CSS APPLIED MATHEMATICS SYLLABUS

Total Marks: 100

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 theorem

• Stokes’ theorem

• Gauss’ theorem

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’s laws

IV. Ordinary Differential Equations — 20%

First-Order Differential Equations

• Equations of first order

• Separable equations

• Exact equations

• First-order linear equations

• Orthogonal trajectories

• Nonlinear equations reducible to linear equations

• Bernoulli equations

• Riccati equations

Higher-Order Differential Equations

• Equations with constant coefficients

• Homogeneous and inhomogeneous equations

• Cauchy–Euler equations

• Variation of parameters

Special Differential Equations

• Ordinary and singular points of a differential equation

• Solution in series

• Bessel equations

• Legendre equations

• Properties of Bessel functions

• Legendre polynomials

V. Fourier Series and Partial Differential Equations — 20%

Fourier Series

• Trigonometric Fourier series

• Sine and cosine series

• Bessel inequality

• Summation of infinite series

• Convergence of Fourier series

Partial Differential Equations

• 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

  • Heat equation in Cartesian coordinates

VI. Numerical Methods — 30%

Numerical Solution of Nonlinear Equations

• Bisection method

• Secant method

• Newton–Raphson method

• Fixed-point iterative method

• Order of convergence of a method

Numerical Solution of Linear Systems

• Solution of a system of linear equations

• Diagonally dominant systems

• Jacobi method

• Gauss–Seidel method

Numerical Differentiation and Integration

• Numerical differentiation

• Numerical integration

• Trapezoidal rule

• Simpson’s rules

• Gaussian integration formulas

Numerical Solution of Ordinary Differential Equations

• Euler method

• Modified Euler method

• Runge–Kutta methods

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