Differential Calculus: Polar curves, angle between the radius vector and the tangent, angle between the polar curves, Pedal equations. Curvature and radius of curvature in cartesian, polar, parametric and Pedal forms.

Power series Expansions, indeterminate forms and multivariable calculus: Statement and problems on Taylor’s and Maclaurin’s series expansion for one variable. Indeterminate forms – L’Hospital’s rule. Partial Differentiation: Partial derivatives, total derivative, differentiation of composite functions, Jacobians. Maxima and minima for functions of two variables.

Ordinary Differential Equations (ODE) of first order and first degree and nonlinear ODE: Exact and reducible to exact differential equations — integrating factors of 1/N (∂M/∂y − ∂N/∂x) and 1/M (∂M/∂y − ∂N/∂x) only. Linear and Bernoulli’s differential equations. Orthogonal trajectories, L–R and C–R circuits.

Non-linear differential equations: Introduction to general and singular solutions, solvable for p only, Clairaut’s equations, and equations reducible to Clairaut’s form.

Ordinary differential equations of higher Order: Higher-order linear ordinary differential equations with constant coefficients, homogeneous and non-homogeneous equations involving e^ax, sin(a x + b), cos(a x + b), and xⁿ only. Method of variation of parameters, Cauchy’s and Legendre’s homogeneous differential equations, and L–C–R circuits.

Linear Algebra: Elementary transformations of a matrix, Echelon form, rank of a matrix, consistency of system of linear equations. Gauss elimination and Gauss –Seidel method to solve system of linear equations. Eigen values and eigen vectors of a matrix, Rayleigh’s power method to determine the dominant eigen value and corresponding eigen vector of a matrix. Applications: Traffic flow.