Vector Analysis: A Supplement to Old School Advanced Calculus
by Keith S. Miller With Additions by Karo Maestro!
Kenneth Miller’s wonderfully readable and amazingly
concise primer on modern vector analysis. This brief and
inexpensive text intends to provide an incredibly focused
introduction to vector analysis in R2 and R3.
A “Readers’ Digest” of Vector Analysis
To begin with, the necessity of some
understanding of vector analysis for anyone
studying the hard sciences cannot be disputed. Its’
role in classical mechanics alone would make it
mandatory learning for such students. In fact,
there are so many other roles vector analysis
plays in both pure and applied mathematics that
its’ importance in undergraduate courses cannot
For example, the following are some of the more
obvious areas in pure and applied mathematics
where vector analysis plays a significant role:
Analytic and differential geometry,
modern multivariable calculus, fluid flow, tensor
analysis, multivariable probability and statistics,
electrostatics and electrodynamics,
special relativity, line integrals in complex
analysis, hydrodynamics-the list goes on and on.
But even if one dismissed the enormous range of
applications that vector analysis has, the subject
would still be worth studying on its own merits as
one of the most beautiful branches of
mathematics. There is probably no other
discipline where the connections between
analysis, algebra and geometry in Euclidean
spaces are clearer and more visually expressed
then in vector analysis.
For example, the tangent plane defined at a point
on a surface in three dimensional Euclidean space
is a 2 dimensional vector space consists of all the
tangent vectors to the surface at that point.
A gradient vector at this point is a normal vector
(i.e. the dot product of any tangent vector with a
gradient vector is 0) which points in the direction
of maximal local change at the point in question
on the surface. As a result, a gradient vector can
be generated from any 2 tangent vectors by taking
the cross product. Consequently, this is a branch
of mathematics where careful proof and
geometric intuition go hand in hand.
A Main Course For Vector Calculus Or A Side Dish For Advanced Calculus
Furthermore, it complements the very rigorous
and wonderfully written presentation of classical
analysis in our companion book, Old School
Advanced Calculus by William Benjamin Fite.
Granted that Fite’s book is otherwise very
comprehensive, his presentation of functions of
several variables is rather archaic and purely
analytic. Nevertheless, vector analysis is now a
necessary part of the mathematical training of
both mathematics and physical science students.
Therefore, the absence of modern vector valued
calculus in low dimensional Euclidean spaces in
Fite is a highly problematic void.
Simultaneously, the republishing of this book by
Miller with Fite is specifically intended to rectify
this for both groups of students. In addition,
despite mostly classical language, Miller carefully
connects the material to modern formulations so
he doesn’t alienate pure mathematics majors. For
example, he carefully lays out vector algebra in
the first chapter using the old 19th century
“arrows” language while simultaneously detailing
their algebraic structure as a vector space over
the real or complex numbers. As a result, this
keeps the book’s intended audience very general.
Therefore, this invites not only mathematics
majors, but students of physics, engineering and
other fields that need to either review or learn
In addition, although the book is intended to
supplement Fite, it can certainly be used as a
vector analysis text in its’ own right.
Topics in The Book
• A careful presentation of the algebraic and geometric properties of R2 and R3 from a modern algebra point of view while preserving the classical concrete “arrow” viewpoint
• Scalar and cross product with applications
• Limits and differentiability properties (total derivatives, differentials, partial derivatives, etc.) of functions of 2 and 3 variables, including the Mean Value, Implicit And Inverse Function Theorems in R2 and R3
Further Topics in The Book
• Tangent planes and lines to surfaces
• Gradient, divergence, curl and other important properties of vector valued maps and vector fields
•line and path integrals and Green’s Theorem, change of variables and curvilinear coordinate systems as well as a number of physical applications
• classical differential geometry of curves and surfaces in R2 and R3 with simple applications to mechanics
While Dover Books has made available a number of inexpensive classical books on vector analysis, many of these are quite old fashioned. Therefore, they may be difficult for students to read. Also, most of the current standard books on vector analysis are rather expensive and lengthy. By contrast, Miller gives a brief and clear alternative, particularly for students pressed for time.
Consequently, this book -either by itself or used in conjunction with another text or lecture notes-gives students a very affordable option. Finally, the book’s brevity and low cost make it an indispensable study aid for students who need to learn or review this material quickly and accurately. The hope is that although the book is intended to supplement Fite, it can and should be used as a vector analysis text in its’ own right Indeed, the hope is that because of the book’s brevity and low cost, it will become an indispensable study aid for students who need to either learn or review this material quickly and accurately.
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