I. Vector Calculus (10%)
Vector algebra; scalar and vector products of vectors; gradient divergence and curl of
a vector; line, surface and volume integrals; Green’s, Stokes’ and Gauss theorems.
II. Statics (10%)
Composition and resolution of forces; parallel forces and couples; equilibrium of a
system of coplanar forces; centre of mass of a system of particles and rigid bodies;
equilibrium of forces in three dimensions.
III. Dynamics (10%)
Motion in a straight line with constant and variable acceleration; simple harmonic
motion; conservative forces and principles of energy.
Tangential, normal, radial and transverse components of velocity and
acceleration; motion under central forces; planetary orbits; Kepler laws;
IV. Ordinary differential equations (20%)
Equations of first order; separable equations, exact equations; first order linear
equations; orthogonal trajectories; nonlinear equations reducible to linear
equations, Bernoulli and Riccati equations.
Equations with constant coefficients; homogeneous and inhomogeneous
equations; Cauchy-Euler equations; variation of parameters.
Ordinary and singular points of a differential equation; solution in series; Bessel
and Legendre equations; properties of the Bessel functions and Legendre
V. Fourier series and partial differential equations (20%)
Trigonometric Fourier series; sine and cosine series; Bessel inequality;
summation of infinite series; convergence of the Fourier series.
Partial differential equations of first order; classification of partial differential
equations of second order; boundary value problems; solution by the method of
separation of variables; problems associated with Laplace equation, wave
equation and the heat equation in Cartesian coordinates.
VI. Numerical Methods (30%)
Solution of nonlinear equations by bisection, secant and Newton-Raphson
methods; the fixed- point iterative method; order of convergence of a method.
Solution of a system of linear equations; diagonally dominant systems; the Jacobi
and Gauss-Seidel methods.
Numerical differentiation and integration; trapezoidal rule, Simpson’s rules,
Gaussian integration formulas.
Numerical solution of an ordinary differential equation; Euler and modified Euler
methods; Runge- Kutta methods.