Mathematical Formaulae handbook

Mathematical Formaulae handbook

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Introduction............................................................................................ 1 Bibliography; PhysicalConstants 1. Series.................................................................................................... 2 Arithmeticand Geometricprogressions; Convergenceof series: theratio test; Convergence ofseries: the comparisontest;Binomialexpansion; Taylorand Maclaurin Series; Power series withreal variables; Integer series; Plane waveexpansion 2. Vector Algebra......................................................................................... 3 Scalar product;Equation of aline; Equationof a plane; Vector product; Scalar tripleproduct; Vector tripleproduct;Non-orthogonalbasis;Summationconvention 3. Matrix Algebra ........................................................................................ 5 Unitmatrices;Products; Transpose matrices; Inverse matrices; Determinants; 2
×
2matrices; Productrules; Orthogonalmatrices;Solvingsets oflinear simultaneous equations;Hermitian matrices; Eigenvalues andeigenvectors;Commutators;Hermitianalgebra;Pauli spin matrices 4. Vector Calculus........................................................................................ 7 Notation;Identities;Grad,Div,Curl andthe Laplacian; Transformationof integrals 5. Complex Variables.................................................................................... 9 Complexnumbers; DeMoivre’s theorem;Power series for complexvariables. 6. Trigonometric Formulae............................................................................ 10 Relations between sides and angles ofany planetriangle; Relations between sides and angles ofany sphericaltriangle 7. Hyperbolic Functions ............................................................................... 11 Relations of thefunctions; Inverse functions 8. Limits.................................................................................................. 12 9. Differentiation........................................................................................ 13 10. Integration............................................................................................ 13 Standardforms;Standardsubstitutions;Integrationby parts;Differentiationof an integral; Dirac d -‘function’;Reduction formulae 11. Differential Equations............................................................................... 16 Diffusion (conduction)equation;Wave equation;Legendre’s equation; Bessel’s equation; Laplace’s equation; Sphericalharmonics 12. Calculus of Variations............................................................................... 17 13. Functions of Several Variables..................................................................... 18 Taylorseries for twovariables;Stationarypoints;Changingvariables: the chainrule; Changingvariables in surface andvolumeintegrals – Jacobians 14. Fourier Series and Transforms..................................................................... 19 Fourier series; Fourier series for otherranges;Fourier series for oddandeven functions; Complexformof Fourier series;Discrete Fourier series; Fourier transforms;Convolution theorem; Parseval’s theorem; Fourier transforms in twodimensions; Fourier transforms in threedimensions 15. LaplaceTransforms.................................................................................. 23 16. Numerical Analysis ................................................................................. 24 Findingthezeros of equations;Numerical integrationof differential equations; Central difference notation;Approximatingtoderivatives;Interpolation: Everett’sformula; Numericalevaluation ofde?nite integrals 17. Treatment of Random Errors....................................................................... 25 Range method; Combination oferrors 18. Statistics............................................................................................... 26 Mean and Variance;Probability distributions; Weightedsums of randomvariables; Statisticsof adatasample x1,...,xn;Regression (leastsquares ?tting)
Introduction
This MathematicalFormaulae handbook has been preparedin response to a request from the Physics Consultative Committee,with thehopethatitwillbeusefultothosestudyingphysics. Itistosomeextentmodelledonasimilar document issued by the Department of Engineering, but obviously re?ects the particular interests of physicists. Therewas discussion as to whether it should also include physical formulae such as Maxwell’s equations, etc., but a decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainly because,in its presentform, cleancopies canbe madeavailableto candidatesin exams. There has been wide consultation among the staff about the contents of this document, but inevitably some users will seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments to the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. The Secretarywill also begratefulto be informed ofany (equally inevitable)errorswhich arefound. This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by Dr DaveGreen,using theTEX typesetting package. Version 1.5December 2005.
Bibliography Abramowitz,M.&Stegun, I.A.,Handbook of MathematicalFunctions, Dover,1965. Gradshteyn, I.S.& Ryzhik, I.M.,Tableof Integrals,SeriesandProducts, AcademicPress,1980. Jahnke,E.& Emde,F., Tablesof Functions, Dover,1986. Nordling, C.& ¨Osterman, J.,Physics Handbook, Chartwell-Bratt,Bromley,1980. Speigel,M.R.,MathematicalHandbook ofFormulas andTables. (Schaum’sOutline Series,McGraw-Hill,1968).
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Foreign Affairs.apk

FOREIGN AFFAIRS ANDROID V.

