lesson plan sem 3
TRANSCRIPT
RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS
LESSON PLAN
Name of the Faculty: S.PAULSAMY Designation / Department: Professor/ MathematicsSubject Code : MA2211 Subject Name : Engineering Mathematics - III Year / Sem: II / III
Sl. No. DATELECTURE
TOPICOBJECTIVE
TIME REQUIRED
MODE MEDIACumulative
No. of Hours
BOOKS (REFERENCE /
TEXT)Unit – I Topic : Fourier series Target Period :
01 05.07.12Introduction
Knows the periodic function and Dirichlet conditions
50 Minutes LectureChalk & Board
1
Engineering Mathematics -III
Dr.A.Singaravelu, Meenakshi Agency.
02 06.07.12About Fourier series
To Knows the definition of Fourier series and problems under Fourier series.
50 MinutesLecture Chalk &
Board 2
03 09.07.12 Euler formulaTo prove Euler formula.
50 MinutesLecture
Chalk & Board 3
04 10.07.12 To find Fourier series expansion
Problems in the intervals (0,2l) and (0, 2 ).
50 Minutes Lecture Chalk & Board
4
1
05 11.07.12To find Fourier series expansion
Some problems to find expansion in the interval (-l,l) and (- , )
50 MinutesLecture
Chalk & Board 5
06 12.07.12To find Fourier series expansion
Some Problems under odd and even function in (- , ).
50 Minutes LectureChalk & Board
6
07 13.07.12Half range expansion
To solve problems under Fourier sine series in the interval (0,l) and (0, ).
50 Minutes LectureChalk & Board
7
08 16.07.12Half range expansion
To solve problems under Fourier cosine series in the interval (0,l) and (0, ).
50 Minutes LectureChalk & Board
8
09 17.07.12 Complex or exponential form of Fourier series
Problems under complex form
50 Minutes Lecture Chalk & Board
9
2
10 18.07.12RMS value and
Parseval’s identity
To knows the RMS value and To find Some Problems under parseval’s identity.
50 Minutes LectureChalk & Board
10
11 19.07.12Harmonic Analysis
Problems under Harmonic analysis
50 MinutesLecture Chalk &
Board11
12 20.07.12Revision II Unit I
Tutorial 50 MinutesLecture Chalk &
Board12
Part – A1. Define periodic function with examples.2. When does a function posses a Fourier series expansion interms of trigonometric terms? (or) Explain Dirichlet’s conditions?3. Write the Fourier coefficients a0, an and bn in (0,2 ).4. Write the formula for Fourier constants of f(x) in (c, c + 2l).5. Can tan x be expanded in Fourier series. If so how? If not why?
6. Find the sum of the Fourier series of f(x) = x, 0 < x < 1
2, 1 < x < 2 , at x = 1.
7. Find the constant term in the Fourier series corresponding to f(x) = cos2x expressed in the interval (- , ). 8. Write a0, an in the expansion x + x3 as a Fourier series in (- , ). 9. Find the constant term in the Fourier series corresponding to f(x) = x - x3 in (- , ). 10. Find the constant term in the Fourier expansion of x2 – 2, in the range |x| 2.
3
11. If f(x) = x2 + x is expressed as a Fourier series in the interval (-2, 2) to which value this series converges at x = 2.
12. Find the Fourier constants bn for x sin x in (- , ). 13. Determine the value of an in the Fourier series expansion of f(x) = x3 in - < x <
14. If f(x) = 2x in the interval (0 , 4) then find the value of a2 in the Fourier series expansion.15. Expand f(x) = 1 in a sine series in 0 < x < .16. Expand f(x) = x in (0 , 1) as a Fourier sine series.17. Find the Fourier sine series of f(x) = x in 0 < x < 2.18. State Parseval’s identity for full range expansion of f(x) as Fourier series in (0 , 2l).19. If the Fourier series corresponding to f(x) = x in the interval (0 , 2 ) is
+ [an cosnx + bn sinnx] , with out finding the values of a0, an, bn find the value
of + [a + b ].
20. Define the RMS value of a function f(x) in (a , b).21. Define the RMS value of a function f(x) in (c , c + 2l).22. Define the RMS value of a function f(x) in (0 , 2 ).23. Find the RMS value of the function f(x) = x in the interval (0 , l).24. State Parseval’s identity for the half – range Cosine expansion of f(x) in (0 , l).25. What do you mean by Harmonic Analysis?
