KAS 103T ENGINEERING MATHMATICS I
3L:1T:0P 4 Credits
COURSE OBJECTIVE: The objective of this course is to familiarize the graduate engineers with techniques in calculus, multivariate analysis, vector calculus and linear algebra. It aims to equip the students with standard concepts and tools from intermediate to advanced level that will enable them to tackle more advanced level of mathematics and applications that they would find useful in their disciplines. The students will learn: To apply the knowledge of differential calculus in the field of engineering. To deal with functions of several variables that is essential in optimizing the results of real life problems. Multiple integral tools to deal with engineering problems involving centre of gravity, volume etc. To deal with vector calculus that is required in different branches of Engineering to graduate engineers. The essential tools of matrices and linear algebra, Eigen values and diagonalization in a Comprehensive manner are required.
Unit I Matrices: Types of Matrices: Symmetric, Skew-symmetric and Orthogonal Matrices; Complex Matrices, Inverse and Rank of matrix using elementary transformations, Rank-Nullity theorem; System of linear equations, Characteristic equation, Cayley-Hamilton Theorem and its application, Eigen values and eigenvectors; Diagonalisation of a Matrix 8
Unit II Differential Calculus- I: Introduction to limits, continuity and differentiability, Rolle’s Theorem, Lagrange’s Mean value theorem and Cauchy mean value theorem, Successive Differentiation (nth order derivatives), Leibnitz theorem and its application, Envelope of family of one and two parameter, Curve tracing: Cartesian and Polar co-ordinates
Unit III Differential Calculus-II: Partial derivatives, Total derivative, Euler’s Theorem for homogeneous functions, Taylor and Maclaurin’s theorems for a function of two variables, Maxima and Minima of functions of several variables, Lagrange Method of Multipliers, Jacobians, Approximation of errors
Unit IV Multivariable Calculus-I: Multiple integration: Double integral, Triple integral, Change of order of integration, Change of variables, Application: Areas and volumes, Center of mass and center of gravity (Constant and variable densities)
Unit V Vector Calculus: Vector identities (without proof), Vector differentiation:
Gradient, Curl and Divergence and their Physical interpretation, Directional
derivatives.
Vector Integration: Line integral, Surface integral, Volume integral, Gauss’s
Divergence theorem, Green’s theorem and Stoke’s theorem (without proof)
and their applications