Is all differentiable functions are integrable?

While attending teleconference as a resource person in bangalore. I and panel members was asked to answer the following questions by my fellow lecturers.
Is integrable functions are continous?, Is every continous functions are integrable?, Is all functions are integrable etc.
Really I find these questions are good and we should have to clarify this questions. Here I tried to answer this : students are competent can give your suggestion
<!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;}
–>1. <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –>Are all functions that can be differentiated, integratable?
All differentiable functions are integrable.
True because all differentiable functions are continuous(Because Differentiability implies continuity but continuity need not implies differentiability) and by FTC, fundamental theorem of integral calculus all continuous functions are integrable.

2.Is every continuous functions are integrable?
True by fundamental theorem of integral calculus

3. Is all integrable functions are continuous?
<!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –>This doesn’t follow from the FTC, but I’m having trouble thinking of a counter-example. I looked around on the web and saw a couple people say that this is false, but never explain why. Can you integrate piecewise functions? If so then I can think of an easy counter-example. We’ve never talked about doing so in class. but think!

Some of the following integrals are not integrable:
LIST OF SUCH INTEGRALS <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} p {mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} span.postbody {mso-style-name:postbody;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> 1. int sinx / x dx

2. int cosx/ x dx

3. int root (sinx) dx

4. int root (cosx) dx

5.int sin (x 2 ) dx

6. int cos (x 2 ) dx

7. int e -x 2 dx

8. int e x 2 dx

10. int root(1+x 3 ) dx

11. int x tan x dx

12. int 1/ log x dx . and many more ok dear. try to remember it.





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WHAT IS INTEGRALS?


Calculus is the study of how things change.
In differentiation we are finding a derivative i.e. slope of tangent to the curve i.e how the curve changes at different points for a given function
In integration we are finding a original function whose derivative or slope of the tangent or change at different points is known
Slopes of curves is also can be considered as rate of change at different cases. This is called instantaneous change. Derivative of a curves tell us the  instantaneous rate of change of a curve.
A curvy function changes at different rates throughout its domain—sometimes it’s increasing quickly and the tangent line is steep (causing a high-valued derivative). At other places the curve may be increasing shallowly or even decreasing, causing the derivative to be small or negative, respectively.
Look at the graph of f(x) in Figure 
If instantaneous change is given to find the original state we need to integrate.
To find the change at different points we need to differentiate.
Calculus analyses things that change, and physics is much concerned with changes. For physics, you’ll need at least some of the simplest and most important concepts from calculus.

A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the “derivative”) of various functions. (The process of doing so is called “differentiation”.)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called “integration”.)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.

We know how to find the derivative of a function by different methods, Now we are going back in integration.
If we know that

 dy/dx=2x

and we need to know the function this derivative came from, then we “undo” the differentiation process. (Think: “What would I have to differentiate to get this result?”)
y=x2  is ONE antiderivative of   dy/dx =2x
There are infinitely many other antiderivatives which would also work, for example:
y=x2+4
y=x2+π
y=x2+27.3
In general, we say y=x2+Kis the indefinite integral of 2x. The number K is called the constant of integration.
Note: Most math text books use C for the constant of integration, but for questions involving electrical engineering, we prefer to write “+K“, since C is normally used for capacitance and it can get confusing.
Notation for the Indefinite Integral
We write: ∫2x  dx=x2+K and say in words:
“The integral of 2x with respect to xequals x2 + K.”

The Integral Sign

The sign is an elongated “S”, standing for “sum”. (In old German, and English, “s” was often written using this elongated shape.) Later we will see that the integral is the sum of the areas of infinitesimally thin rectangles.
is the symbol for “sum”. It can be used for finite or infinite sums.
is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
This “long s” notation was introduced by Leibniz when he developed the concepts of integration.
GEOMETRICAL INTERPRETATION OF INDEFINITE INTEGRAL:
Let f (x) = 2x. Then ∫ f (x) dx = x2 + C. For different values of C, we get different integrals. But these integrals are very similar geometrically.
Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis.

Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin.
The curve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along y-axis in positive direction.
For C = – 1, y = x2 – 1 is obtained by shifting the parabola y = x2 one unit along y-axis in the negative direction.
Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig
 


Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x2,
y = x2 + 1, y = x2 + 2, y = x2 – 1, y = x2– 2 at P0, P1, P2, P-1, P-2etc., respectively, then dy / dx at these points equals 2a.
This indicates that the tangents to the curves at these points are parallel. Thus, ∫2x2 dx = x + C = FC (x) (say), implies that the tangents to all the curves y = FC (x), C R, at the points of intersection of the curves by the line x = a, (a R), are parallel.

Further, the following equation (statement) ∫ f (x) dx = F (x) + C = y (say) , represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral.
∫ f (x) dx = F (x) + C = Family of all curves which are geometrically similar and the tangents drawn to all curves by the line x=a (at any particular point) are parallel.

NOW THE QUESTIONS?
1.      Is All differentiable functions are integrable.
Ans: True because all differentiable functions are continuous and by FTC all continuous functions are integrable.
2.      Is All integrable functions are continuous.?
Ans:  This doesn’t follow from the FTC, Now for this it is difficult to answer at this level. We have to learn advance mathematics
Can you integrate piecewise functions? If so then I can think of an easy counter-example. We’ve never talked about doing so in class.
3.      Is All integrable functions are differentiable.
Ans: Even though 1 is true this doesn’t follow from it. Same difficulty as 2.

{Differentiable functions} {Continuous functions} {Integrable functions}

In order for some function f(x) to be continuous at x = c, then the following two conditions must be true:
i)                    f(c) is defined and the limit of f(x) as x approaches c is equal to f(c).
In order for some function f(x) to be differentiable at x = c, then it must be continuous at x = c and it must not be a corner point (i.e., it’s right-side and left-side derivatives must be equal).
Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists.
While all continuous functions are integrable, not all integrable functions are continuous. To understand this idea we need to study advanced mathematics i.e Reimann integral.
Hope this will satisfy your needs    : Happy learning   : by KHV, SAGAR

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CHRNOLOGY OF LIFE OF SRINIVASA RAMANUJAN