PART – B1. Find the Fourier series expansion of period 2l for the function f(x) = ( l –x)2
in the range (0, 2l). Deduce the sum of the series .
2. Find the Fourier series for f(x) = x, in 0 ≤ x ≤ 3
6-x, in 3 ≤ x ≤ 6. 3. Express f(x) = x sin x as a Fourier series in 0 ≤ x ≤ 2 . 4. Expand f(x) = x - x2 as a Fourier series in –L < x < L. 5. Obtain Fourier series for f(x) of period 2L and defined as follows.
f(x) = L + x, in (-L , 0)
4
L – x in (0 , L) . Hence deduce that + 23
1+
25
1+……. = .
6. Find the Fourier series for f(x) = -K, - < x < 0 K, 0 < x < . Hence deduce that
1 - + - +……. = .
7. Prove that for - < x < , = - + - ……
8. Obtain the Fourier series expansion of f(x) given by f(x) = 1+ , - ≤ x ≤
1- , 0 ≤ x ≤ and hence
deduce that + 23
1+
25
1+……. = .
9. Find the Fourier series for f(x) = |cos x| in the interval (- , ).
Obtain the sine series for the function f(x) = x, in 0 ≤ x ≤
-x, in ≤ x ≤
10. Find the half – range sine series of f(x) = 1 – x in (0 , 1).
11. Find the half – range sine series for f(x) = x ( - x) in (0 , ).
Deduce that - + -……. = .
12. Obtain a half – range cosine series of the function f(x) = kx, in 0 ≤ x <
k( -x), in ≤ x ≤ .
5
13. Find a cosine series for the function f(x) = x, in 0 ≤ x <
- x, in ≤ x < .
14. Obtain the Fourier expansion of x sin x as a cosine series in (0, ) and hence deduce the value of 1 + - + - ………...
15. Find the complex form of Fourier series f(x) = cos ax in - < x < .16. Find the complex form of Fourier series f(x) = in - < x < .
17. Find the Fourier series for f(x) = x2 in - < x < . Hence show that
+ + +……. = .
18. If for 0 < x < , the function f(x) has the expansion f(x) = , show that
19. The following table gives the variation of a periodic function over the period T.
x 0 T
f(x) 1.98 1.3 1.05 1.3 -0.88 -0.25 1.98
Show that f(x) = 0.75 + 0.37 cos + 1.004 sin , where = .
20. Find the Fourier series as far as the second harmonic to represent the function given in the following data.
x 0 1 2 3 4 5f(x) 9 18 24 28 26 20
ASSIGMENT TOPICS:1. Obtain the constant term and the first harmonic in the Fourier seriews expansion for f(x)
6
where f(x) is given in the following data.
x 0 1 2 3 4 5 6 7 8 9 10 11f(x) 18.0 18.7 17.6 15.0 11.6 8.3 6.0 5.3 6.4 9.0 12.4 15.7
TEXT BOOKS:
1.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition, Khanna Publications, Delhi, (2007).
REFERENCES:
1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 7th edition,Laxmi Publication (P) Ltd.,(2007).
2.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).
3.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).
4. Erwin Kreyszig, “Advanced Engineering Mathematics” , 8th edition, Wiley india, (2007).
RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS
LESSON PLAN
Name of the Faculty: K.SUBHA Designation / Department: Lecturer / MathematicsSubject Code : MA2211 Subject Name : Engineering Mathematics - III Year / Sem: I / III
7
Sl. No. DATELECTURE
TOPICOBJECTIVE
TIME REQUIRED
MODE MEDIACumulative
No. of Hours
BOOKS (REFERENC
E / TEXT)Unit – I1 Topic : Fourier Transforms Target Period :
01 23.07.12Introduction
To know Definitions for Fourier integral transform and complex form of Fourier integrals.
50 Minutes LectureChalk & Board
1
Engineering Mathematics -III
Dr.A.Singaravelu, Meenakshi Agency.
02 24.07.12
Fourier sine and cosine integrals
To solve Problems based on Fourier sine and cosine integrals.
