Syllabus (Session 2016-17) Department of Mathematics nstitute of Applied Sciences & Humanities
AHM-1101: ENGNEERNG MATHEMATCS Course Objective: To make the students understand the concepts of Calculus, Differential Equations and Matrices by giving more emphasis to their applications in engineering. Credits: 04 Semester L T P: 3 1 0 Prerequisites: Differential Calculus: Partial differentiation, Euler s theorem for homogeneous functions, Composite Functions, Total derivatives, Expansion of functions of several variabl es, Asymptotes and Curve Tracing (in Cartesian coordinates), Jacobian and its properties, Extrema of functions of several variables using Lagrange s multipliers. Matrices: nverse by elementary transformations, rank of a matrix, solution of system of linear equations, linear dependence and independence of vectors, complex matrices, eigen values, eigen vectors. Cayley Hamilton theorem, Diagonalization, Reduction to Quadratic form, rank, ndex, signature; definite and semi - definite matrices. Ordinary Differential Equations (ODEs): ntroduction, exact and reducible to exact differential equations, n th order linear differential equations with constant coefficients, Euler-Cauchy Eqns., Simultaneous differential equations, method of variation of parameters, Applications of order ODEs in engineering probl ems involving SHM, Electrical circuits and Mechanical systems. Jain, yengar and Jain: Advanced Engg. Mathematics, Narosa Publishing House, Delhi N. P. Bali & M. Goyal: A Text Book of Engg. Mathematics, Laxmi Publication, Delhi Hari Kishan: A Text Book of Matrices, Atlantic Publishers and Dist., Delhi G. B. Thomas & R. Finney: Calculus & Analytic Geometry (9 th Ed. ) Addison Wesley W. E. Boyce and R. Di Prima, Elementary Diff. Equations (8 th Ed.), John Wiley T. M. Apostol, Calculus, Volumes 1 and 2, Wiley Eastern, 1980 Understand Partial differentiation and its applications Trace the curves given in cartesian coordinates Determine the linear dependence of functions Find the inverse of a square non singular matrix by various methods Solve the ordinary differential eqns. of higher order and grasp their applications
AHM-2101: ENGNEERNG MATHEMATCS Course Objective: To make the students understand the concepts of Calculus, convergence, vectors and Fourier series by giving more emphasis to their applications in engineering. Credits: 04 L T P: 3 1 0 Semester Convergence of nfinite Series: ntroduction, Geometric series test, n th term test, Leibnitz test, Comparison test, p-test, Cauchy s root test, Ratio test, Raabe s test, Logarithmic test, Cauchy condensation test, Special p - test, De Morgan and Bertrand s test, Special Logarithmic test. Multiple ntegrals: Beta and Gamma functions, Double and triple integrals, change of order of integration, Applications to area and volume, change of variabl es, Dirichlet integral and its Liouville extension. Vector Calculus: Gradient, Divergence and curl, Vector dentities, Line, surface and volume integrals, Work done by a force, Green, Gauss' divergence and Stoke's theorem (without proof). Fourier series: Fourier series of period 2, Even and Odd functions, Half range series, Change of interval. Jain, yengar & Jain: Advanced Engg. Mathematics, Narosa Publishing House, Delhi M. Goyal & N. P. Bali: A Text Book of Engg. Maths, Laxmi Publications, Delhi Hari Kishan: Vector Algebra and Calculus, Atlantic Publishers & Dist., Delhi Hari Kishan: Sure Success in Convergence, Atlantic Publishers & Dist., Delhi G. B. Thomas & R. Finney: Calculus & Analytic Geometry (9 th Ed. ) Addison Wesley T. M. Apostol, Calculus, Volumes 1 and 2, Wiley Eastern, 1980 R. V. Churchill & J. W. Brown : Fourier series & boundary value problems (7 th ed.), McGraw Hill (2006). Understand the concept of convergence and divergence. Apply different tests for determining convergence of an infinite series. Evaluate double, triple integrations and study their applications. Analyze the Fourier series expansion of a discontinuous function. Find integration and differentiation of vectors.
