MATHEMATICS-II SUB.CODE: MA1151

UNIT-I LAPLACE TRANSFORMS

Laplace Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer He formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in applied mathematics, is also named after him.

Transforms of elementary functions Basic properties Transforms of derivatives and integrals Initial and final value theorems Inverse Laplace transforms Convolution theorem Solution of ordinary Differential equations using Laplace transforms Transform of periodic function Solution of integral equations

PERIODIC FUNCTION. Periodic function is a function that repeats its values in regular intervals or periods. For example, the sine function is periodic with period 2π, since for all values of x. This function repeats on intervals of length 2π

APPLICATIONS. Circuit theory Solving n-th order differential equations. Signal transformations. Wave transformations.

UNIT -II VECTOR CALCULUS

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration

Gradient, Divergence and Curl Directional derivative Irrotational and solenoidal vector fields Vector integration Problem solving using Green’s theorem, Gauss divergence theorem,Stoke’s theorem

LINE INTEGRAL

GAUSS DIVERGENCE THEOREM Consider the following volume enclosed by a surface we will call S.

Now we will embed S in a vector field:

We will cut the the object into two volumes that are enclosed by surfaces we will call S1 and S2.

Again we embed it in the same vector field.

It is clear that flux through S1 + S2 is equal to flux through S It is clear that flux through S1 + S2 is equal to flux through S.This is because the flux through one side of the plane is exactly opposite to the flux through the other side of the plane:

We could subdivide the surface as much as we want and so for n subdivisions Therefore We can subdivide the volume into a bunch of little cubes

APPLICATIONS Differential geometry Partial differential equations Electromagnetic field Gravitational field Fluid dynamics

UNIT -III ANALYTIC FUNCTIONS

Necessary and sufficient conditions Cauchy-Riemann equations Properties of analytic functions Harmonic conjugate Construction of Analytic function Conformal mapping and Bilinear transformation.

CONFORMAL MAPING

Bilinear transformation

APPLICATIONS Non-linear dynamic system Special functions Number theory Digital signal processing Discrete time Control theory Image Processing

UNIT -IV MULTIPLE INTEGRALS

Double integration Cartesian and polar co-ordinates Change of order of integration Area as a double integral Change of variables between Cartesian and polar co-ordinates Triple integration Volume as a triple integral.

APPLICATIONS Probability theory Mathematical Physics(moment of inertia) To find the area of the surface To find the volume of the surface

UNIT -V COMPLEX INTEGRATON

Problems solving using Cauchy’s integral theorem and integral formula Taylor’s and Laurent’s expansions Residues Cauchy’s residue theorem Contour integration over unit circle Semicircular contours with no pole on real axis

APPLICATIONS Aero dynamics Elasticity Two dimensional fluid flow

TEXT BOOKS

Text Books 1.Grewal B. S “Higher engineering mathematics”,38th edition Khanna Publishers New Delhi, 2005. 2.Venkatraman M.K.,”Engineering Mathematics”,volume-I &II 4th edition .The National Publishing company ,chennai,2004. 3.Veerarajan.T ,”Engineering Mathematics”, 4th edition Tata Mcgraw Hill publishing company limited,New Delhi,2005. 4.Bali N.P & Manish Goyal, “Text book of Engineering Mathematics” 3rd edition,Laxmi publications (p)ltd 2008

to be continued…