CSS Applied Mathematics Syllabus PAPER: 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.

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.