardar; differential Calculus

7. S. L. Ross; Differential equations

 

Differential Calculus: Differentiation of explicit and implicit functions and parametric equations, sucfessive differentiation of various types of funnctions. Leibnitz’s theorem, Rolls theorem, Mean Value Theorem. Taylor’s theorem in finite and infinite forms. Laclaurin’s theorem in finite and infinite forms, lagrange’s form of remainder, Cauchy’s form of remainder. Expansion of function by differentiation and integration. Partial diffferentiation. Euler’s theorem. Tangent and normal, subtangent and subnormal in cartesian and polar coordinates. Determination of maximum and minimum values of functions, point ofinflexion, its applications. Evaluation of indeterminant forms by L’ Hospital’s rule. Curvature, radius of curvature, centre of curvature and chord of curvature. Evolute and involute. Asymptotes , Envelopes, Curve tracing.

Vectors: Definitions of vectors Equality of vectors, Addition and multiplication of Vectors, Triple products and multiple products.

1.Thomas and Finney: Calculus and Analytic Geometry

2. Swokowski, E. W.: Calculus with Analytic Geometry

3. Mohammad and Bhattacharjee: Defferential Calculus

4. Spiegel, M. R.: Vector Analysis

5.Bell, R. J. T.: An Elementary Treatise on Coordinate Geometry of Three Dimension.

6.Rahman and Bhattacharjee: A Text Book on Coordinate Geometry

 

Differential Calculus: Differentiation of explicit and implicit functions and parametric equations, sucfessive differentiation of various types of funnctions. Leibnitz’s theorem, Rolls theorem, Mean Value Theorem. Taylor’s theorem in finite and infinite forms. Laclaurin’s theorem in finite and infinite forms, lagrange’s form of remainder, Cauchy’s form of remainder. Expansion of function by differentiation and integration. Partial diffferentiation. Euler’s theorem. Tangent and normal, subtangent and subnormal in cartesian and polar coordinates. Determination of maximum and minimum values of functions, point of inflexion, its applications. Evaluation of indeterminant forms by L’ Hospital’s rule. Curvature, radius of curvature, centre of curvature and chord of curvature. Evolute and involute. Asymptotes , Envelopes, Curve tracing.and symmetry

Three dimensional coordinate geometry : System of coordinates, distance between two points, Sections formula, Projections, Direction cosines and direction ratios. Equations planes and straight lines.

1.Thomas and Finney: Calculus and Analytic Geometry

2. Swokowski, E. W.: Calculus with Analytic Geometry

3. Mohammad and Bhattacharjee: Defferential Calculus

4. Spiegel, M. R.: Vector Analysis

5.Bell, R. J. T.: An Elementary Treatise on Coordinate Geometry of Three Dimension.

6.Rahman and Bhattacharjee: A Text Book on Coordinate Geometry

 

Integral calculus: Difinition of integration,integration by method of substitution , integration by parts , standard integrals , method of successive reduction. Definite integral ,its properties and use in summing series. Walli’s formulae. Improper integral, Beta and Gamma function. Area under a plane curve in cartesian and polar coordinates , area of the region enclosed by two curves in cartesian and polar coordinates, Trapezoidal rule .Simpson’s rule, Arc length of curves in cartesian and polar coordinates, parametric and pedal equations , intrisic equation. Volumes of solid of revolution Volumes of hollow solid of revolution by shell method , area of surface of revolution .

Differential Equation : Ordinary differential equation and formation of differential equations , Solution of first order differential equations with various method. Solutions of general linear equations of second and higher order with constant coefficients. . Solutions of homogeneous linear equations , applications . Solution of differential equations of the higher order when the dependent and independent variables are absent . solutions of differential by the method based on factorization of the operators.

Thomas and finney ,. Calculus and Analytic Geometry .

Swokowski , E.W., Calculus and Analytic Geometry

Mohammed & Bhattacharjee , Integral Calculus .

