Matrices

Rank of a matrix by Echelon form and Normal form

Inverse of Non-singular matrices by Gauss-Jordan method

System of linear equations

• Solving system of Homogeneous and Non-Homogeneous equations by Gauss elimination method

Gauss Seidel Iteration Method

Linear Transformation and Orthogonal Transformation

• Eigenvalues, Eigenvectors and their properties

Diagonalization of a matrix

Cayley-Hamilton Theorem

• Finding inverse and power of a matrix by Cayley-Hamilton Theorem

Quadratic forms and Nature of the Quadratic Forms

• Reduction of Quadratic form to canonical forms by Orthogonal Transformation

Mean value theorems

• Rolle’s theorem
• Lagrange’s Mean value theorem with their Geometrical Interpretation and applications
• Cauchy’s Mean value Theorem
• Taylor’s Series

Applications of definite integrals to evaluate surface areas and volumes of revolutions of curves (Only in Cartesian coordinates)

• Definition of Improper Integral: Beta and Gamma functions and their applications

Definitions of Limit and continuity

Partial Differentiation

• Euler’s Theorem
• Total derivative
• Jacobian
• Functional dependence & independence

Applications

• Maxima and minima of functions of two variables and three variables using method of Lagrange multipliers

Evaluation of Double Integrals

• Cartesian and polar coordinates
• change of order of integration (only Cartesian form)

Evaluation of Triple Integrals

• Change of variables (Cartesian to polar) for double and (Cartesian to Spherical and Cylindrical polar coordinates) for triple integrals

Applications

• Areas (by double integrals) and volumes (by double integrals and triple integrals)