**Vector Analysis: A Supplement to Old School Advanced Calculus**

**by Keith S. Miller With Additions by Karo Maestro! **

**Vector Analysis: A Supplement to Old School Advanced Calculus**

#### **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. T****here 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 *

*Old School*

*Advanced Calculus* by William Benjamin Fite.

*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**

**Buy The Paperback Version At:**

## Amazon

**For The Ebook Version:**

**Kindle **