Ayres , F., Differential equation

Edward , J.,Integral Calculus

 

Integral calculus: Difinition of integration,integration by method of substitution , integration by parts , standard integrals , method of successive reduction. Definite integral ,its properties and use in summing series. Walli’s formulae. Improper integral, Beta and Gamma function. Area under a plane curve in cartesian and polar coordinates , area of the region enclosed by two curves in cartesian and polar coordinates, Trapezoidal rule .Simpson’s rule, Arc length of curves in cartesian and polar coordinates, parametric and pedal equations , intrisic equation. Volumes of solid of revolution Volumes of hollow solid of revolution by shell method , area of surface of revolution .

Differential Equation : Ordinary differential equation and formation of differential equations , Solution of first order differential equations with various method. Solutions of general linear equations of second and higher order with constant coefficients. . Solutions of homogeneous linear equations , applications . Solution of differential equations of the higher order when the dependent and independent variables are absent . solutions of differential by the method based on factorization of the operators.

1. Thomas and finney ,. Calculus and Analytic Geometry .

2. Swokowski , E.W., Calculus and Analytic Geometry

3. Mohammed & Bhattacharjee , Integral Calculus .

4. Ayres , F., Differential equation

5. Edward , J.,Integral Calculus

 

Matrix: Definition, elementary properties and solution of system of linear equations with the help of matrices. Differential Calculus: limit, continuity, differentiation of functions, partial differentiation, leibniitz’s theorem and its applications, maxima and minima of a function of one variable. Integral Calculus: Methods of integration, integration by parts, definite integral, area and volumes. Vector Analysis: Scalars and vectors, algebraic operations on vectors, Scalar and vector product of two vectors. derivative of vectors with respect to scalars, vector operators gradient, divergence and Curl. Out lines of line, volume and surface integration.

Integral calculus: Definition of integration, integration by method of substitution, integration by parts, standard integrals, method of successive reduction. Definite integral, its properties and use in summing series. Walli’s formulae. Improper integral, Beta and Gamma function. Area under a plane curve in Cartesian and polar coordinates, area of the region enclosed by two curves in Cartesian and polar coordinates, Trpezoidal rule, Simpson’s rule, Arc length of curves in Cartesian and polar coordinates, parametric and pedal equations, intrinsic equation. Volumes of solid of revolution, volumes of hollow solid of revolution by shell method, area of surface of revolution.

Differential Equation: Ordinary differential equation and formation of differential equations, solutions of first order differential equations with various method. Solution of general linear equations of second and higher order with constant coefficient. Solutions of homogeneous linear equations, applications. Solutions of differential equations of the higher order when the dependent and independent variables are absent, solutions of differential by the method based on factorization of the operators.

1. Thomas and finney ,. Calculus and Analytic Geometry

2. Swokowski , E.W., Calculus and Analytic Geometry

3. Mohammed & Bhattacharjee , Integral Calculus

4. Edward , J.,Integral Calculus

5. Ayres , F., Differential equation

 

Matrix: Definition of a matrix, different types of matrices, addition and multipication of matrices. adjoint and inverse of matrix, Cramer’s rule, application of inverse matrix and Cramer’s rule. Elementary row operations and Echelon forms of matrices, rank, row rank, column rank of a matrix and their equivalenc, use of rank and Echelon forms in solving system of homogeneous and nonhomogeneous equations. Vector space and subspace over reals and direct sum, linear combination linear dependence and independence on vectors, basis and dimension of vector space, quotient space and isomorphism theorems, Linear transformations, kernel, rank and nullity nonsingular transformations and matrix representation , Changes of basis, Eigenvector. eigenvalues, characteristic equations and Cayley-Hamilton theorem. Similar matrices, canonical forms orthogonal and hermitian matrices, inner product, orthogonal vectors and orthonormal bases, Gram-schmidt orthogonalization process. Bilinear and quadratic forms.