50 Minutes
Lecture Chalk & Board
2
03 25.07.12Fourier transform
To know Definition and properties
50 MinutesLecture
Chalk & Board 3
04 26.07.12
Fourier transform
To solve Problems based on Fourier transform
50 Minutes
LectureChalk & Board
4
05 27.07.12 Convolution of two functions in Fourier transform
To solve Problems based on convolution theorem.
50 Minutes Lecture Chalk & Board
5
8
06 30.07.12Parseval’s identity
To solve problems based on Parseval’s identity.
50 Minutes LectureChalk & Board
6
07 31.07.12Fourier sine and cosine transforms
To know definition for Fourier sine and cosine transform.
50 Minutes LectureChalk & Board
7
08 01.08.12
Properties of Fourier sine and cosine transforms
To know the properties of Fourier sine and cosine transform.
50 Minutes LectureChalk & Board
8
09 02.08.12Inversion formula
To solve problems based on Fourier cosine transform.
50 Minutes LectureChalk & Board
9
10 03.08.12Fourier sine transform
To solve problems based on Fourier sine transform
50 Minutes LectureChalk & Board
10
11 06.08.12Revision I Unit I
Tutorial 50 MinutesLecture Chalk &
Board11
12 07.08.12Revision II Unit I
Tutorial 50 MinutesLecture Chalk &
Board12
PART – A1. State Fourier integral theorem.2. Show that f(x) = 1, 0 < x < can not be represented by a Fourier integral.3. Define Fourier transform pair. (or) Define Fourier transform and its inverse transform.4. If Fourier transform of f(x) = F(s) then what is Fourier transform of f(ax).
9
5. Find the Fourier transform of f(x) = 1 for |x| < a
0 for |x| > a > 0 6. What is the Fourier cosine transform of a function. Write down the Fourier cosine transform pair of formulae.
7. Find the Fourier cosine transform of f(x) = cos x if 0 < x < a 0 if x a
8. Find the Fourier cosine transform of e-x.9. Find the Fourier cosine transform of e-3x.10. Find the Fourier sine transform of e-x.11. Find the Fourier sine transform of e-3x.12. Find the Fourier sine transform of 3e-2x.
13. Find the Fourier sine transform of .
14. Find Fc[xe-ax] and Fs[xe-ax].
15. Prove that Fc[f(x)cosax] = [f(s+a) + f(s-a)] where Fc denote the Fourier cosine
transform of f(x).17. IF F(s) is the Fourier transform of f(x), then show that the Fourier transform of eiaxf(x) is F(s+a).
18. Given that e- is self reciprocal under Fourier cosine transform, find Fourier Sine transform of x e- .
19. If Fc(s) is the Fourier cosine transform of f(x). prove that the Fourier cosine transform of f(ax) is Fc( ).
20. If F(s) is the Fourier transform of f(x), then find the Fourier transform of f(x-a).21. If Fs(s) is the Fourier sine transform of f(x),
Show that Fs[f(x)cosax] = [Fs(s+a) + Fs(s-a)].
22. Find F[xnf(x)] in terms of Fourier transform of f(x).
23. Find F[ ]in terms of Fourier transform of f(x).
10
24. State the Convolution theorem for Fourier transforms.25. State Parseval’s identity for Fourier transform.
PART – B
1. Show that the Fourier transform of f(x)= a2-x2, |x|< a 0, |x| > a is
. Hence deduce that .
Using Parseval’s identity show that = .
2. Find the Fourier transform of f(x) = 1 – x2, |x| < 1 0 , |x| > 1.
Hence prove that .
3. Find the Fourier transform of f(x) = 1 , |x| < a 0, |x| > a > 0.
Hence deduce that (i) (ii) .
4. Find the Fourier transform of f(x) = 1-|x|, |x| < 1 0 , |x| > 1.
Hence deduce that and .
5. Find the Fourier cosine and sine transform of the function f(x) = e-ax , a > 0.
6. Evaluate , if a > 0 using parseval’s identity.
7. Evaluate , if a > 0 using parseval’s identity.
11
8. Evaluate , if a, b > 0 using parseval’s identity.
9. Evaluate , if a, b > 0 using parseval’s identity.
10. Evaluate , using Transform method.
11. Find the Fourier cosine and sine transform of f(x) = xe-ax.
12. Find the Fourier cosine transform of f(x) = .
13. Find the Fourier sine transform of f(x) = .
14. Find the Fourier cosine transform of f(x) = .
15. Find the Fourier cosine transform of , a > 0 and hence deduce that sine transform of x .
16. Show that the Fourier sine transform of x is self reciprocal.
17. Find the Fourier sine and cosine transform of e-x also find the Fourier sine transform of and Fourier cosine transform of .
