SUB.CODE: MA1151. UNIT-I LAPLACE TRANSFORMS Laplace Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomerFrench.

MATHEMATICS-II

SUB.CODE: MA1151

UNIT-I

LAPLACE TRANSFORMS

LaplacePierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomerHe 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 functionsBasic propertiesTransforms of derivatives and integralsInitial and final value theoremsInverse Laplace transformsConvolution theoremSolution of ordinary Differential equations

using Laplace transformsTransform of periodic functionSolution 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 theorySolving 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 CurlDirectional derivativeIrrotational and solenoidal vector fieldsVector integrationProblem solving using Green’s theorem, Gauss divergence theorem,Stoke’s theorem

LINE INTEGRAL

GAUSS DIVERGENCE THEOREMConsider 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.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 geometryPartial differential equationsElectromagnetic fieldGravitational fieldFluid dynamics

UNIT -III

ANALYTIC FUNCTIONS

Necessary and sufficient conditions Cauchy-Riemann equationsProperties of analytic functionsHarmonic conjugateConstruction of Analytic functionConformal mapping and Bilinear

transformation.

CONFORMAL MAPING

Bilinear transformation

APPLICATIONSNon-linear dynamic systemSpecial functions Number theoryDigital signal processingDiscrete time Control theoryImage Processing

UNIT -IV

MULTIPLE INTEGRALS

Double integrationCartesian and polar co-ordinatesChange of order of integrationArea as a double integralChange of variables between Cartesian and

polar co-ordinatesTriple integrationVolume as a triple integral.

APPLICATIONSProbability theoryMathematical Physics(moment of inertia)To find the area of the surfaceTo 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 expansionsResiduesCauchy’s residue theoremContour integration over unit circleSemicircular contours with no pole on real

axis

APPLICATIONS

Aero dynamicsElasticityTwo dimensional fluid flow

TEXT BOOKS

Text Books1.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…