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The present book is the first Issue of a Series explaining various
mathematical terms and concepts. It introduces the topics,
definitions, main results and theorems avoiding proofs of the
results. It may serve as a reference book. The first Issue consists
of topics in Mathematical Analysis. The remaining ones covering
topics of other mathematical disciplines will follow in the sequel.
The contents are divided into Sections - numbered chapter wise. The
discussion within the Sections is presented in the form of
Definitions, Theorems, Notes and Examples. These subtitles within
the Sections are numbered in decimal pattern. For instance, the
equation number (c.s.e) refers to the eth equation in the sth
section of Chapter c. When the number c coincides with the chapter
at hand, it is dropped. Adequate references to the previously
quoted results are made in the text avoiding their unnecessary
repetition. For brevity, some set-theoretic notations and symbols
are frequently used, e.g. the symbol means implies. The logarithm
of a number to the exponential base e is denoted by ln. All the
Latin mathematical symbols are normally italicized, while their
Greek counterparts are in normal fonts."
The book offers a second course on Integral Calculus (of functions
of real variables) for Graduate and Engineering students.
Convergence (including uniform convergence) of improper integrals
and various tests of Abel, Dirichlet and Weierstrass for the same
are discussed. Improper integrals of quotient functions of various
forms are evaluated. Integrations of continuous functions of 2
variables are discussed in relation with differential and integral
properties of parameters in the functions. Eulerian integrals: Beta
and Gamma functions, their transformation properties, relations
connecting them, reflection and duplication formulae for the Gamma
function and Frullani's integral are given. Double and triple
integrals giving volume and areas of surfaces are discussed in the
last 3 chapters. Numerous examples are solved illustrating the
methods of change of order of integration. Dirichlet's integrals of
2nd, 3rd and pth orders are evaluated. Transfor- mations of
integrals into 2 and 3-dimensional polar coordinates including
Dirichlet's and Liouville's integrals are given. A short
bibliography of the subject and an alphabetical index are added at
the end.
Astrology has always been a fascinating subject to the mankind and
the Cosmos ever remained a great mystery in spite of having been
widely explored by the scientists. Taking birth in a traditional
Vaishnavite family of India, I am also motivated towards this
subject right since my childhood. Coming in contact with Mr.
Yogendar Nath Dixit, a teacher of Mathematics at Allahabad (India)
with profound knowledge of Indian astrology, in 1963, I learnt many
more interesting characteristics of the subject from him during my
long association with him. It was October 1995 when I was compelled
to give a talk in an "International Conference on Integrated
Systems of Medicine" organized at the Banaras Hindu University,
Varanasi (India) by its organizers, I selected this topic and
prepared a lecture (in English) with due consultations with many
scholars of the subject. Varanasi has been well-known since ages
for its scholars in astrology too. With my long expertise of
mathematical training, I have tried to present here various aspects
of astrology (with special reference to Indian astrology) in brief.
Many explanations are given here in of mathematical tabular form
that makes comprehension of the complex subject easier.
Infinitesimal transformations defining motions, affine motions,
projective motions, conformal transformations and curvature
collineations in various types of Finslerian spaces are discussed
here. The notation and symbolism used is mainly based on 60] and
author's works 24] - 42]. The present article offers an exposition
of the axiomatic definition of tensors and their further
developments from this very standpoint. Various types of tensor
sand their examples have been included. A systematic study of
manifolds endowed with a metric defined by the positive fourth-root
of a 4th degree differential form was considered by P. Finsler in
1918, after whom such manifolds were eventually named. Thereafter,
several geometers: E. Cartan, L. Berwald, J.A. Schouten, J.
Douglas, W. Barthel, H. Rund, A. Lchnerowicz, A. Kawaguchi, H.
Busemann, A. Moor, K. Takano, S.S. Chern, M.S. Knebelman etc.
explored this domain extensively. The first treatise on the subject
(in English) was published by Rund in 1959. Main aspects of the
theory are presented here more elegantly and briefly.