18. State and prove Convolution theorem in Fourier transform.
ASSIGMENT TOPICS:
1.Find the Fourier transform of f(x) = a- |x|, |x| < a 0, |x| > a > 0.
Deduce that (i) .
TEXT BOOKS:
1.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition, Khanna Publications, Delhi, (2007).
REFERENCES:
12
1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 7th edition,Laxmi Publication (P) Ltd.,(2007).
2.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).
3.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).
4. Erwin Kreyszig, “Advanced Engineering Mathematics” , 8th edition, Wiley india, (2007).
RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS
LESSON PLAN
Name of the Faculty: Dr.S.Paulsamy, K.Subha, N.Ananthan Designation / Department: Professor/Lecturer/ Lecturer / Mathematics
Subject Code : MA2111 Subject Name: Engineering Mathematics -II Year / Sem: I / II
13
Sl. No. DATELECTURE
TOPICOBJECTIVE
TIME REQUIRED
MODE MEDIACumulative
No. of Hours
BOOKS (REFERENCE /
TEXT)Unit – I11 Topic :ANALYTIC FUNCTIONS Target Period :
01 07.02.12Complex Variables
To know the properties of Arithmetic Operations and limit to the functions
50 Minutes LectureChalk & Board
1
Engineering Mathematics -II
Dr.A.Singaravelu, Meenakshi Agency.
02 08.02.12Functions of a Complex Variables
To know the Continuity of a Function and some problems for that Continuity functions
50 MinutesLecture Chalk &
Board 2
03 14.02.12
The necessary condition for f(z) to be analytic
To know the Cauchy Riemann equations and to find some problems
50 MinutesLecture
Chalk & Board 3
04 15.02.12 Cauchy – Riemann Equations
Using Cauchy Riemann equations to find some problems
50 Minutes Lecture Chalk & Board
4
14
05 21.02.12Sufficient condition of f(z) to be analytic
To know the sufficient conditions and Polar form of Cauchy Riemann equations and to find some problems
50 MinutesLecture
Chalk & Board 5
06 22.02.12 Properties of analytic functions
To know some properties and using to find some problems
50 Minutes LectureChalk & Board
6
07 28.02.12Constructions of analytic functions
To know the Milne – Thomson method and using some problems
50 Minutes LectureChalk & Board
7
08 29.02.12 Harmonic Functions
To know the Harmonic functions and using some problems
50 Minutes Lecture Chalk & Board
8
15
09 13.03.12Conformal Mappings
To know what is Conformal mapping and using Transformations to find some problems
50 Minutes LectureChalk & Board
9
10 14.03.12Bilinear Transformation
To known what is bilinear transformations and to find some problems
50 Minutes LectureChalk & Board
10
11 20.03.12Revision I Unit I
Tutorial 50 MinutesLecture Chalk &
Board11
12 21.03.12Revision II Unit I
Tutorial 50 MinutesLecture Chalk &
Board12
PART –A1.State the Cauchy – Riemann equation in polar coordinates.2.Give an example that the Cauchy – Riemann equation are necessary but not sufficient for a function to be analytic..3.State the sufficient condition for the function f(z) to be analytic.4.If w = ez, find dw/dz using complex variable.5. Choose the correct answer, w = f(z) is analytic function of z,then------------.6. The function f(z) = u+iv is analytic only if ----------7. Show that the function f(z) = z z¯ is not analytic at z = 0.8. State the necessary and sufficient condition for f(z) to be analytic.9. Verify whether w = sin xcos hy+ i cos x sin hy is analytic or not.10. The function f(z) = √׀xy׀ is not regular at the origin.11. The function w = log z is analytic every where in the complex plane.12. A certain function u(x,y) can be the real part of an analytic function if----------.13. If Cauchy Riemann equation are satisfied by real and imagionary parts of a complex function f(z) then f(z) should be analytic.
16
14. Write down the necessary condition for w = f(z) = f(reiɵ) to be analytic.15. State two properties of analytic function.16. Show that the function x4-6x2y2+y4 is harmonic.17. Examine whether the function xy2 can be real part of an analytic function.18. Define Mobius transformation.19. Define isogonal transformation.20. What is invariant point in a mapping?PART –B 1 .a. Find the analytic function f(z) whose real part is u(x,y) = 3x2y+2x2 – y3 – 2y2
b. Find the bilinear transformation that maps 1,i,-1 of the z- plane onto 0,1,∞ of the w – plane.2 .a. Find the image of 1 < x < 2 under the mapping w = 1/z. b. If u+v = (x-y)(x2+4xy+y2) and f(z) = u+iv, find f(z) interms of z.3.a. Show that the transformation w = z-i/ 1-iz maps (i) the interior of the circle׀ z 1= ׀ on to the lower half of the w- plane and (ii) the upper half of the z-plane on to the interior of the circle ׀w 1= ׀ . b. Find the analytic function u+iv, if u = (x-y)(x2+4xy+y2). Also find the conjugate harmonic function v.4. a. Find the image of the hyperpola x2-y2 = 1 under the transformation w = 1 / z. b. Prove that the transformation w = z / 1-z maps the upper half of z – plane onto the upper half of w-plane. What is the image of ׀z 1 =׀ under this transformation?.ASSIGMENT TOPICS:1. If a,b, are the two fixed points of a bilinear transformation, show that it can be written in the form, w-a/w-b = K(z-a/z-b), K is Constant, a ≠ b.2. If f(z) is a regular function of z, prove that (∂2/∂x2 + ∂2/∂y2) ׀f(z) ׀4 = 2׀ f’(z) 2׀ .
TEXT BOOKS:
1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 3rd edition,Laxmi Publication (P) Ltd.,(2008).
2.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition,Khanna Publications, Delhi, (2007).
REFERENCES:
1.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).
2.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).
17
3. Erwin Kreyszig, “Advanced Engineering Mathematics” , 7th edition, Wiley india, (2007).
2.Jain R.K and Iyengar S.R.K, “Advanced Engineering Mathematics” , 3rd edition,Narosa Publishing House Pvt.Ltd.,(2007).
RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS
LESSON PLAN
Name of the Faculty: Dr.S.Paulsamy, K.Subha, N.Ananthan Designation / Department:Professor/Lecturer/ Lecturer / MathematicsSubject Code : MA2111 Subject Name: Engineering Mathematics- II Year / Sem: I / II
Sl. No. DATELECTURE
TOPICOBJECTIVE
TIME REQUIRED
MODE MEDIACumulative
No. of Hours
BOOKS (REFERENC
E / TEXT)Unit – 1V Topic : COMPLEX INTEGRATION01 03.04.12 Properties of
complex integrals
To know the some properties and to find some problems
50 Minutes Lecture Chalk & Board
1 Engineering Mathematics -II
18
Dr.A.Singaravelu, Meenakshi Agency.
02 04.04.12Cauchy’s integral theorem
To know the statement of Cauchy’s Integral theorem and to find some problems
50 MinutesLecture Chalk &
Board 2
03 10.04.12Cauchy’s integral formula
Some problems using to find Cauchy’s Integral formula for derivatives
50 MinutesLecture
Chalk & Board 3
04 11.04.12 Taylor Series
To know some important results and using to find some problems
50 MinutesLecture
Chalk & Board 4
05 17.04.12 Laurent’s Series
To know the Laurent’s Series expantions and to find some problems
50 MinutesLecture
Chalk & Board 5
06 18.04.12 Singularities To know types of singularities and using to find some problems
50 Minutes Lecture Chalk & Board
6
19
07 02.05.12Residue at a Pole
To know definitions for Residue and Pole and using to find some problems
50 Minutes LectureChalk & Board
7
08 04.05.12Cauchy’s Residue theorem
To know the Statement of Cauchy’s Residue theorem and using to find some problems
50 Minutes LectureChalk & Board
8
09 09.05.12
Evaluation of integrals using Rasidue theorem
Some problems using to find Residue Theorem
50 Minutes LectureChalk & Board
9
10 14.05.12
Real definite integrals by contour integration
To know the Definition for Definite integral and Contour integrations and using to find some problems
50 Minutes LectureChalk & Board
10
11 15.05.12Revision I Unit I
Tutorial 50 MinutesLecture Chalk &
Board11
12 16.05.12Revision II Unit I
Tutorial 50 MinutesLecture Chalk &
Board12
20
PART - A1.What is the value of ∫ 3z2+7z+1/ z+1 if C is ׀z׀= ½ over the region C. 2. State Cauchy’s Integral Formula.3. Find the value of ∫ z / z2-1 where C is ׀ z׀ = ½ over the region C.4. State Cauchy’s Integral Formula.5. The integral value of ∫ (x2 – iy) dz along the path y = x is 1/6(5-i) and the limits from 0 to 1+i say true or false?.6. Find the Laurent’s expansion of f(z) = 1/ z(1-z)2 in ׀z 1 >׀ .7. Expand cos z in a Taylor’s series at z = π / 4.8. Expand the function sin z / z-π about z = π9. Laurent’s series expansion of f(z) over a given annular is not unique, say true or false?.10. Expand 1 / z-2 at z = 1 as a Taylor’s series.11. What are the poles of cot z?12. Define essential singular point of f(z)?.13. Find the zero’s of z3-1 / z3+114. If f(z) = 1 / z2+1 is analytic everywhere except ----------15. For a simple pole at z = a the residue of f(z) at z = a, where f(z) = P(z) / Q(z) is ---------16. Evaluate ∫ cos π z / z-1 dz where C is ׀z 1.5 = ׀ over the region C.17. The residue at the pole of the function z / z2 +1 are equal.18. If f(a) = ∫ z2+1 / z-a dz where C is the ellipse x2 + 4y2 = 4, then the value of f(1) is ----------------, over the region C.19. State the function f(z) and the region contour to evaluate the integral ∫ x sin x / x4 + a4 dx, and the limits from 0 to ∞.20. Express ∫ dɵ / 1+ a cos ɵ as a contour integral around the circle ׀z 1 = ׀ , and the limits from 0 to 2π.
PART –B1.Evaluate ∫ dx / (x2+a2)3, a > 0, using contour integrtion, and the limits from 0 to ∞.2.Evaluate ∫ dɵ / 2+ cos ɵ , and the limits from 0 to 2π.3. Expand 1 / (z-1)(z-2) in Laurent’s series valid for ׀ z 1 <׀ and 1<׀ z 2 < ׀ .4.Evaluate ∫ dɵ / a+b cos ɵ where a>b by contour integration, and the limits from 0 to 2π.5.Using residue calculus prove that ∫ dɵ / 5+3 cos ɵ = π / 2 and the limits from 0 to ∞.
ASSIGMENT TOPICS:
1.Evaluate ∫x2dx / (x2+a2)(x2+b2) using contour integration, where a > b> ɵ, and the limits from -∞to ∞.
2.Evaluate ∫ dɵ / 2-cos ɵand the limits from 0 to 2π.
21
TEXT BOOKS:
1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 3rd edition,Laxmi Publication (P) Ltd.,(2008).
2.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition,Khanna Publications, Delhi, (2007).
REFERENCES:
1.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).
2.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).
3. Erwin Kreyszig, “Advanced Engineering Mathematics” , 7th edition, Wiley india, (2007).
2.Jain R.K and Iyengar S.R.K, “Advanced Engineering Mathematics” , 3rd edition,Narosa Publishing House Pvt.Ltd.,(2007).
RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS
LESSON PLAN
Name of the Faculty:Dr. S.Paulsamy, K.Subha, N.Ananthan Designation / Department:Professor/Lecturer/ Lecturer / MathematicsSubject Code : MA2111 Subject Name : Engineering Mathematics -II Year / Sem: I / II
Sl. No. DATELECTURE
TOPICOBJECTIVE
TIME REQUIRED
MODE MEDIACumulative
No. of Hours
BOOKS (REFERENC
E / TEXT)
22
Unit – V Topic : LAPLACE TRANSFORM
01 02.04.12
Conditions for existence of Laplace transforms
Know that Laplace definition and some problems based on exponential orderand sine, cosine method
50 Minutes LectureChalk & Board
1
Engineering Mathematics -II
Dr.A.Singaravelu, Meenakshi Agency.
