MATH1 (B-TECH Mechanical Engineering 1st-2nd)

Section A

Infinite series: Convergence and divergence, comparison tests, D' Alembert's ratio test, integral test, Raabe’s test,

logarithmic and Cauchy root tests, Gauss’s Test, alternating series, absolute and conditional convergence.

Section B

Matrices & its Applications: Rank of a matrix, elementary transformations, elementary matrices, inverse

using elementary transformations, normal form of a matrix, linear dependence and independence of vectors,

consistency of linear system of equations, linear and orthogonal transformations, eigenvalues and eigenvectors,

properties of eigenvalues, Cayley - Hamilton theorem and its applications, diagonalization of matrices, similar

matrices, quadratic forms.

Section C

Differential Calculus: Successive differentiation, Leibnitz Theorem and applications, Taylor's and Maclaurin's

series, curvature, asymptotes, curve tracing. Functions of two or more variables, limit and continuity, partial

derivatives, total differential and differentiability, derivatives of composite and implicit functions, Jacobians, higher

order partial derivatives, homogeneous functions, Euler’s Theorem and applications. Taylor's series for functions of

two variables (without proof), maxima-minima of function of two variables, Lagrange's method of undetermined

multipliers, differentiation under integral sign (Leibnitz rule).

Section D

Integral Calculus: Beta and gamma functions and relationship between them.

Applications of single integration to find volume of solids and surface area of solids of revolution. Double integral,

change of order of integration, double integral in polar coordinates, applications of double integral to find area

enclosed by plane curves, triple integral, change of variables, volume of solids, Dirichlet’s integral.