Prayag is the ancient Sanskrit name of Allahabad - a city in North
India, situated on the confluence of three rivers: the Ganges,
Yamuna and Saraswati - the Goddess of Learning. Sant Tulsidas
describes this Prayag as Tirthraj ("King of all Tirthas") in his
classical epic Ram Charit Manas. When the sun crosses the line of
Capricorn all the Tirthas, saints and other Rishis visit the
Tirthraj Prayag. The University there can be regarded as the second
oldest University in the entire Indian sub-continent. The first
three Universities (of Calcutta, Bombay and Madras) were
established at the same time in 1857 by a decree of the British
Government. After three decades, the entire northern jurisdiction
of the University of Calcutta was carved out in the name of the
University of Allahabad on 23rd September, 1887. The author is
privileged to have taught there for more than two decades. Learned
teachers of mathematics at this University contributed a lot for
its academic growth. Their names and their mathematical texts
figure in this reportage. Some of them like Dr. Gorakh Prasad,
Prof. B.N. Prasad and Prof. R.S. Mishra became legendary figures
for their valuable services towards the subject.
The book comprises of 7 expository articles of different nature.
The first one "Metrics of curved surfaces and spaces" deals with
various possibilities of formulaefor measurement of distance
between two infinitesimal points in different geometrical models
especially Finslerian structures and their generalizations.
"Differen- tial geometry: its past and future" describes briefly
the development of the subject. Various techniques used therein are
given. References, some standard books on "Finsler geometry and its
applications" and recent "International conferences on Finsler
geometry" held are also listed. "Physical field theories"
characterizes Finslerian physics and Thermodynamic Finsler spaces.
The smallest number which, on division by a, a + md, a + 2md leaves
the remainders r, r + d, r + 2d respectively; where a, m, r and d
are positive integers is discussed in the article on "Some
properties of numbers." "Mathematics and computer science education
in developing countries" and "Operations research" offer exposition
of the subjects as indicated. Major events in the life of Professor
Albert Einstein are presented in chronological order in the last
article.
Summary 1. Projectively flat Finsler spaces were introduced by
Douglas 2], and studied by Berwald 1] for 2-dimension. Okumura 11]
gave certain properties of projectively flat non-Riemannian spaces
admitting concircular and torse-forming vector fields in
association with symmetric and recurrent properties of their
curvature tensors. Meher 3] studied projective flatness in a
Finsler space when Berwald's curvature tensor is recurrent. He
derived relations connecting the curvature tensor and the
recurrence vector. Pandey 13] derived a necessary and sufficient
condition for the projective flatness of a Finsler space in terms
of its isotropic property. Currently, projective flatness of
Finsler spaces Fn, n > 2, is studied in association with their
sectional curvature, symmetric and recurrent character of their
curvature, and normal projective curvature tensor. The notation and
symbolism used here are mainly based on the works 10] and 14]. 2.
Concircular transformations, introduced by Yano 13] in Riemannian
geometry, were extended by Takano 12] to affine geometry with
recurrent curvature. Okumura 7] extended these to different types
of Riemanna
The book, having eight chapters, offers one semester course (with
four credit hours per week) on diverse topics usually included in
the mathematics syllabus of the first year Engineering class of
most Universities. The first chapter deals with the introductory
topics in 2-dimensional coordinate geometry: straight lines and
conic sections. A brief discussion of theory of equations is given
in the second chapter. Vectors (mostly in 3-dimensional space) are
introduced in the third chapter. Vector algebra and products of
vectors are included. The next chapter deals with the matrices and
determinants. Matrix algebra including multiplication of matrices
is given. Further concepts such as rank of a matrix etc. are
avoided. A comprehensive course on convergence of infinite series
forms the subject mater of Chapter 5. Presuming that the students
already had a first course on calculus only main concepts and
results of differentiation and integration of functions of single
variable have been given in the chapter 6. Power series,
Maclaurin's and Taylor's expansions are explained in detail in the
same chapter.