Books Recommended:

1. Hamilton A.G Linear Algebra

2. AYres, F. Matrices

3. Kolman B, Elementary Linear Algebra

4. Bering E.D Linear Algebra and Matrix Theory

5. Lipschutz S, Linear Algebra

6. Morris AO, Linear Algebra

7.Khanna M I, Linear Algebra

8. Rahman M A, College Linear Algerba

 

Sets: Elementary idea of set, subsets, power set of a set, product set. Basic set operations and related theorems on sets, Venn diagrams, countable and uncountable sets, cardinality of a set. Real Number system: set of natural numbers, rational numbers, irrational numbers and real numbers along with their geometrical representation, idea of open & closed interval, product set of real numbers and their geometric representation, Idea of absolute value of real number. Axioms of real number system and their application in solving algebraic equations. Equation and Inequality: Elementary idea of law of inequality, solution of equations and inequalities. Relations and Functions: binary relations, reflexive, symmetry anti-symmetry and transitive relations, Pictorial representations of relations, properties of relation. Variable of a set, functions of a variable, domain and range of a function Polynomial, graph of single polynomial functions, exponential, logarithmic, trigonometric functions and their graphs, algebra of functions, inverse of functions and its graph. Vertical line test for a function and test for symmetry of functions, test for continuity of a function from its graph. Complex Number system: Geometrical representation and properties.

Seymour Lipschutz: Set Theory

R. David Gustafson & Peter D. Frisk: Functions and Graphs

Earl W. Swokowski: Calculus with Analytic Geometry

George B. Thomas Jr. & Ross L. Finney: Calculus with Analytic Geometry

 

Introduction : Definitions and identities of trigonometric and hyperbolic functions with their inverses, Demoivre’s Theorem and its application. Summation of series (algebraic and trigonometric): Arithmetic and geometric series, method of difference and C + i S method (for trigonometric series), Inequalities: Inequalities involving mean, inequalities of Weirstrass, Cauchy, Tchebyshev, Holder and Minskowski. Theory of equations: Polynomials and division algorithms, fundamental theorem of algebra. multiple roots, transformation of equations, relations between roots and coefficients. Descarte’s rule of signs, symmetric functions of the roots. solutions of cubic and biquadratic equations. Sturm’s theorem.

Lipschutz, S : Set Theory and Related Topics

Bernard & Child : Higher Algebra

Spiegel, M. R. : Vector Analysis

 

Two dimensional geometry: Set of coordinates for a plane, straight line in a plane. increments, distance of two lines, slope of a line, tangent and normal on a curve, pair of straight lines, basic properties of Circle, Parabola, Ellipse and Hyperbola. Change of coordinates and axes, invariant. General equation of second degree, Reduction of general equation of second degree to standard form and identification of Conic. Polar and parametric equations of conic., poles, polars, chords in terms of middle points, director circle, eccentric angles and conjugate diameters of conic. Functions: limit and its properties of functions, continuity of functions. Derivative of Functions: Intermediate forms and L’Hospital rules, successive differentiations and Leibnitz Theorem. Integration: Introduction, Indefinite integrals, applications, determining constants of integration, basic integration formulas, integration by parts, products and powers of trigonometric functions, even powers of sine’s and cosines, trigonometric substitutions, partial fractions. Definite integrals, calculating areas as limits, the fundamental theorems of integral calculus, integration by substitution, differentials, rules for approximating definite integrals.

Thomas and Finney: Calculus and Analytic Geometry

J. Stewart, Calculus

Swokowski, E. W.: Calculus with Analytic Geometry

Smith, C.: The Analytical Geometry of Conic Sections

 

Matrix: Introduction to matrices, addition and multiplication of matrices, determinant of matrix, Cramer’s rule, adjoint and inverse of a matrix, elementary row operations and echelon forms of matrix, rank, row rank, column rank of a matrix and their equivalence, use of rank and echelon form in solving system of homogeneous and non-homogeneous equations. Vector space and subspace over real numbers and direct sum, linear combination, linear dependence and independence of vectors, basis and dimension of vector space, quotient space and isomorphism theorems. Linear transformations, kernel, rank and nullity, matrix representation, change of basis, eigenvalues and eigenvectors, characteristic equations and Caley-Hamilton theorem, diagonalization of matrices, similar matrices, canonical forms. Orthogonal and Hermitian matrices, inner product, orthogonal vectors and orthonormal basis, Gram-Schmidt orthogonalization process, bilinear and quadratic forms.

Hamilton, A. G.: Linear Algebra

Anton, H. and Rorres, C. Elementary Linear algebra with Applications.

