Advanced Engineering Mathematics

Part 1 Linear algebra: introduction; vector spaces; linear transformations; The solution of simultaneous linear algebraic equations; schems for solution of linear equations; partitioned matrices. Part 2 Eigenvalue problems: algebraic determination of eigenvalues; further results on eigenvalues; quadratic forms and their reduction; boundary-value problems; finding the eigenvalue of largest modulus; determination of other eigenvalues. Part 3 Optimization: linear programming - graphical solution; the simplex algorithm; non-linear optimization; search techniques in one variable; functions of several variables - direct search methods; calculas approach to functions of several variables; methods using the gradient of a function. Part 4 Ordinary differential equations: one-step and multistep methods; predictor-corrector methods; linear difference equations; stability of numerical procedures; case study - surge tank; phase-plane diagrams; boundary-value problems. Part 5 Special functions: a problem in heat transfer; series solution of ordinary differential equations; the gamma function; bessel functions of the first and second kind; modifeied bessel functions; transformations of bessel's equation; an intoduction to legendre polynomials; solution of partial differential equations. Part 6 Fourier series approximations: approximation of a function by a trigonometic series; examples of fourier series; odd and even functions - half-range series; further features of fourier series; trigonometric series approximation of discrete data. Part 7 Partial differential equations: steady state temperature distribution in a plate; some basic ideas; separation of variables method; origin of some partial differential equations; parabolic equations - finite difference methods; elliptic equations. Part 8 Integral transforms: basic results on the laplace transform; finite fourier transforms; infinite fourier transforms; solution of heat conduction equation. Part 9 Integration and vector field theory: scalar and vector fields - differentiation and integration of vectors; the gradient of a scalar field; divergence of a vector field; line integrals; curl of a vector field; vector identities; double integration; further features of double integrals; triple integrals; green's theorem in the plane; surface integral-stokes' theorem; gauss divergence theorem. Part 10 Functions of a complex variable: analytic functions - the cauchy-riemann equations; standard functions of a complex variable; complex potential and conformal mapping; further conformal mappings; complex integrals; taylor and laurent series; the residue theorem; evaluation of real integrals; further applications of contour integration. Part 11 Statistical methods: tests of hypotheses; mean of a small sample - the t test; test of sample variance - the x2 distribution; sample variances - the f test; comparison of sample means; introduction to analaysis of variance; introduction to simple linear regression.