Vector Analysis Text Here At Blue Collar Scholar: Vector Analysis: A Supplement to Old School Advanced Calculus!

Vector Analysis: A Supplement to Old School Advanced Calculus
by Keith S. Miller With Additions by Karo Maestro! 




%name    Here at Blue Collar Scholar is another unique textbook:

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

be overstated.



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.

        Graphic Knowledge

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.

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

this material.

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.

Preview Pages 13-23

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