Kolman, B.: Elementary Linear Algebra

Nering, E. D.: Linear Algebra and Matrix Theory

Lipschutz, S.: Linear Algebra

 

Applications of derivatives: Curve sketching, the significance of the first derivative, increasing & decreasing functions, concavity and point of inflection, asymptotes and symmetry, maxima and minima. involute, evolute, envelop. Rolle's theorem, Mean Value theorem, Taylor’s Theorem in different forms, Maclaurin Series and their application for the expansion of functions, extending the Mean Value theorem to Taylor’s formula, estimating approximate errors. Applications of definite integrals: Area between two curves, calculating volumes by slicing , volumes modeled with shells and washers, length of a plane curve, area of a surface of revolution, average value of a function, moments and centre of mass, evaluation of improper integrals. Gamma and Beta functions, reduction formulas. Transcendental functions: Derivatives of trigonometric functions and related integrals, the natural logarithm and its derivative, graph of exponential and logarithmic functions. Applications of exponential and logarithmic functions. Hyperbolic function: Derivatives and integrals, hanging cables, polar coordinates, graphs of polar equations, polar equations of conic and other curves, integrals.

Thomas and Finney: Calculus and Analytic Geometry

Swokowski, E. W.: Calculus with Analytic Geometry

Spiegel, M. R.: Advanced Calculus

Stewart, J. Calculus

Trigonometry: Trigonometric functions and their inter relations, Trigonometric identities.

Set Theory: Concepts of sets and subsets , operations on sets, Cartesian products , functions and relations, binary operations on the sets N,Q,R as algebraic systems, equivalence relations, equivalent sets, countable and uncountable sets.

Vector Space: Defination, linear sum, inner product space.

Calculas: Concept of Integration, ibndefinite and definite integrals, methods of integration, applications from a marginal to total functions, investment and capital formation, consumers and producer surplus.

Calculus of Variation: Eeuler’s equation and its application. Differential and Difference equations: Equations of the first and second order, simple cases of linear differential equations with constant, co-efficients, economic applications.

1. A.C. Chiang, Fundamental methods of Mathematical Economics( 3^rd edition )

2. W.J Baumol, Economic dynamics

3. W.J. Baumol, Economic theory and Operations Analysis (4^th edition)

4. A.K. Dixit Optimization in Economic Theory

5. P.J. Iamberte Advanced Mathematics for Economists

6. taro Yamane Mathematics for Economists

7. E.T.Dowling Mathematics for Economists

8. Akira Takayama Mathematical Economics

Vector Analysis: Scalars and vectors, equality of vectors. Addition and subtraction of vectors by scalars. Position vector of a point. Resolution of vectors, Scalar and vector products and multiple products. Application to geometry and mechanics. Linear dependence and independence of vectors. Differentiation and integration of vectors together with elementary applications. Definition of line, surface and volume integrals. Gradient divergence and curl of point functions. Various formulae. Gauss’s theorem, Stoke’s theorem, Green’s theorem and their applications.

Numerical Analysis: Interpolation: Simple difference, Newton’s formulae for forward and backward interpolation. Divided differences. Tables of divided differences. Relation between divided differences and simple differences. Newtonís general interpolation formulae. Lagrangeís interpolation formulae. Inverse interpolation by Lagrangeís formula and by successive approximations. Numerical differentiation of Newton’s forward and backward formula. Numerical integration. General quadrature formula for equidistant ordinates. Trapezoidal rule. Simpsonís rule. Weddleís rule. Calculation of errors. Relative study of three rules. Gaussís quadrature formula. Legendre polynomials. Newtonís Cotes formula. Principles of least squares. Curve fitting. Solution of algebraic and transcendental equations by graphical method. Regula-Falsi method. Newton-Rapson method, Geometrical significance. Convergence of iteration and Newton-Rapson methods. Newton-Rapson method and iteration method for the solution of simultaneous equations. Solution of ordinary first order differential equations by Picardís and Eularís method. Runge-Kuttaís methods for solving differential equation.