1887
Dec. 22 Born of Komalathamal and K.Srinivasa Iyangar at Erode in his maternal grandmother¡¯s house. He was the first of the three surviving sons. They are Lakshmi Narashiman and Tirunarayanan.
1882
Oct.1, on a Vijayathasami day, he was enrolled in the Thinnai Palli
Koodam (Pial school)
1884
Enrolled in Kangeyam Primary School in Kumbokonam. He was quiet and contemplative. He was fond of questions, and liked to be by himself. Discouraged by his mother from going out to play, he talked to his friends only from the window overlooking the street. He lacked interest in sports, so he was fat and plump.
1897
Stood first in the Tanjore District Primary examination and earned a half-fee concession for a high school study.
1898 Joined Town High School, Kumbokonam in I Form.
1900
During his school days, he impressed everyone by his extraordinary intuition and astounding proficiency in several branches of mathematics. In III form, studied by himself Algebra, Geometric series, Loney¡¯s Part II Trigonometry, and learnt sine and cosine series before knowing them as ratios.
1903
In VI form, obtained G.S.Carr¡¯s: A Synopsis of Elementary Results, a book on Pure Mathematics, which contained propositions, formulae and methods of analysis with abridged demonstrations published in 1886. It was this book which awakened the genius in him.
1904
Passed Madras Matriculation examination and joined the F.A., class in the Government Arts College in Kumbakonam. Obtained Junior Subramaniam scholarship.
1905
He did not pass in English and hence was not promoted to senior F.A. class, so lost his scholarship.
1906
Joined Pachaiappa¡¯s College, after about three months, he fell ill and discontinued his studies.
1907
Appeared privately for F.A., examination, secured centum in mathematics, but failed to secure pass marks in other subjects. Never tried again.
1909 July 14 Married to 9 year old Janaki in Karur.
1911
His first contribution to the Journal of the Indian Mathematical Society appeared.
1912 Jan. 12 Joined as an officiating clerk in A.G¡¯s Office on Rs.20 per month.
Feb. 09 Applied to Madras Port Trust for a job.
Mar. 01 Joined Madras Port Trust in the Account Section in Class III, Grade IV on Rs.30 per month.
Wrote to E.W.Hobson, a Senior Wrangler in Cambridge, but received no reply.
Wrote to Henry Frederick Backer, an elected Fellow of the Royal Society and Cayley Lecturer in Mathermatics at Cambridge. He also did not acknowledge.
1913
Jan. 14 Ramanujan tumbled upon G.H.Hardy¡¯s Tract on Orders of Infinity through Prof. Seshu Aiyar. He was excited, since he had already discovered the result which gave the order of p(x). This made Prof. Seshu Aiyar suggest communication of this and other results to G.H.Hardy.
Jan. 16 Wrote his first letter to Prof. G.H.Hardy, a Fellow of the Royal Society and Cayley Lecturer in Mathermatics at Cambridge.
Feb. 08 Hardy acknowledged and asked for rigorous proofs.
Feb. 27 Ramanujan wrote to Hardy, stating that he has found in him a friend.
Hardy wrote to the Secretary of the Indian students, in the India Office, London, suggesting that some means be found to get Ramanujan to Cambridge.
Ap. 07 The Government sanctioned the proposal of the Syndicate of the Madras University to grant Ramnaujan a special scholarship of Rs. 75 per month for two years. This sanction gave Ramanujan free access to the Mathematical books in the University Library.
May 01 Ramanujan was granted two years leave on loss of pay and started working as the first research scholar of the Madras University.
Aug. 05 Sent his first Quarterly report to the University.
Nov. 07 Sent his second Quarterly report to the University.1915
1914
Jan. 14 Ramanujan expressed his willingness to leave for Cambridge.
Feb. 12 Government sanctioned the appropriation of a sum not exceeding Rs.10,000 from the University Vacation Lecturer¡¯s Fund for the grant to Ramanujan of a scholarship of £250 a year tenable in England for a period of two years, a free passage and a reasonable outfit.
Mar. 17 Sailed to England by s.s.Nevasa. The cost of the II Class ticket was Rs.440.
Ap. 14 Arrived in London, admitted in Trinity College and stays in Prof.Nevilli¡¯s house.
Ap. 17 Went to Cambridge.
1915 Jan. 07 Worked on Arithmetical Functions.
Dec. 15 On recommendation of Francis Spring to Dewsbury, Ramanujan¡¯s scholarship was extended up to the end of March 1918.
1916 Mar. 15 Ramanujan was conferred upon, B.A., degree by research.
1917 Mar. 12 Fell ill and hospitalized in Sanatoria at Wells, Matlock and London.
Oct. 12 Showed improvement and resumed active work.
Oct. 18 Conferred upon, Trinity Fellowship consisting of £ 250 a year with no conditions about duties or residence.
1918 Feb. 28 Elected Fellow of the Royal Soceity.
Nov.26 Hardy wrote to Dewsbury, about Ramnaujan¡¯s continued illness and need for his short return to India.
Dec. 09 On the recommendation made by Dewsbury, the Syndicate granted Ramanujan a sum of £ 250 a year for 5 years from April 1, 1919 in recognition of his service to the Science of Mathematics.
1919 Mar. 13 Left England by s.s.Nagoya.
Mar. 27 Arrived in India and given a warm reception at Bombay.
Ap. 02 Reached Madras by train.
Aug.12 Based on the report of Dr.Pires, C.F.Fearnside, the D.M.O., shifted Ramanujan to Coimbatore.
1920 Jan. 20 Returned to Madras.
Ap. 26 Passed away leaving behind his wife Janaki Ammal aged jut 20 years. He had no issue

