Calculus

Hyperbolic functions, Leibniz rule and its applications to problems of type eax+bsinx, eax+bcosx, (ax+b)n sinx, (ax+b)n cosx, Reduction formulae, Techniques of sketching conics, reflection properties of conics, rotation of axes and second degree equations, etc.

The main aim of this course is to make the students acquainted with the basic concepts of calculus and analytic geometry through theoretical teaching and practicals.

Algebra

Polar representation of complex numbers, nth roots of unity, De Moivre’s theorem for rational indices and its applications, Equivalence relations, Functions, Composition of functions, Systems of linear equations, Introduction to linear transformations, matrix of a linear transformation, etc.

This paper focuses on the concepts of algebra and complex numbers along with Graph theory and applications of linear algebra.

Real Analysis

Review of Algebraic and Order Properties of R, ߜ-neighborhood of a point in R, Idea of countable sets, uncountable sets and uncountability of R, Sequences, Bounded sequence, Convergent sequence, Limit of a sequence, Infinite series, convergence and divergence of infinite series, Cauchy Criterion, etc.

This paper deals with the concepts of real analysis.

Differential Equations

Differential equations and mathematical models, Introduction to compartmental model, exponential decay model, lake pollution model etc., General solution of homogeneous equation of second order, principle of super position for homogeneous equation, Equilibrium points, Interpretation of the phase plane, predatory-prey model and its analysis, etc.

The paper deals with the computing and modeling of differential equations and its practical approach using Maple and MATLAB.

Theory of Real Functions

Limits of functions (߳െߜ approach), sequential criterion for limits, divergence criteria, Differentiability of a function, Caratheodory’s theorem, Cauchy’s mean value theorem, Riemann integration, Riemann conditions of integrability, Improper integrals, Pointwise and uniform convergence of sequence of functions, Limit superior and Limit inferior. Power series, radius of convergence, etc.

This paper gives the elementary understanding of the real functions and their analysis.

Group Theory

Definition and examples of groups including permutation groups and quaternion groups (illustration through matrices), Properties of cyclic groups, classification of subgroups of cyclic groups, External direct product of a finite number of groups, Group homomorphisms, properties of homomorphisms, Cayley’s theorem, Characteristic subgroups, Commutator subgroup and its properties, etc.

This course deals with topics related to abstract algebra and theory of groups.

PDE and Systems of ODE

Partial Differential Equations – Basic concepts and definitions, Derivation of Heat equation, Wave equation and Laplace equation, Systems of linear differential equations, types of linear systems, differential operators, etc.

Through this paper the students are acquainted with the linear partial differential equations and differential equations in general.

Multivariate Calculus

Functions of several variables, limit and continuity of functions of two variables, Chain rule for one and two independent parameters, directional derivatives, Double integration over rectangular region, Triple integrals, Triple integral over a parallelepiped and solid regions volume by triple integrals, Line integrals, Applications of line integrals, Green’s theorem, surface integrals, integrals over parametrically defined surfaces, etc.

The focus of the paper is calculus and analytical geometry involving basic multivariable calculus, its concepts and contexts and also an understanding of advanced calculus.

Complex Analysis

Limits, Limits involving the point at infinity, continuity, Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, An extension of Cauchy integral formula, consequences of Cauchy integral formula, Liouville’s theorem, Laurent series and its examples, absolute and uniform convergence of power series, uniqueness of series representations of power series etc.

The paper deals with the complex variables and its application and the theory of complex variables.

Rings and Linear Algebra

Definition and examples of rings, properties of rings, integral domains and fields, characteristic of a ring. Ideals, ideal generated by a subset of a ring, operations on ideals, prime and maximal ideals. Ring homomorphisms, properties of ring homomorphisms, polynomial rings over commutative rings, division algorithm, Eisenstein criterion. Vector spaces, subspaces, algebra of subspaces, quotient spaces, etc., Linear transformations, null space, range, rank and nullity of a linear transformation, etc., Dual spaces, dual basis, double dual, transpose of a linear transformation and its matrix in the dual basis, annihilators etc.

The paper is about the concepts of abstract algebra, linear algebra and its applications and geometric approaches.

Mechanics

Moment of a force about a point and an axis, couple and couple moment, Moment of a couple about a line, resultant of a force system etc., Laws of Coulomb friction, application to simple and complex surface contact friction problems, transmission of power through belts, screw jack, wedge, first moment of an area and the centroid, other centers, etc., Conservative force field, conservation for mechanical energy, work energy equation, kinetic energy and work kinetic energy expression based on center of mass, etc.

The course is of engineering mechanics and deals with its statistics and dynamics.

Numerical Methods and Programming

Algorithms, Convergence, Bisection method, False position method, Fixed point iteration method, Newton’s method, Secant method, LU decomposition, Gauss-Jacobi, Gauss-Siedel and SOR iterative methods. Lagrange and Newton interpolation: linear and higher order, finite difference operators. Numerical differentiation: forward difference, backward difference and central difference. Integration: trapezoidal rule, Simpson’s rule, Euler’s method.

The paper is about the numerical analysis and numerical methods for scientific and engineering computation.

Integral Equations and Calculus of Variation

Preliminary Concepts: Definition and classification of linear integral equations. Conversion of initial and boundary value problems into integral equations, Fredholm Integral Equations: Solution of integral equations with separable kernels, Eigen values and Eigen functions, Classical Fredholm Theory: Fredholm method of solution and Fredholm theorems, Volterra Integral Equations: Successive approximations, Neumann series and resolvent kernel. Equations with convolution type kernels. Solution of integral equations by transform methods: Singular integral equations, Hilberttransform, Cauchy type integral equations. Calculus of Variations: Basic concepts of the calculus of variations such as functionals, extremum, variations, function spaces, the brachistochrone problem, Necessary condition for an extremum, Euler`s equation with the cases of one variable and several variables, etc., General Variation: Functionals dependent on one or two functions, Derivation of basic formula, Variational problems with moving boundaries, etc.

The course deals with concepts of integral equations calculus of variations with applications to physics and engineering.

Laplace Transform

Laplace Transform: Laplace of some standard functions, etc,. Finite Laplace Transform: Definition and properties, Shifting and scaling theorem. Z-Transform: Z–transform and inverse Z-transform of elementary functions, etc., Hankel Transform, Hankel Transform, Fourier series, Fourier Transforms.

The topics covered are from advanced engineering mathematics.

Government Degree College

Jammu, Jammu and Kashmir

Guru Ghasidas Vishwavidyalaya

Bilaspur, Chhattisgarh

Mayur College

Kapurthala, Punjab