1. Speigel M R.: Vector analysis

2. Freeman H, Finite Difference for Actuarial Students

3. Francis Scheid: Numerical Analysis

4. Hilderman, F.B.: Introduction to Numerical Analysis

5. Noble B.: Numerical Methods Vol. I & II

 

Matrix: Type of matrix: null and unit matrices, algebraic operations on matrices, determinent of squre matrices, matrices equivalance, adjoint and inverse of a matrix, orthogonal and unitary matrices, linear dependence of vectors, system of linear equations. Complex Variables: analytic functions, Cauchy- Rieman equations, Complex integration. Furier series: Periodic functions, furier series of odd and even functions. Special Functions: Hermite and Bessel equations,Legendre and associated Legendre equations. Laplace transfor,mation.

Geometry: Two Dimensional: Locus of a point, equations for straight lines, circles, parabola, ellipse, hyperbola, pair of straight lines. General equations of second degree. Three dimensional: Coordinates in three dimensions, locus of a point, equations for straight lines & planes in space, spheres, cylinders, cones, spheroids & ellipsoids.

Numerical Analysis: Interpolation and extrapolation. Shifting operators, difference operators and their relationships. Newton’s interpolation formulae, Lagrange’s formula, Newton’s divided difference formula, central difference formulae (Stirling and Bessel’s) Relationship between divided difference and simple difference. and simple difference. Inverse interpolation formula. Numerical differentiation. Numerical integration by different formulas. Numerical solution of equations by various methods. Convergence of these methods and their inherent errors. Numerical solution of simultaneous Linear equation. solution by determinants by inverse matrices, by iteration and by successive elimination of the unknowns.

Freeman H, Finite Difference for Actuarial Students

Francis Scheid: Numerical Analysis

Hilderman, F.B.: Introduction to Numerical Analysis

Noble B.: Numerical Methods Vol. I & II

Loney, S. L.: Coordinate Geometry of Two dimensions

Rahman and Bhattacharjee: Text Book on Coordinate Geometry

Kar, J. M.: Conic Section

Bell, R. J. T.: An Elementary Treatise on Coordinate Geometry of Three Dimension.

Askwith, R. E. H. : Analytical Geometry of Conic Sections

Loney, S. L.: Coordinate Geometry of Two dimensions

Smith, C.: The Analytical Geometry of Conic Sections

 

Complex Variable : Complex number system, general functions of a complex variable limits and continuity of a function of complex variable and related theorems. Complex differential and the Cauchy, Riemann equations. Mapping by elementary functions. Line integral of a complex function. Cauchy’s integral formula. Kiouville’s theorem. Taylor’s and Laurent’s theorem. Singular points. Residue. Cauchy’s residue theorem. Evaluation of residues. Contour integration. Conformal mapping.

Fourier series: Real and complex form. Finite transformation. Fourier integral. Fourier transforms and their uses in solving boundary value problems.

Partial Differential Equation: Introduction. Equation of the linear and non-linear first order. Standard forms. Linear equations of higher order. Equations of the second order with variable coefficients.

Laplace Transform: Definition. Laplace transforms of some elementary functions. Sufficient conditions for existence of laplace transform. Inverse Laplace transforms. Laplace transforms of derivatives. The unit step function. Periodic function. Some special theorems on Laplace transforms. Partial fraction. Solutions of differential equations by Laplace transform. Evaluation of improper integrals

Churchill: Introduction to Complex Variable and Applications

Freeman H: Finite Difference for Actuarial Students

Macrobeat: Complex Variable.

Spiegel, M.R. Complex Variable

Stephenson : Mathematical Methods

Ross, S. L. : Differential Equations

Spiegel, M. R.: Laplace Transform

Khanna, M. L. : Partial Differential Equations

Khanna, M. L. : Laplace Transforms

 

Complex Variables: DeMoivre’s theorem and its application,Locus problem. Complex numbers and their properties, functions of a complex variable, limits and continuity of a function of complex variable. Analytical functions, the Cauchy-Riemann equations, Cauchy’s theorem, Singularity and Poles, Residues , Simple contour integration, and their uses in solving boundary value problems

Laplace Transformations: Definition of Laplace transform, Laplace transform of different functions, first shift theorem, inverse transform, linearity, use of first shift theorem and partial functions. Transform of derivative, transform of an integral, the Heaviside unit function, the unit impulse function, the second shift theorem, periodic functions, convolutions, solution of ordinary differential equations by Laplace Transform.

