Applied Mathematics

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.

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.

• 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's laws.

• 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 polynomials.

• 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.

• 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.