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Western Region, New Admn. Building,2nd Floor, ATI Campus,Sion-Trombay Road, Sion, MUMBAI – 400 022.
86
Board of Practical Training (BOPT), Eastern Region
Block EA, Sector I (OPP. Labony Estate) PO Salt Lake City, Kolkata – 700 064.
87
Board of Apprenticeship
Training (BOAT), Plot No.16, Block-1-A, Lakhanpur, GT Road, Kanpur – 208024.
88
Board of Apprenticeship Training,
(BOAT) CIT Campus, Taramani, Chennai – 600 113.
89
Indian School of Mines University
Dhanbad – 826004, JHARKHAND
90
National Institute of Foundry and Forge Technology (NIFFT)
P.O. Hatia, Ranchi – 834003, Jharkhand.
91
National Institute of Industrial Engineering
Vihar Lake, PO- NITIE, MUMBAI – 400 087.
92
School of Planning & Architecture
I.P. Estate, New Delhi – 110 002.
93
School of Planning & Architecture
Bhopal. Temporary at NIT Campus, Bhopal.
94
National Institute of Technology
CALICUT – 673601.
95
School of Planning & Architecture
Vijayawada. (Mentored by SPA, New Delhi)
96
Sant Longowal Institute of Engineering & Technology (SLIET)
Village Longowal, Distt: Sangrur Punjab 148106
97
North Eastern Regional Institute of Science & Technology (NERIST)
Nirjuli – 79110 (Itanagar), Arunachal Pradesh
98
National Institute of Technology
Hazaratbal, SRINAGAR – 190006, J&K.
99
Motilal Nehru National Institute of Technology
ALLAHABAD – 211004, (UP).
100
Rashtriya Sanskrit Sansthan
56-57, Institutional Area, Pankha Road, Janak Puri, NEW DELHI.
01
National Institute of Technology
DURGAPUR – 713209, (WEST BENGAL).
102
Shri Lal Bahadur Shastri Rashtriya Sanskrit Vidyapeeth
Katwaria Sarai, Near Qutub Hotel, New Mehrauli Road, NEW DELHI-110067.
103
Rashtriya Sanskrit Vidyapeetha
TIRUPATI, (A.P.).
104
National Institute of Technology
JAMSHEDPUR-831014, JHARKHAND
105
Maharshi Sandeepani Rashtriya Veda Vidya Pratishthan
Ujjayini Development Authority, Administrative Building, Bharatpur, Ujjain – 456010.
106
Visvesvaraya National Institute of Technology
NAGPUR – 440001.
107
National Institute of Technology
Srinivasanagar SURTHAKAL – 575 025.
108
Kendriya Hindi Sansthan
Hindi Sansthan Marg, AGRA – 282005.
109
National Council for Promotion of Urdu Language
West Block No.I, R.K. Puram, New Delhi – 110 066.
110
National Institute of Technology
WARANGAL – 506004, (AP)
111
Malaviya National Institute of Technology
JAIPUR – 302017. Rajasthan
112
National Council for Promotion of Sindhi Language
5th Floor, Darpan Building, R.C.Dutt Road, Alkapuri, Vadodra – 390005.
113
Central Institute of Classical Tamil
Plot.No.40, IRT Campus,100 Ft Road,Taramani,Chennai 600 005
114
National University of Educational Planning and Administration (NUEPA
17-B, Sri Aurobindo Marg, NIE Camp, NEW DELHI – 110016.
115
Auroville Foundation
Bharat Nivas, P.O. Auroville, Distt. Villupuram, AUROVILLE – 605101, Tamil Nadu.
116
Educational Consultants of India Limited (EdCIL)
Plot No. 18A,Sector – 16A, NOIDA – 201301, (UP).
117
Central Institute of Indian Languages
Manasagangotri, Mysore – 570 006
118
Central Hindi Directorate
R.K.Puram, New Delhi.
119
Commission for Scientific and Technical Terminology
R.K.Puram, New Delhi.
120
National Book Trust of India
A-15, Green Park, NEW DELHI – 110016.
National Institute of Technology
ROURKELA – 769008, ORISSA
122
Maulana Azad National Institute of Technology
BHOPAL – 462007.
123
All India Council of Technical Education (AICTE)
7th Floor, Chanderlok Building, Janpath, New Delhi – 110 001.
124
Indian Institute of Advanced Studies (IIAS)
Rashtrapati Nivas, Shimla – 171 005.
125
National Institute of Technology
Tiruchirapalli- 620 015, TAMIL NADU
126
National Commission for Minority Educational Institutions
1st Floor, Jeevan Tara Building, 5, Sansad Marg, Patel Chowk, New Delhi – 110 001
127
National Institute of Technology
Kurukshetra – 132119, HARYANA
128
Council of Architecture
India Habitat Centre, Core-6-A, Ist Flooor, Lodhi Road, New Delhi – 110 003.
129
National Institute of Technology
Silchar – 788010, ASSAM
130
National Institute of Technology
Hamirpur – 177001 , HIMACHAL PRADESH
131
National Institute of Technology
Patna – 800 005, BIHAR
132
Dr. B.R. Ambedkar National Institute of Technology
G.T. Road, Bye Pass, Jallandhar – 144 011, PUNJAB.
133
National Institute of Technology
Raipur, Chhattisgarh
134
National Institute of Technology
Agartala, Tripura
135
National Institute of Technology
Sikkim, Barfung Block, Ravangla Sub-Division, South Sikkim – 737 139. 
136
National Institute of Technology
Yupia, District Papum Pare, Arunachal Pradesh – 791112.
137
National Institute of Technology
Meghalaya
138
National Institute of Technology
Nagaland
139
National Institute of Technology
Manipur
140
National Institute of Technology
Mizoram, Chaltlang, Aizawl-796012.
National Institute of Technology
Govt. Polytechnic, Srinagar (Garhwal) – 246174, Uttarkhand
142
National Institute of Technology
Goa
143
National Institute of Technology
Delhi
144
National Institute of Technology
Puducherry
145
Central Board of Secondary Education
2 Community Centre, Preet Vihar New Delhi – 110092
146
National Council for Educational Research and Training (NCERT)
Sri Aurbindo Marg New Delhi – 1100162
147
National Institute of Open Schooling (NIOS)
A-24/25, Institutional Area Sector – 62 NOIDA-201309 Uttar Pradesh
148
Central Tibetan School Administration (CTSA)
Plot No. 1, Community Centre Sector-3, Rohini Delhi – 110085
149
Navodaya Vidyalaya Samiti (NVS)
A-28, Kailash Colony New Delhi – 110048 Sector-3, Rohini Delhi – 110085
150
Kendriya Vidyalaya Sangathan (KVS
18 – Shaheed Jeet Singh Marg New Delhi – 110016
151
Central Institute of Education Technology (CIET)
NCERT Campus Sri Aurbindo Marg New Delhi – 110016
152
National Bal Bhawan
Kotla Road, New Delhi-110002
153
Sardar Vallabhbhai National Institute of Technology
Ichchhanath, SURAT-395 007 (Gujarat)