Fourier Series: Fourier series, Convergence of Fourier Series, Fourier Analysis, Fourier transforms.

1. KK Kodaira: Introduction to Complex analysis

2. H Jaffreys & B Jaffreys: Methods of Mathematical Physics

3. Spiegel, M. R.: Laplace Transform

4. Khanna, M. L. : Laplace Transforms

 

Matrix: Types of matrices, null and unit matrices, algebraic operations of matrices, determinant of square matrix, matrix equivalence, adjoint and inverse of matrices, orthogonal and unitary matrices, linear dependence and independence of vectors, system of linear equations. Complex variables: Analytic functions, Cauchy-Riemann equations, complex integration. Differential equations: Definition, solution of differential equations, basic theory of linear differential equations, homogeneous differential equations of the 2nd and higher order with constant coefficients, power series solution about ordinary and regular points, non-homogeneous differential equations, solutions by the methods of undetermined coefficients and variation of parameters, Hermite and Bessel equations, Legendre and associated Legendre equations, partial differential equations, linear and non-linear partial differential equations of 1st order. Laplace transforms: Definition of Laplace transform, elementary transformations and properties, convolutions, solution of differential equations by Laplace transforms, evaluation of integrals by Laplace transforms.

1. Ayres, F.: Matrices

2. Stephenson : Mathematical Methods

3. Ross, S. L. : Differential Equations

4. Spiegel, M. R.: Laplace Transform

5. Khanna, M. L. : Partial Differential Equations

6. Khanna, M. L. : Laplace Transforms

 

Group A: Advanced Calculus: Improper Integrals, Gamma and Beta functions, their incompleteness and other properties, functions of several variables and limit and continuity, Taylor’s expansion of such functions, maxima and minima of functions of more than one variables, Lagrange’s multipliers, multiple integrals, jacobians of transformation, Dirichlet integral and its extension, Laplace transformation, concept of Fourier series.

Group B: Differential equations: Definition, solution of differential equations, basic theory of linear differential equations, basic theory of linear differential equation, equation of the first order and their solution, homogeneous differential equations, linear differential equations of the second and higher order and their solution.

1. Ayres F, Differential Equations

2. Edward, Differential and Integral Calculus

3. Maxwell E H G, Analytical Calculus, Vol-II & Vol-II

4. Piaggio H TH, An Elementary Treaties of Differential Equations and Their Application

5. Ross S L, Differential Equations

6. Widder, Advanced Calculus

 

Vector Calculus: Differentiations and integration of vectors togather with elementary applications. Line, surface and volume integrals. Gradient of scalar functions. Divergence and curl of vector functions. Physical significance of gradient, divergence and curl. Stoke’s theorem. Green’s theorem and their applications. Matrices: Types of matrices and algebraic properties. Rank and elementary transformations of matrices. Solution of linear equations by mtrix methods. Linear dependence and independence of vectors. Quadratic forms, matrix polynomials. Determination of characteristic roots and vectors. Laplace transforms: Definition of Laplace transforms. Elementary transformations and properties. Convolution. Solution of differential equations by Laplace transforms. Evaluation of integrals by Laplace transforms.

1. Spiegel, M.R. Advanced Calculus

2. Spiegel, M.R. Vector Analysis and and Introduction to Tensor Analysis

3. Lass, H. Vector and Tensor Analysis

4. Ayres, F: Matrices

5. A G Hamilton: Linear Algebra

6. Spiegel, M. R.: Laplace Transform

7. Khanna, M. L. : Laplace Transforms

 

Vector Calculus:Definitions of vectors Equality of vectors, Addition and multiplication of Vectors, Triple products and multiple products. Differentiations and integration of vectors togather with elementary applications. Line, surface and volume integrals. Gradient of scalar functions. Divergence and curl of vector functions. Physical significance of gradient, divergence and curl. Stoke’s theorem. Green’s theorem and their applications. Matrices: Types of matrices and algebraic properties. Rank and elementary transformations of matrices. Solution of linear equations by mtrix methods. Linear dependence and independence of vectors. Quadratic forms, matrix polynomials. Determination of characteristic roots and vectors. Laplace transforms: Definition of Laplace transforms. Elementary transformations and properties. Convolution. Solution of differential equations by Laplace transforms. Evaluation of integrals by Laplace transforms.

