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Basic Analysis - Volumes I-V (Hardcover)
Loot Price: R15,428
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Basic Analysis - Volumes I-V (Hardcover)
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Basic Analysis: Volumes I-V is written with the aim of balancing
theory and abstraction with clear explanations and arguments, so
that students and researchers alike who are from a variety of
different areas can follow this text and use it profitably for
self-study. The first volume is designed for students who have
completed the usual calculus and ordinary differential equation
sequence and a basic course in linear algebra. This is a critical
course in the use of abstraction, but is just first volume in a
sequence of courses which prepare students to become practicing
scientists. The second volume focuses on differentiation in
n-dimensions and important concepts about mappings between finite
dimensional Euclidean spaces, such as the inverse and implicit
function theorem and change of variable formulae for
multidimensional integration. These important topics provide
background in important applied and theoretical areas which are no
longer covered in mathematical science curricula. Although it
follows on from the preceding volume, this is a self-contained
book, accessible to undergraduates with a standard course in
undergraduate analysis. The third volume is intended as a first
course in abstract linear analysis. This textbook covers metric
spaces, normed linear spaces and inner product spaces, along with
many other deeper abstract ideas such a completeness, operators and
dual spaces. These topics act as an important tool in the
development of a mathematically trained scientist. The fourth
volume introduces students to concepts from measure theory and
continues their training in the abstract way of looking at the
world. This is a most important skill to have when your life's work
will involve quantitative modeling to gain insight into the real
world. This text generalizes the notion of integration to a very
abstract setting in a variety of ways. We generalize the notion of
the length of an interval to the measure of a set and learn how to
construct the usual ideas from integration using measures. We
discuss carefully the many notions of convergence that measure
theory provides. The final volume introduces graduate students in
science with concepts from topology and functional analysis, both
linear and nonlinear. It is the fifth book in a series designed to
train interested readers how to think properly using mathematical
abstractions, and how to use the tools of mathematical analysis in
applications. It is important to realize that the most difficult
part of applying mathematical reasoning to a new problem domain is
choosing the underlying mathematical framework to use on the
problem. Once that choice is made, we have many tools we can use to
solve the problem. However, a different choice would open up
avenues of analysis from a different, perhaps more productive
perspective. In this volume, the nature of these critical choices
is discussed using applications involving the immune system and
cognition. Features: Can be used as a supplementary text for anyone
whose work requires that they begin to assimilate more abstract
mathematical concepts as part of their professional growth Function
as a traditional textbook as well as a resource for self-study
Suitable for mathematics students and for those in other
disciplines such as biology, physics, and economics and others
requiring a careful and solid grounding in the use of abstraction
in problem solving Emphasizes learning how to understand the
consequences of the underlying assumptions used in building a model
Regularly uses computation tools to help understand abstract
concepts.
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