COMMON FOR ALL BRANCHES This is a crash course on Engineering Mathematics designed with crisp informative concepts which are common for all branches. This course is designed to crack critical problems both in the GATE entrance as well as in the I and II semister course in engineering. Course duration - 3 months

Topics Covered

1) Matrix Theory - Elementary row and column operations on a matrix, Rank of matrix – Normal form – Inverse of a matrix using elementary operations –Consistency and solutions of systems of linear equations using elementary operations, Gauss Seidal iteration method - linear dependence and independence of vectors - Characteristic roots and vectors of a matrix - Caley-Hamillton theorem and its applications, Calculation of dominant eigen value by iteration - Reduction to diagonal form - Reduction of a quadratic form to canonical form – orthogonal transformation and congruent transformation. 2) Differential Calculus: Rolle’s theorem; Mean value theorem; Taylor’s and Maclaurin’s theorems with remainders, Expansions; Indeterminate forms; Asymptotes and curvature; Curve tracing; Functions of several variables, Partial Differentiation, Total Differentiation, Euler’s theorem and generalization, maxima and minima of functions of several variables (two and three variables) – Lagrange’s method of Multipliers; Change of variables – Jacobians. 3) Ordinary differential equations of first order: Formation of differential equations; Separable equations; equations reducible to separable form; exact equations; integrating factors; linear first order equations; Bernoulli’s equation; Orthogonal trajectories. 4) Ordinary linear differential equations of higher order : Homogeneous linear equations of arbitrary order with constant coefficients - Non-homogeneous linear equations with constant coefficients; Euler and Cauchy’s equations; Method of variation of parameters; System of linear differential equations. 5) Laplace Transformation: Laplace transform - Inverse Laplace transform - properties of Laplace transforms - Laplace transforms of unit step function, impulse function and periodic function - convolution theorem - Solution of ordinary differential equations with constant coefficients and system of linear differential equations with constant coefficients using Laplace transform. 6) Integral Calculus: Fundamental theorem of integral calculus and mean value theorems; Evaluation of plane areas, volume and surface area of a solid of revolution and lengths. Convergence of Improper integrals – Beta and Gamma integrals – Elementary properties – Differentiation under integral sign. Double and triple integrals – computation of surface areas and volumes – change of variables in double and triple integrals. 7) Vector Calculus : Scalar and Vector fields; Vector Differentiation; Level surfaces - directional derivative - Gradient of scalar field; Divergence and Curl of a vector field - Laplacian - Line and surface integrals; Green’s theorem in plane; Gauss Divergence theorem; Stokes’ theorem. 8) Fourier Series: Expansion of a function in Fourier series for a given range - Half range sine and cosine expansions 9) Fourier Transforms : Complex form of Fourier series - Fourier transformation - sine and cosine transformations - simple illustrations. 10) Z-transforms : Inverse Z-transfroms – Properties – Initial and final value theorems – convolution theorem - Difference equations – solution of difference equations using z-transforms 11) Partial Differential Equations: Solutions of Wave equation, Heat equation and Laplace’s equation by the method of separation of variables and their use in problems of vibrating string, one dimensional unsteady heat flow and two dimensional steady state heat flow including polar form. 12) Complex Variables: Analytic function - Cauchy Riemann equations - Harmonic functions - Conjugate functions - complex integration - line integrals in complex plane - Cauchy’s theorem (simple proof only), Cauchy’s integral formula - Taylor’s and Laurent’s series expansions - zeros and singularities - Residues - residue theorem, evaluation of real integrals using residue theorem, Bilinear transformations, conformal mapping. 13) Fourier Series: Expansion of a function in Fourier series for a given range - Half range sine and cosine expansions 14) Fourier Transforms : Complex form of Fourier series - Fourier transformation - sine and cosine transformations - simple illustrations. 15) Z-transforms : Inverse Z-transfroms – Properties – Initial and final value theorems – convolution theorem - Difference equations – solution of difference equations using z-transforms 16) Partial Differential Equations: Solutions of Wave equation, Heat equation and Laplace’s equation by the method of separation of variables and their use in problems of vibrating string, one dimensional unsteady heat flow and two dimensional steady state heat flow including polar form. 17) Complex Variables: Analytic function - Cauchy Riemann equations - Harmonic functions - Conjugate functions - complex integration - line integrals in complex plane - Cauchy’s theorem (simple proof only), Cauchy’s integral formula - Taylor’s and Laurent’s series expansions - zeros and singularities - Residues - residue theorem, evaluation of real integrals using residue theorem, Bilinear transformations, conformal mapping.

Who should attend

1) Engineering Students (1st and 2nd year students) 2) GATE - 2014 Aspirants 3) Students pursuing masters (Mtech, ME)

Pre-requisites

1) Engineering Students (1,2,3,4 year) 2) Students appearing for GATE 2014

What you need to bring

Notebook for assignments

Key Takeaways

1) Understanding the mathematics as applied in engineering 2) Stress free solving mathematical problems 3) Develop a mathematical aptitude to solve engineering problems for placements and GATE entrance 4) Key notes especialy designed for easy reference and strong rememberance.