sub.code: ma1151. unit-i laplace transforms laplace pierre-simon, marquis de laplace (23 march 1749...
TRANSCRIPT
![Page 1: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/1.jpg)
MATHEMATICS-II
SUB.CODE: MA1151
![Page 2: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/2.jpg)
UNIT-I
LAPLACE TRANSFORMS
![Page 3: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/3.jpg)
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.
![Page 4: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/6.jpg)
APPLICATIONS.Circuit theorySolving n-th order differential equations.Signal transformations.Wave transformations.
![Page 7: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/7.jpg)
UNIT -II
VECTOR CALCULUS
![Page 8: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/9.jpg)
Gradient, Divergence and CurlDirectional derivativeIrrotational and solenoidal vector fieldsVector integrationProblem solving using Green’s theorem, Gauss divergence theorem,Stoke’s theorem
![Page 10: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/10.jpg)
LINE INTEGRAL
![Page 11: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/11.jpg)
GAUSS DIVERGENCE THEOREMConsider the following volume enclosed by a surface we will call S.
![Page 12: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/12.jpg)
Now we will embed S in a vector field:
![Page 13: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/13.jpg)
We will cut the the object into two volumes that are enclosed by surfaces we will call S1 and S2.
![Page 14: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/14.jpg)
Again we embed it in the same vector field.
![Page 15: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/15.jpg)
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:
![Page 16: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/17.jpg)
APPLICATIONS
Differential geometryPartial differential equationsElectromagnetic fieldGravitational fieldFluid dynamics
![Page 18: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/18.jpg)
UNIT -III
ANALYTIC FUNCTIONS
![Page 19: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/19.jpg)
Necessary and sufficient conditions Cauchy-Riemann equationsProperties of analytic functionsHarmonic conjugateConstruction of Analytic functionConformal mapping and Bilinear
transformation.
![Page 20: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/20.jpg)
CONFORMAL MAPING
![Page 21: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/21.jpg)
![Page 22: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/22.jpg)
![Page 23: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/23.jpg)
Bilinear transformation
![Page 24: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/24.jpg)
APPLICATIONSNon-linear dynamic systemSpecial functions Number theoryDigital signal processingDiscrete time Control theoryImage Processing
![Page 25: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/25.jpg)
UNIT -IV
MULTIPLE INTEGRALS
![Page 26: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/26.jpg)
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.
![Page 27: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/27.jpg)
![Page 28: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/28.jpg)
APPLICATIONSProbability theoryMathematical Physics(moment of inertia)To find the area of the surfaceTo find the volume of the surface
![Page 29: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/29.jpg)
UNIT -V
COMPLEX INTEGRATON
![Page 30: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/31.jpg)
APPLICATIONS
Aero dynamicsElasticityTwo dimensional fluid flow
![Page 32: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/32.jpg)
TEXT BOOKS
![Page 33: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/33.jpg)
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
![Page 34: 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](https://reader033.vdocuments.net/reader033/viewer/2022061614/5697c01c1a28abf838ccfdda/html5/thumbnails/34.jpg)
to be continued…