# Applied Mathematics

### CSS Syllabus Applied Mathematics – 100 Marks

###### 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 polynomials.

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.