Determining Eigen values and Eigenvectors – Properties of Eigen values and Eigenvectors - Some applications of Eigen value problems – Eigen value problems arising from population models (Leslie model) – Elastic deformations – Cayley Hamilton Theorem (statement only) and its applications.

Diagonalization of a real symmetric matrix–Quadratic form– Canonical form – Nature of the quadratic form – Reduction of quadratic form to canonical form by orthogonal transformation – Some Applications: Transformation to Principal axes- Conic sections – Solving first order linear system using diagonalization.

Radius of Curvature in Cartesian co-ordinates – Centre and circle of curvature in Cartesian co-ordinates – Evolutes – Envelopes.

Partial derivatives –Total derivatives – Jacobians – Taylor’s expansion of functions of two variables – Errors and Approximations – Maxima and Minima of functions of two variables – Lagrange’s method of undetermined multipliers.

Homogeneous Linear ODEs with constant coefficients – Linear ODE with variable coefficients - Cauchy’s and Legendre’s Equations – Method of variation of parameters – Methods of undetermined coefficients - Applications: Modelling of Free Oscillations of a Mass-Spring system.