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The book aims to give a mathematical presentation of the theory of
general relativity (that is, spacetime-geometry-based gravitation
theory) to advanced undergraduate mathematics students.
Mathematicians will find spacetime physics presented in the
definition-theorem-proof format familiar to them. The given precise
mathematical definitions of physical notions help avoiding
pitfalls, especially in the context of spacetime physics describing
phenomena that are counter-intuitive to everyday experiences.In the
first part, the differential geometry of smooth manifolds, which is
needed to present the spacetime-based gravitation theory, is
developed from scratch. Here, many of the illustrating examples are
the Lorentzian manifolds which later serve as spacetime models.
This has the twofold purpose of making the physics forthcoming in
the second part relatable, and the mathematics learnt in the first
part less dry. The book uses the modern coordinate-free language of
semi-Riemannian geometry. Nevertheless, to familiarise the reader
with the useful tool of coordinates for computations, and to bridge
the gap with the physics literature, the link to coordinates is
made through exercises, and via frequent remarks on how the two
languages are related.In the second part, the focus is on physics,
covering essential material of the 20th century spacetime-based
view of gravity: energy-momentum tensor field of matter, field
equation, spacetime examples, Newtonian approximation, geodesics,
tests of the theory, black holes, and cosmological models of the
universe.Prior knowledge of differential geometry or physics is not
assumed. The book is intended for self-study, and the solutions to
the (over 200) exercises are included.
The book aims to give a mathematical presentation of the theory of
general relativity (that is, spacetime-geometry-based gravitation
theory) to advanced undergraduate mathematics students.
Mathematicians will find spacetime physics presented in the
definition-theorem-proof format familiar to them. The given precise
mathematical definitions of physical notions help avoiding
pitfalls, especially in the context of spacetime physics describing
phenomena that are counter-intuitive to everyday experiences.In the
first part, the differential geometry of smooth manifolds, which is
needed to present the spacetime-based gravitation theory, is
developed from scratch. Here, many of the illustrating examples are
the Lorentzian manifolds which later serve as spacetime models.
This has the twofold purpose of making the physics forthcoming in
the second part relatable, and the mathematics learnt in the first
part less dry. The book uses the modern coordinate-free language of
semi-Riemannian geometry. Nevertheless, to familiarise the reader
with the useful tool of coordinates for computations, and to bridge
the gap with the physics literature, the link to coordinates is
made through exercises, and via frequent remarks on how the two
languages are related.In the second part, the focus is on physics,
covering essential material of the 20th century spacetime-based
view of gravity: energy-momentum tensor field of matter, field
equation, spacetime examples, Newtonian approximation, geodesics,
tests of the theory, black holes, and cosmological models of the
universe.Prior knowledge of differential geometry or physics is not
assumed. The book is intended for self-study, and the solutions to
the (over 200) exercises are included.
The book constitutes a basic, concise, yet rigorous first course in
complex analysis, for undergraduate students who have studied
multivariable calculus and linear algebra. The textbook should be
particularly useful for students of joint programmes with
mathematics, as well as engineering students seeking rigour. The
aim of the book is to cover the bare bones of the subject with
minimal prerequisites. The core content of the book is the three
main pillars of complex analysis: the Cauchy-Riemann equations, the
Cauchy Integral Theorem, and Taylor and Laurent series. Each
section contains several problems, which are not drill exercises,
but are meant to reinforce the fundamental concepts. Detailed
solutions to all the 243 exercises appear at the end of the book,
making the book ideal for self-study. There are many figures
illustrating the text.The second edition corrects errors from the
first edition, and includes 89 new exercises, some of which cover
auxiliary topics that were omitted in the first edition. Two new
appendices have been added, one containing a detailed rigorous
proof of the Cauchy Integral Theorem, and another providing
background in real analysis needed to make the book self-contained.
The book constitutes a basic, concise, yet rigorous first course in
complex analysis, for undergraduate students who have studied
multivariable calculus and linear algebra. The textbook should be
particularly useful for students of joint programmes with
mathematics, as well as engineering students seeking rigour. The
aim of the book is to cover the bare bones of the subject with
minimal prerequisites. The core content of the book is the three
main pillars of complex analysis: the Cauchy-Riemann equations, the
Cauchy Integral Theorem, and Taylor and Laurent series. Each
section contains several problems, which are not drill exercises,
but are meant to reinforce the fundamental concepts. Detailed
solutions to all the 243 exercises appear at the end of the book,
making the book ideal for self-study. There are many figures
illustrating the text.The second edition corrects errors from the
first edition, and includes 89 new exercises, some of which cover
auxiliary topics that were omitted in the first edition. Two new
appendices have been added, one containing a detailed rigorous
proof of the Cauchy Integral Theorem, and another providing
background in real analysis needed to make the book self-contained.
