L 3 C 3
Teachers Continuous Evaluation: 40 marks. Term-End Semester Examination: 60 marks.
Course outcomes
- Apply multivariable differential calculus to analyze optimization problems and engineering models involving functions of several variables.
- Solve ordinary differential equations using analytical and power series methods, and interpret their applications in engineering systems.
- Analyze systems of linear equations and apply matrix methods, eigenvalue techniques, and matrix factorizations in engineering computations.
- Apply vector differential and integral calculus, including Green's, Gauss', and Stokes' theorems, to model and analyze engineering and physical phenomena.
Unit I
Functions of several variables, Partial derivatives, Chain rule, Differentiation of implicit functions, Exact differentials, Jacobians and transformation of coordinates, Taylor's and Maclaurin's series, Maxima, minima and saddle points, Method of Lagrange multipliers, Differentiation under the integral sign. Ordinary differential equations: Basic concepts, Geometrical interpretation of first-order differential equations, Direction fields, Euler's method, Separable differential equations, Exact differential equations, Integrating factors, Linear differential equations, Bernoulli equation, Orthogonal trajectories, Applications.
Unit II
Higher-order linear differential equations with constant coefficients, Differential operator method, Euler–Cauchy equations, Wronskian, Method of variation of parameters, Free oscillations of mass–spring systems. Power series solution of ordinary differential equations, Bessel's equation and Bessel functions of the first kind, Legendre's equation and Legendre polynomials, Recurrence relations and important properties, Gamma and Beta functions and their properties.
Unit III
Matrices and determinants, Gauss elimination, Gauss–Jordan elimination, Rank of a matrix, Linear independence, Vector spaces, Systems of linear equations, Existence and uniqueness of solutions, Cramer's rule, Eigenvalues and eigenvectors, Symmetric, skew-symmetric and orthogonal matrices, Eigenbases, Matrix diagonalization, Quadratic forms, Gram–Schmidt orthogonalization, Cayley–Hamilton theorem (without proof), LU decomposition and Cholesky decomposition with engineering applications.
Unit IV
Vector and scalar point functions, Space curves, Arc length, Curvature, Gradient, Directional derivative, Divergence, Curl, Line integrals, Path independence, Double and triple integrals, Surface integrals, Green's theorem, Gauss divergence theorem, Stokes' theorem, Engineering applications.
Textbooks
- E. Kreyszig, Advanced Engineering Mathematics, 10th ed. Hoboken, NJ, USA: John Wiley & Sons, 2011.
References
- K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. Cambridge, U.K.: Cambridge University Press, 2013.
- K. A. Stroud and D. J. Booth, Engineering Mathematics, 8th ed. London, U.K.: Macmillan International Higher Education, 2020.
- L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. New York, NY, USA: Oxford University Press, 1998.
- L. Turyn, Advanced Engineering Mathematics. Boca Raton, FL, USA: CRC Press (Taylor & Francis Group), 2014.
- D. G. Zill, Advanced Engineering Mathematics, 7th ed. Burlington, MA, USA: Jones & Bartlett Learning, 2020.
- D. G. Duffy, Advanced Engineering Mathematics with MATLAB, 4th ed. Boca Raton, FL, USA: CRC Press (Taylor & Francis Group), 2017.
- M. C. Potter, J. L. Lessing, and E. F. Aboufadel, Advanced Engineering Mathematics, 4th ed. Cham, Switzerland: Springer, 2019