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AN FP SPECIAL REPORT: In May, FOREIGN POLICY’s Africa editor, Ty McCormick, traveled with photojournalist Nichole Sobecki to Agadez, Niger, an age-old trading post and a gateway to the Sahara that has become the epicenter of the modern human- smuggling trade. Their story, “Highway Through Hell,” shows how European e?orts to halt migration through Niger have only made the journey more perilous for those who attempt it. “It’s clear from our conversations with smugglers that the EU-funded crackdown hasn’t stopped the ?ow of people,” McCormick says. “It’s just pushed it into remote parts of the desert where there is no water and no margin for error. If you break down out there, you die. And so will all of your passengers.”
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Benazir Bhutto! DAUGHTER OF DESTINY (East)

DAUGHTER OF  DESTINY 

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I have always believed in the importance of historic record. When the government of my father, Zulfikar Ali Bhutto, was overthrown in 1977, I encouraged many of those who had worked closely with him to write about the Bhutto period. But in the difficult years of Martial Law that followed in Pakistan, many from my father's government were busy fighting the persecution and the false cases brought against them by the military regime. Others had fled into exile, and had no access to their personal papers. My own involvement in the struggle to return democracy to Pakistan, and the years I consequently spent imprisoned without charge, made it impossible for me to write a book on my father's government myself. 
  More than a million of my countrymen came out to greet me when I returned to Pakistan from two years of exile in April 1986, catapulting me into the glare of international publicity. Suddenly I received several offers to write not my father's story, but my own. I hesitated. It was one matter to write about my father, who had become the democratically elected prime minister of Pakistan and had lasting achievements to his name, and quite another to write about myself, whose most important political battles were still to be fought. It seemed presumptuous. I thought autobiographies were written in the autumn of one's life, looking back. 
  A friend's chance remark changed my mind. "What is not recorded is not remembered," she told me. I saw her point. Like many in Pakistan, I had experienced the dark years of Martial Law. Unlike many, I had the opportunity to put those experiences on record. It is important that the world remember the repression we in Pakistan had to bear following General Zia's coup d'etat. 
  Writing the book has been difficult. It has meant reliving the pain of the past. But it has also been cathartic, forcing me for the first time to come to terms with memories I had been trying to escape. 
  This is my story, events as I saw them, felt them, reacted to them. It is not an in-depth study of Pakistan, but a glance into the transformation of a society from democracy to dictatorship. Let it also be a call for freedom. 
Benazir Bhutto June 1988  Karachi, Pakistan.....More Read.....

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Zulfikar Ali Bhutto ! If i am Assassinated.apk

Zulfikar Ali Bhutto ! If i am Assassinated.

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Introduction   :

“More than My Life Is At Stake”       Five Omens
Rao Abdur Rashid, a former Director of the Intelligence Bureau of Pakistan, has reported to the Supreme Court of Pakistan that Brigadier Abdul Naeem, a trusted member of the military group which rules Pakistan had the following conversation with him one day about Zulfikar Ali Bhutto, President and Prime Minister of Pakistan until he was deposed in a military coup , put into jail and sentenced to death:
Brigadier Abdul Naeem: “Do you think that the army can afford to see Mr. Bhutto back in power?” 
Rao Abdur Rashid, a trusted Special Secretary to Bhutto, keeps a discreet silence.
Brigadier Naeem answers his own question and says:
“Obviously it can’t”.
Before departing, Brigadier Naeem advises Rao Abdur Rashid:
“Please cooperate with the army”.
This conversation sums up the dire threat that hangs on the head of Mr. Bhutto “in this narrow, dark, sinking death cell, where I am confined 22 or 23 hours every day.” This is how Bhutto describes what has been his residence for almost a year in the District Jail, Rawalpindi............... Read........

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