02 04.04.12 Linear Property
Know that the linear property definition and solve some problems
50 MinutesLecture Chalk &
Board 2
03 09.04.12First Shifting Theorem
Know that First shifting theorem and solve some problems
50 MinutesLecture
Chalk & Board 3
04 11.04.12 Second Shifting Theorem
Know that Second shifting theorem and solve some problems
50 Minutes Lecture Chalk & Board
4
23
05 12.04.12
Laplace transforms of Derivatives And integrals
Know that the derivative and integral theorem, Dirac delta function and solve some problems
50 MinutesLecture
Chalk & Board 5
06 16.04.12
Laplace Transform of periodic functions
Know that the periodic functions and solve some problems
50 Minutes LectureChalk & Board
6
07 18.04.12The Inverse laplace transform
Know that some formula for inverse Laplace transform and solve some problems
50 Minutes LectureChalk & Board
7
08 25.04.12Convolution Theorem
Know that Convolution theorem and using some problems
50 Minutes LectureChalk & Board
8
09 26.04.12 Initial value Problem and Final Value Theorem
Solve some problems based on initial value problem and final value theorem
50 Minutes Lecture Chalk & Board
9
24
10 27.04.12
Laplace Transforms for differential equations
Solve some problems based on differential equations
50 Minutes LectureChalk & Board
10
11 30.04.12Simultaneous Differential Equations
Solve some problems based on simultaneous differential equations
50 Minutes LectureChalk & Board
11
12 02.05.12Revision I Unit I
Tutorial 50 MinutesLecture Chalk &
Board12
13 03.05.12Revision II Unit I
Tutorial 50 MinutesLecture Chalk &
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PART –A
1. If L(sin at / t) = cot -1 (s / a) then ∫ sin at / t dt = ---------, and the limits from 0 to ∞.2. Say true or false L[t(f(t)] = Lf(t) / s
3. L[f(t)] = ϕ(s) then L[f (t/2)] = 2ϕ(2s).
4. Find the Laplace transform of cos3t / t.
5. Find the Laplace transform ׀ sin wt׀.
6. Find L[sin √t].
7. If L [f (t)] = s2-s+1 / (2s+1)2(s-1), applying the change of scale property shown that L [f (2t)] = s2-2s+1 / 4(s+1)2(s-2).
8. L (u (t-a)) = ------------ (or) Give the Laplace transform of unit step function.
9. Find the Laplace transform of Dirac Delta Function.
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10. If f(t) = e-2t sin2t, find L[f’(t)]
11. L[f(t)] = 1/ s(s+1) = f(s) find f(0) and f(∞).
12. Find L-1(1 / s2+4s+4).
13. Find L[cos √t / √t]
14. Find the inverse transform of tan-1s.
15. Find L-1[1 / s4-1]
16. Find L-1(1 / s2(s2+81)).
17. Find L-1(e-πs / s2)
18. Solve the integral equation.
19. Solve y+ ∫ydt = t2+2t and the limits from 0 to t.
20. Show that [1*1*1*……*1] n times = tn-1 / (n-1)!.
PART – B 1. Solve the initial value problem. y” – 3y’+2y = 4t, y(0) = 1, y’(0) = -1.2. Using convolution theorem, find L-1[s / (s2+1)2(s2+4)].3.a. Find L(e-2t t sin2t), b. find L-1[tan-1(2 / s2)]4. Using convolution theorem find the inverse Laplace transform of 1 / (s2+4)2.5. Find L-1(s/ s4+4a4).
ASSIGMENT TOPICS:1 Find the Laplace transform of the following triangular wave function given by f(t) = { t, 0 ≤ t ≤ π and 2-t, π ≤ t ≤ 2π f(t+2) = f(t).
2.Verify initial and final value theorems for the function f(t) = 1+e-t(sin t + cos t).
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TEXT BOOKS:
1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 3rd edition,Laxmi Publication (P) Ltd.,(2008).
2.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition,Khanna Publications, Delhi, (2007).
REFERENCES:
1.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).
2.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).
3. Erwin Kreyszig, “Advanced Engineering Mathematics” , 7th edition, Wiley india, (2007).
2.Jain R.K and Iyengar S.R.K, “Advanced Engineering Mathematics” , 3rd edition,Narosa Publishing House Pvt.Ltd.,(2007).
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