The present book offers a first course on Complex Analysis. It
deals with the differential and integral properties of functions of
a single complex variable. It includes conformal transformations
and applications of complex analysis to potential theory and flow
problems.The subject matter is presented here in five chapters. It
starts with a historical note for the introduction and subsequent
development of the subject right from the initial stage to the
present modern complex analysis. The first chapter includes the
introduction of complex numbers, their geometric representation, De
Moivre's theorem, roots and logarithm of a complex number and real
and imaginary parts of a complex function.The concepts of
differential calculus for functions of a complex variable are
outlined in the second chapter. It gives the concept of limit and
differentiability of a function. Analytic properties and
Cauchy-Riemann equations form the main subject matter of this
chapter. The third chapter deals with the integration of functions
of a complex variable.
The first chapter deals with the transformations (more precisely
translation, reflection, rotation, dilation) in a plane while
similar transformations in a 3-dimensional Euclidean space are
dealt with in the second chapter. Non-linear, affine and locally
affine transformations and rigid motions are included. Isometries
are introduced in the third chapter. Chapter 4 offers a detailed
account of product transformations which form an algebraic group.
Infinitesimal transformations are introduced in Chapter 5. Cases
when they may form a motion (preserving distances), translation
(displacing each point through a fixed distance), conformal motion
(preserving the angle between two directions), homothetic motion (a
special case of conformal motion) and geodesic preserving
transformations (called projective motions) are dealt therein. The
last chapter defines a projective space in general and projective
transformations: both singular and non-singular are studied. The
chapter concludes with the projective transformations of a
projective space onto itself, called collineations. Certain
properties of collineations are studied.
The book has three chapters and an Appendix on different methods to
calculate approximate areas of plain regions enclosed by curves.
Tutorials, Tests and Examination Papers of different Universities
where the author taught the course at the faculties of Engineering
and Business Studies are provided. A short bibliography of books on
the subject and alphabetical index of the topics covered are given
in the end.The first chapter accounts the different methods for
numerical solutions of ordinary differential equations. Picard's,
Taylor's, Euler's, Runge-Kutta, Milne's and Adams-Bashforth's
methods are given. The problems of curve fittings and spline
fittings are explained in the second chapter. The 'Gaussian method
of least squares' for the 'curve of best fit' is included. The
concepts of Linear Programming are studied in the third chapter.
Solution(s) of linear relations obtained analytically are also
discussed by graphical methods. General problems of Linear
Programming (LPP), canonical and standard forms of LPP and Simplex
methods are discussed. Trapezoidal rule and Simpson's rules for
approximate areas of plain regions are included in the Appendix
The book presents an introduction to mathematics starting right
from the natural numbers. Covering a vast range of topics in
Arithmetic, Algebra, Trigonometry, Coordinate Geometry, Theory of
Equations, Sets, Matrices and Determinants, Solid Geometry,
Calculus (both Differential and Integral), Analysis the book
concludes with a chapter on Complex Numbers. Altogether, it has
eleven chapters. Generally, the proofs are avoided but many results
especially in Algebra, Trigonometry, Coordinate Geometry and
Complex Numbers have been established. Fundamental theorems of
Arithmetic, Algebra and Calculus are also given. The book is not
intended to be a comprehensive text rather it offers a crash course
of basic mathematics to the beginners and as a reference book to
the advanced learners. Infinity ( ) is generally mistaken as a
number or the logarithm of a negative number is just undefined
concept at least to the beginners. These misconceptions are removed
here. Almost all the authors claim r2 as the area of a circle of
radius r (which, indeed, is zero) for circle having length (i.e.
its circumference 2 r) only and is supposed to be of width zero.
None takes pains to call r2 as the area of a circular (plane)
region of radius r.
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