1. Spiegel, M.R. Advanced Calculus

2. Spiegel, M.R. Vector Analysis and and Introduction to Tensor Analysis

3. Lass, H. Vector and Tensor Analysis

4. Ayres, F: Matrices

5. A G Hamilton: Linear Algebra

6. Spiegel, M. R.: Laplace Transform

7. Khanna, M. L. : Laplace Transforms

 

Group A: Numerical Methods: Interpolation and extrapolation. Shifting operators, difference operators and their relationships. Newton’s interpolation formulae, Lagrange’s formula, Newton’s divided difference formula, central difference formulae (Stirling and Bessel’s) Relationship between divided difference and simple difference. and simple difference. Inverse interpolation formula. Numerical differentiation. Numerical integration by different formulas. Numerical solution of equations by various methods. Convergence of these methods and their inherent errors. Numerical solution of simultaneous Linear equation. solution by determinants by inverse matrices, by iteration and by successive elimination of the unknowns. Group-B Complex functions. elementary single and many valued functions of complex variables. differentiable functions. analytic functions. Cauchy’s theorem for simple contours. Taylor’s theorem. Laurent’S theorem. Liouville’s theorem, different types of singularity, Cauchy,s residue theorem, evaluation of integral by contour integration.

1. Churchill Introduction to Complex Variable and Applications

2. Freeman H, Finite Difference for Actuarial Students

3. Macrobeat, Complex Variable.

 

Sets: Functions, relations,equivalence relations, real valued functions, open set, dense set, countability, compact and connected sets,monotonic class of sets, aditive class of sets. Sequence: Convergence of a sequence, monotonic sequence, upper limit and lower limit. Infinite Series: Concept of sum, series of positive terms, alternating series, absolute and conditional convergence, test for convergence. Limit points, Bolzano-Weierstrass theorem, properties of continuous functions, uniform continuity, Heine-Borel theorem. Derivatives: Rolle’s theorem, Mean value theorem and Taylor’s theorem with remainder in Lagrange’s and Cauchy’s forms. Expansions of functions. Power series: Interval and radius of convergence, differentiation and integration of power series, Abels’ continuity theorem. Riemann integration: Definition of Riemann integration. Fundamental theorem and mean value theorem of integral calculus. Improper integral and their tests for convergence.

1. Ruddin, W.: Principle of mathematical analysis

2. Aposstal, I.: Mathematical Analysis

3. Bartle,: Real Analysis

4. Royden,: Real Analysis

5. Hobson, E.: The Theory of Functions of Real Variable and Theory of Fourier Series

6. Burkill,J. G.: A First Course in Mathematical Analysis

7. Binmore, K. G.: Mathematical Analysis

 

Vectors: Vectors in the plane, Vectors in space, vector algebra, Dot and Cross products of vectors, Vector differentiation: Vector and scalar fields, directional derivatives, gradient, divergence, curl and Laplacian operator. Vector integration: Line, surface and volume integrals, theorems of Green, Gauss and Stokes and their applications. Curvilinear Coordinates, Concept of Tensors: Transformation of coordinates, covariant and contravariant tensor. The fundamental metric tensor: Christoffel symbols, covariant differentiation and contraction of tensor. Parallesism and geodesics. Riemann- Christoffel Tensor, curvature tensor, tensor and Bianchi identity.

1.Thomas and Finney: Calculus and Analytic Geometry

2. Swokowski, E.W. Calculus with Analytic Geometry

3. Spiegel, M.R. Advanced Calculus

4. Spiegel, M.R. Vector Analysis and and Introduction to Tensor Analysis

5. Lass, H. Vector and Tensor Analysis

6. Spain, B.: Tensor Calculus

 

Introduction to differential equations. Ordinary differential equations and their solutions: Ordinary differential equations of first order and first degree, ordinary differential equations of 1st order but of higher degree, initial value problem, orthogonal trajectories, general solution of linear ordinary differential equations (homogeneous and non-homogeneous) with constant coefficients, methods of undetermined coefficients and variation of parameters, reduction of order. Simple cases of non-linear differential equations, system of linear ordinary differential equations. linear system and fundamental matrix solutions of linear systems with constant coefficients. Partial differential equations of the 1st order: Charpit’s method, total differential equations of three variables. Existence and uniqueness theorem (statement) of solutions of differential equations, successive approximations, existence theorem of 1^st order equations.

Ayres, F.: Differential Equations

Piaggio, H. T. H.: Differential Equations

Forsyth: Differential Equations

Ross, L.: Introduction to Differential Equations

Boyce and D’Prima: Differential Equations

 

Coordinates in three dimensions: Different systems of coordinates and transformation of coordinayes, direction cosine, direction ratios, planes and straight lines in three dimensions, general equation of second degree in three variables, redunction to stadard forms and identification of conicoid, sphere, cylinder, cone, ellipsoid, paraboloid and hyperboloid. Curve tracing: Polar Coordinate systems and tracing the curves using the ideas of Calculus. Partial derivatives: Functions of two or more variables, limits and continuity, partial derivatives, chain rule, gradients, directional derivatives and tangent planes, higher order derivatives, partial differentials, linear approximation and increment estimation, maxima, minima and saddle points, Lagrange multipliers, exact differentials, Taylor’s theorem. Multiple integrals: Double integrals, areas and volumes, physical applications, changing to polar coordinates, triple integral in rectangular coordinates, physical application in three dimensions.

Smith, C.: An Elementary Treatise on Solid Geometry.

Bell, R. J. T.: An Elementary Treatise on Coordinate Geometry of Three Dimension.

Stewart, J. Calculus

Thomas and Finney: Calculus and Analytic Geometry

Swokowski, E. W.: Calculus with Analytic Geometry

 

Real number system: The completeness axioms and and Dedekind’s axioms. Limit points of a set of real numbers, Bolzano-Weierstrass theorem. Sequence of real numbers: Definition, convergence of a sequence, subsequence, monotonic sequence bounded sequence, Cauchy sequence, Cauchy criteria for convergence of sequences. Infinite series: Concept of sum and convergence, series of positive terms, alternating series, absolute and conditional convergence, tests for convergence., Functions: Limits and continuity of functions with their properties, uniform continuity, Heine-Borel theorem, differentiability of functions, Rolle’s theorem, Intermediate value theorem, Darboux theorem, Mean value theorem, Taylor’s theorem with remainder in Lagrange’s and Cauchy’s forms, expansions of functions. Power series: Interval and radius of convergence, differentiation and integration of power series, Abel's continuity theorem. Riemann integrals: Definition of Riemann integration, Riemann integration with related theorems, conditions for integrability, fundamental theorem and mean value theorem of integral calculus, Riemann-Stieltjes integrals.

Rudin, W.: Principle of mathematical analysis

Apostol, I.: Mathematical Analysis

Bartle,: Real Analysis

Marsden, J.E. and Hoffman, M.J. Elementary Classical Analysis

Burkill,J. G.: A First Course in Mathematical Analysis

 

Complex variables: Geometry and topology of the complex plane, elementary functions of a complex variable (including the general power and the logarithm). Limit, continuity and differentiability of functions of a complex variable, analytic functions and their properties, harmonic functions, meromorphic function and entire functions. Complex integrals: Line integral over rectifiable curves, Cauchy’s theorem for simple contours, Cauchy’s integral formula, theorems of Liouville and Morera, fundamental theorem of algebra. Zeros, singularities, poles, and residues. Taylor’s and Laurent’s series, expansion of functions. Cauchy’s residue theorem, Rouche’s theorem, the maximum modulus principle, evaluation of real integrals by contour integration. Conformal mappings, bilinear transformations and their properties.

Churchill and Brown: Complex variables and Applications

Stewart and Tall: Complex Analysis

Spiegel, M. R.: Complex Variable

Copson, E. I.: Theory of Function of Complex Variables