APPLIED MATHEMATICS CSS SYLLABUS (100 MARKS)

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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

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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

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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

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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

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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
  • Heat equation in Cartesian coordinates

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VI. Numerical Methods (30%)

• Solution of nonlinear equations:
  • Bisection method
  • Secant method
  • Newton-Raphson method
  • Fixed-point iterative method
  • Order of convergence of a method
• Solution of a system of linear equations:
  • Diagonally dominant systems
  • 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