'The book is unusual among functional analysis books in devoting a
lot of space to the derivative. The aEURO~friendlyaEURO (TM) aspect
promised in the title is not explained, but there are three things
I think would strike most students as friendly: the slow pace, the
enormous number of examples, and complete solutions to all the
exercises.'MAA ReviewsThis book constitutes a concise introductory
course on Functional Analysis for students who have studied
calculus and linear algebra. The topics covered are Banach spaces,
continuous linear transformations, Frechet derivative, geometry of
Hilbert spaces, compact operators, and distributions. In addition,
the book includes selected applications of functional analysis to
differential equations, optimization, physics (classical and
quantum mechanics), and numerical analysis. The book contains 197
problems, meant to reinforce the fundamental concepts. The
inclusion of detailed solutions to all the exercises makes the book
ideal also for self-study.A Friendly Approach to Functional
Analysis is written specifically for undergraduate students of pure
mathematics and engineering, and those studying joint programmes
with mathematics.
'The book is unusual among functional analysis books in devoting a
lot of space to the derivative. The aEURO~friendlyaEURO (TM) aspect
promised in the title is not explained, but there are three things
I think would strike most students as friendly: the slow pace, the
enormous number of examples, and complete solutions to all the
exercises.'MAA ReviewsThis book constitutes a concise introductory
course on Functional Analysis for students who have studied
calculus and linear algebra. The topics covered are Banach spaces,
continuous linear transformations, Frechet derivative, geometry of
Hilbert spaces, compact operators, and distributions. In addition,
the book includes selected applications of functional analysis to
differential equations, optimization, physics (classical and
quantum mechanics), and numerical analysis. The book contains 197
problems, meant to reinforce the fundamental concepts. The
inclusion of detailed solutions to all the exercises makes the book
ideal also for self-study.A Friendly Approach to Functional
Analysis is written specifically for undergraduate students of pure
mathematics and engineering, and those studying joint programmes
with mathematics.
The book constitutes an elementary course on Plane Euclidean
Geometry, pitched at pre-university or at advanced high school
level. It is a concise book treating the subject axiomatically, but
since it is meant to be a first introduction to the subject,
excessive rigour is avoided, making it appealing to a younger
audience as well. The aim is to cover the basics of the subject,
while keeping the subject lively by means of challenging and
interesting exercises. This makes it relevant also for students
participating in mathematics circles and in mathematics
olympiads.Each section contains several problems, which are not
purely drill exercises, but are intended to introduce a sense of
'play' in mathematics, and inculcate appreciation of the elegance
and beauty of geometric results. There is an abundance of colour
pictures illustrating results and their proofs. A section on hints
and a further section on detailed solutions to all the exercises
appear at the end of the book, making the book ideal also for
self-study.
The book constitutes an elementary course on Plane Euclidean
Geometry, pitched at pre-university or at advanced high school
level. It is a concise book treating the subject axiomatically, but
since it is meant to be a first introduction to the subject,
excessive rigour is avoided, making it appealing to a younger
audience as well. The aim is to cover the basics of the subject,
while keeping the subject lively by means of challenging and
interesting exercises. This makes it relevant also for students
participating in mathematics circles and in mathematics
olympiads.Each section contains several problems, which are not
purely drill exercises, but are intended to introduce a sense of
'play' in mathematics, and inculcate appreciation of the elegance
and beauty of geometric results. There is an abundance of colour
pictures illustrating results and their proofs. A section on hints
and a further section on detailed solutions to all the exercises
appear at the end of the book, making the book ideal also for
self-study.
The book constitutes a basic, concise, yet rigorous course in
complex analysis, for students who have studied calculus in one and
several variables, but have not previously been exposed to complex
analysis. The textbook should be particularly useful and relevant
for undergraduate students in joint programmes with mathematics, as
well as engineering students. The aim of the book is to cover the
bare bones of the subject with minimal prerequisites. The core
content of the book is the three main pillars of complex analysis:
the Cauchy-Riemann equations, the Cauchy Integral Theorem, and
Taylor and Laurent series expansions.Each section contains several
problems, which are not purely drill exercises, but are rather
meant to reinforce the fundamental concepts. Detailed solutions to
all the exercises appear at the end of the book, making the book
ideal also for self-study. There are many figures illustrating the
text.
The book constitutes a basic, concise, yet rigorous course in
complex analysis, for students who have studied calculus in one and
several variables, but have not previously been exposed to complex
analysis. The textbook should be particularly useful and relevant
for undergraduate students in joint programmes with mathematics, as
well as engineering students. The aim of the book is to cover the
bare bones of the subject with minimal prerequisites. The core
content of the book is the three main pillars of complex analysis:
the Cauchy-Riemann equations, the Cauchy Integral Theorem, and
Taylor and Laurent series expansions.Each section contains several
problems, which are not purely drill exercises, but are rather
meant to reinforce the fundamental concepts. Detailed solutions to
all the exercises appear at the end of the book, making the book
ideal also for self-study. There are many figures illustrating the
text.
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