Polar Curves and Curvature: Polar coordinates, Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal equations. Curvature and radius of curvature – Cartesian, parametric, polar and pedal forms.

Series Expansion, 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, total derivative – differentiation of composite functions. Jacobian. Maxima and minima for the function of two variables.

Ordinary Differential Equations of First Order: Linear and Bernoulli’s differential equation. Exact and reducible to exact differential equations with integrating factor: 1 / (∂M/∂y − ∂N/∂x) and 1 / (∂N/∂x − ∂M/∂y). Orthogonal trajectories, Law of natural growth and decay.

Linear Algebra-1: Elementary row transformation of a matrix, Row echelon form and Rank of a matrix. Inverse of matrix by Jordan method. Consistency and Solution of system of linear equations – Gauss- elimination method, LU decomposition method and approximate solution by Gauss-Seidel method. Application to traffic flow.

Linear Algebra -2: Eigenvalues and Eigenvectors, Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector. Model matrix, Diagonalization of the matrix, inverse of a matrix by Cayley-Hamilton theorem, Characteristic and minimal polynomials of block matrices, Moore-Penrose pseudoinverse.