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.