1. MATHEMATICS-II
SUB.CODE: MA1151

2. UNIT-I
LAPLACE TRANSFORMS

3. 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.

4. 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

5. 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

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

7. UNIT -II
VECTOR CALCULUS

8. 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

9. 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

10. LINE INTEGRAL

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

12. Now we will embed S in a vector field:

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

14. Again we embed it in the same vector field.

15. 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:

16. 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

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

18. UNIT -III
ANALYTIC FUNCTIONS

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

20. CONFORMAL MAPING

23. Bilinear transformation

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

25. UNIT -IV
MULTIPLE INTEGRALS

26. 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.

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

29. UNIT -V
COMPLEX INTEGRATON

30. 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

31. APPLICATIONS
Aero dynamics
Elasticity
Two dimensional fluid flow

32. TEXT BOOKS

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

34. to be continued…