AHM-3101 : ENGNEERNG MATHEMATCS Course Objective: To make the students understand the concepts of Partial Differential equations, Laplace Transforms, and Complex analysis by giving more emphasis to their applications in engineering. Credits: 04 L T P : 3 1 0 Semester Partial Differential Equations (PDEs): ntroduction, order Lagrange's linear PDEs, n th order linear PDEs, Classification of order PDEs, Method of separation of variables, One dimensional wave equation, D Alembert solution, One dimensional heat flow equation. Laplace Transforms: Properties of Laplace transform, Laplace transform of derivatives and integrals, Unit step, Dirac - delta and periodic function, Properties of inverse Laplace transform, convolution theorem, Application to ordinary differential equations. Complex Analysis: Analytic functions, C R equations, Harmonic Functions, Line integral in a complex plane, Cauchy s integral theorem and formula, Cauchy integral formula for derivatives, Taylor and Laurent series (without proof), Singularities, Residue at a pole, Residue theorem and its application in evaluation of real integrals (excluding pol es on the real axis). Jain, yengar & Jain: Advanced Engg. Mathematics, Narosa Publishing House, Delhi Manish Goyal and N. P. Bali: A Text Book of Engg. Maths, Laxmi Publications, Delhi R. V. Churchill and J. W. Brown, Complex variables and applications (7 th Ed.), McGraw Hill (2003). J. M. Howie, Complex analysis, Springer Verlag (2004) Solve Partial Differential Equations of and higher orders. Apply Fourier series in applications of PDEs to wave and heat flow equations. Know about the use of transforms in solving differential equations. Understand the use of special function like unit step and dirac delta. Grasp the concept of Analytic function and its applications in engineering.
AHM-4001: COMPUTER BASED NUMERCAL & STATSTCAL TECHNQUES (For CE Branch) Course Objective: To make the students understand the concepts of Numerical Methods by giving more emphasis to their applications in engineering. Credits: 04 L T P : 3 1 0 Semester V NUMERCAL TECHNQUES : Solution of algebraic and transcendental equations: ntroduction to errors, Convergence of iterative methods, Newton Raphson and Bairstow s methods. Solution of a system of simultaneous linear equations: Cholesky and Gauss Seidel iterative method. Eigen value Problem : Rayleigh s power Method. nterpolation: Finite differences, Newton s forward, backward & divided difference formulae, Gauss central difference methods and Lagrange s method (without proof). NUMERCAL TECHNQUES : Numerical ntegration : Trapezoidal and Simpson s rules. Numerical solution of ordinary differential equations : Runge Kutta V order method. Numerical solution of partial differential eqns. (PDE) : Finite difference approximations to derivatives. Elliptic PDE: Jacobi method & Leibmann s iteration process. Parabolic PDE: Bender Schmidt & Crank Nicolson methods. Hyperbolic PDE: Explicit finite difference method. Teaching Hours STATSTCAL TECHNQUES: Linear and non linear least square curve fitting,testing of statistical hypothesis, Level of significance, Test of significance of small samples, Student s t test, Snedecor s variance ratio F - test, Degree of freedom, Chi square test as a test of goodness of fit and as a test of independence, ANOVA (one way classification). Manish Goyal: Comp. Based Numerical & Statistical Tech., Laxmi Publications, DL S. S. Sastry: ntroductory methods of Numerical Analysis, PH S. P. Gupta & V. K. Kapoor: Fundamentals of Mathematical Statistics, Sultan Ch. Sons Jain, yenger, Jain: Numerical Methods for Sci. & Engg. Comp.., New Age nt., Delhi Conte & D. Boor: Elementary Numerical Analysis, An Algorithmic Approach, TMH Solve algebraic and transcendental equation numerically Solve system of linear equation, Eigen value problem Apply numerical technique to solve integration, differentiation and PDE`s Application of testing of hypothesis, curve fitting based on data for research purposes