(An Autonomous Institute Affiliated to Savitribai Phule Pune University)
This is compulsory subject offered in first semester of first year engineering students particularly with which I am mostly associated. The maximum part of the course focuses on differential calculus and linear algebra.
Unit I:- Linear Algebra-Matrices, System of Linear Equations : Rank of a Matrix, System of Linear Equations, Linear Dependence and Independence, Linear and Orthogonal Transformations, Application to problems in Engineering.
Unit II:- Linear Algebra : Eigen Values and Eigen Vectors, Diagonalization Eigen Values and Eigen Vectors, Cayley Hamilton theorem, Diagonalization of a matrix, Reduction of Quadratic forms to Canonical form by Linear and Orthogonal transformations.
Unit III:- Differential Calculus: Rolle’s Theorem, Mean Value Theorems, Taylor's Series and Maclaurin's Series, Expansion of functions using standard expansions, Indeterminate Forms, L' Hospital's Rule, Evaluation of Limits and Applications.
Unit VI :- Differential Equations And It's Application : Exact differential equations, Equations reducible to exact form. Linear differential equations, Equations reducible to linear form, Bernoulli’s equation. Applications of Differential Equations to Orthogonal Trajectories, Kirchhoff’s Law of Electrical Circuits, Rectilinear Motion.
Unit V:- Fourier Series : Definition, Dirichlet’s conditions, Full range Fourier series, Half range Fourier series, Harmonic analysis, Parseval’s identity and Applications to problems in Engineering.
UNIT VI:- Complex Number And it's Application :Exponential and circular functions, De’Moivre’s theorem and its application to find roots of algebraic equations.
Definition of hyperbolic functions, inverse hyperbolic functions, real and imaginary parts of circular and hyperbolic functions, logarithmic function of complex variables.
This is compulsory subject offered in second semester of first year engineering students particularly with which I am mostly associated. The course focuses on Partial differential equations and it's applications, integral calculus, curve tracing .
Unit I:- Partial Differentiation : Introduction to functions of several variables, Partial Derivatives, Euler's Theorem on Homogeneous functions, Partial derivative of Composite Function, Total Derivative, Change of Independent variables
Unit II:- Applications of Partial Differentiation : Jacobian: Jacobian for explicit and implicit functions, properties, partial differentiation of implicit functions, functional dependence.Errors and approximation, maxima and minima of functions with two variables, Lagrange’s method of undetermined multipliers.
Unit III:- Integral Calculus : Reduction formulae: Reduction formulae for trigonometric, algebraic and exponential functions.
Special functions: Gamma function and its properties, beta function and its properties, relation between gamma and beta functions.
Leibnitz’s rule: Leibnitz’s rule of differentiation under integral sign, evaluation of integrals.
Unit IV:- Curve Tracing : Tracing of curves: Tracing of cartesian curves, polar curves, and parametric curves, rules/steps in curve tracing (viz. symmetry, boundedness, point of intersection, tangents, existence of special points, asymptotes-vertical, horizontal, oblique, convexity, concavity, region of absence).Rectification of curves: Cartesian curves, polar curves, parametric curves.
Unit V:- Multiple Integrals : Double integration in cartesian and polar co-ordinates, change of order of integration,Triple integration in cartesian, cylindrical, spherical coordinates.
Unit VI:- Application Of Multiple Integrals:-Area bounded by curve, mass of lamina, volume of solids, moment of inertia, centre of gravity.
This course is compulsory in third or fourth semester depending on what branch of engineering student have selected. It mainly focuses on higher order differential equations, Transforms(Fourier transforms, Laplace Transform, Z-transform), Vector Calculus(branch specific), Numerical methods, Complex Variables(branch specific), Statistics and probability, applications of partial differential equations, etc.