LINKS TO USEFUL WEBSITES FOR EDUCTION

Links to useful websites

. Ministry Of Education, Government Of India
. National Center for Technology in Education, Dublin City University, Dublin
. Center for Research & Studies on Educational Technology, Open & Distance Education and Teacher Training
. National Council of Teachers of English, USA
. Support for New Teachers Ideas that work
. Education sources for theory, research, policy and practice
. Serving educators with ideas, tools, and resources for integrating technology into classroom and curriculum
. On Line Learning Resources for Teachers and students
. Project Based Learning
. Global School House
. Good Education Research Material
. Better Teachers, Better Schools
. REFORM : VOICE OF A SCHOOL
. US DEPARTMENT OF EDUCATION
. Society for Information Technology and Teacher Education international association of teacher educators who are interested in the use of IT in teacher education
. ASOCIATION OF TEACHER EDUCATORS – TEACHER EDUCATORS SPECIAL INTEREST GROUP
. Society for ADVANCEMENT OF COMPUTING IN EDUCATION
. ISATT – THE INTERNATIONAL STUDY ASSOCIATION ON TEACHERS AND TEACHING
. Teacher Learning Home Page
. GATEWAY TO SEARCH EDUCATION MATERIALS OF THE WEB
. Education Initiatives
. High-Performing, High-Poverty Schools
. DISNEY : Learning Partnership ALL INDIA ASSOCIATION FOR EDUCATIONAL RESEARCH
. A VISIT TO VIGYAN PRASAR (For Daily Science News From India) Read ComCom February Issue : INTERVIEW with Prof. E.V. Chinis (Let Us Reach 50,000 Science Teachers) MASS Higher EDUCATION (but with a difference) By Dr. D.K. Passi
. Indian News Papers
. World News papers   
LINKS TO EDUCATION DEPARTMENTS OF INDIA: