The Vector Analysis Problem Solver (Problem Solvers)

Rating:
Summary: They worked hard at it, but it's unacceptable
Review: This book apparently has two primary goals: to cover a large amount of material and to do so at as low a price as possible. Those two goals are astonishingly well achieved, but the deficiencies are so severe that they render the book unacceptable.

First, the printing is substandard: unadorned Courier or something similar, not right-justified, and is so primitive that the publishers cannot use much of the usual notation. Since vector analysis uses mathematical symbolism heavily, this is a major disadvantage. The same on-the-cheap approach applies to the diagrams. There is no color, and the diagrams are composed almost entirely of compass-and-ruler constructions and notation in the same typeface as the text.

Second, the index is inadequate. This book is likely to be used as a reference rather than read sequentially, so the index is important. This index doesn?t include some of the most fundamental terms or topics, such as "component" or "dimension". There are scores (if not hundreds) of references to derivatives in the book, and yet the index includes only five references to derivatives.

There is not a single reader exercise in the book (unless I missed something). The book is intended to be a demonstration of how Vector Analysis is done, so all the problems are worked out by the authors. If the readers have their own textbooks, this is not necessarily a big disadvantage. But this book, in the introduction, actually criticizes other texts for requiring the students to do exercises, and goes on to make the following stunning statement: "The staff of REA considers vector analysis a subject that is best learned by allowing students to view the methods of analysis and solution techniques themselves."

I can't overemphasize what an incredible statement that is. There are not two schools of thought on this issue. The idea that a student can learn Vector Analysis (or any mathematical topic) by watching someone else do it, makes as much sense as recommending that someone learn to play the guitar by listening to Eric Clapton. That one point of educational philosophy alone demonstrates conclusively that the book is incompetent.

The Problem Solver series is apparently produced with another eccentric belief, and one that is annoying for anyone accustomed to reading traditional math books. The publishers apparently believe (according to their own words in the introduction) that one of the shortcomings of other books is that they present material without first explicitly stating what the goal of each paragraph or section is. (In my experience, this criticism is sometimes valid, but usually not.) Their solution to this is to make every word in the text, part of the statement or solution to a "Problem". However, these are not problems in the usual sense. For example, the second "problem" in the book asks for the dimensions of five vectors which are presented in component form. At that point in the book (page 3), the word "component" has not been defined, no vectors have yet been presented in component form, and the equality between dimension and the number of components has not yet been mentioned. So this is not a "problem" that anyone would be able to solve if his knowledge were limited to what has been presented in the book up to that point. The book usually presents those necessary definitions in the course of "solving" the problem (although in this case, they don't define the word component, and as far as I can see they NEVER define that word. As I mentioned earlier, "component" and "dimension" are both missing from the index.) This entire quirky approach reminds me of that TV game where the contestants are required to respond with a question instead of an answer. It requires a pointlessly awkward presentation just to cram the material into their ideological system. The definitions and formulas are not designated separately, as in other books, but imbedded in the problem "solutions".

Most important, the book is full of mathematical errors. Page 22 refers to "vectors of the same magnitude" when they mean "vectors of the same dimension". Page 23 has a problem involving vector addition and subtraction, in which they present a1, b1, c1, a2, b2, and c2 as components of vectors (using subscripts), and proceed to use a1c1 as the DIFFERENCE between those two components (or the distance between the terminal points of the horizontal projections of vectors a and c, which is the same thing). Of course, they should have used c1 - a1. This is not just non-standard notation; it is simply mathematically wrong. And I believe they repeat that same error in other problems later on.

In proving an important theorem (the equivalence of the algebraic and geometric/trigonometric definitions of dot products), they suddenly pull out a formula from analytic geometry that very few calculus students in the US will ever have seen. Since the use of that formula renders the rest of the proof entirely trivial, the student will learn nothing at all about how to prove theorems from reading that proof. The same sort of thing happens in Problem 3-6, where they use a formula for finding the components of a projection - a formula which has never been mentioned before, and which they neither derive nor explain.

The book constantly rounds off intermediate results. This should never be done in these days of 12-digit calculators, and ESPECIALLY when the calculated number is to be used as input into an inverse trigonometric or exponential function. In fact, they don't even really round those numbers off - they just truncate them! This causes their final answers to be wrong in many cases, such as in Problem 3-11, where the answer should be the vector (0.279, 0.885, -0.373). This is amateurish. If the authors don't know better than that then they shouldn't be writing math books.

After reading "Problem" 3-16 on pages 81-82, I finally gave up entirely on this book. The problem statement asks for the cross product of two vectors, while sort of half-defining the cross product within the problem statement itself. Then, in the "solution", they don't find the cross-product at all! They find only the magnitude! They simply forgot to finish the problem!

This book was never competently proofread, peer-reviewed, or edited. Presumably that would have been too expensive. But this amateurish production is unusable for learning Vector Calculus. Not recommended.

Rating:
Summary: They worked hard at it, but it's unacceptable
Review: This book apparently has two primary goals: to cover a large amount of material and to do so at as low a price as possible. Those two goals are astonishingly well achieved, but the deficiencies are so severe that they render the book unacceptable.

First, the printing is substandard: unadorned Courier or something similar, not right-justified, and is so primitive that the publishers cannot use much of the usual notation. Since vector analysis uses mathematical symbolism heavily, this is a major disadvantage. The same on-the-cheap approach applies to the diagrams. There is no color, and the diagrams are composed almost entirely of compass-and-ruler constructions and notation in the same typeface as the text.

Second, the index is inadequate. This book is likely to be used as a reference rather than read sequentially, so the index is important. This index doesn?t include some of the most fundamental terms or topics, such as "component" or "dimension". There are scores (if not hundreds) of references to derivatives in the book, and yet the index includes only five references to derivatives.

There is not a single reader exercise in the book (unless I missed something). The book is intended to be a demonstration of how Vector Analysis is done, so all the problems are worked out by the authors. If the readers have their own textbooks, this is not necessarily a big disadvantage. But this book, in the introduction, actually criticizes other texts for requiring the students to do exercises, and goes on to make the following stunning statement: "The staff of REA considers vector analysis a subject that is best learned by allowing students to view the methods of analysis and solution techniques themselves."

I can't overemphasize what an incredible statement that is. There are not two schools of thought on this issue. The idea that a student can learn Vector Analysis (or any mathematical topic) by watching someone else do it, makes as much sense as recommending that someone learn to play the guitar by listening to Eric Clapton. That one point of educational philosophy alone demonstrates conclusively that the book is incompetent.

The Problem Solver series is apparently produced with another eccentric belief, and one that is annoying for anyone accustomed to reading traditional math books. The publishers apparently believe (according to their own words in the introduction) that one of the shortcomings of other books is that they present material without first explicitly stating what the goal of each paragraph or section is. (In my experience, this criticism is sometimes valid, but usually not.) Their solution to this is to make every word in the text, part of the statement or solution to a "Problem". However, these are not problems in the usual sense. For example, the second "problem" in the book asks for the dimensions of five vectors which are presented in component form. At that point in the book (page 3), the word "component" has not been defined, no vectors have yet been presented in component form, and the equality between dimension and the number of components has not yet been mentioned. So this is not a "problem" that anyone would be able to solve if his knowledge were limited to what has been presented in the book up to that point. The book usually presents those necessary definitions in the course of "solving" the problem (although in this case, they don't define the word component, and as far as I can see they NEVER define that word. As I mentioned earlier, "component" and "dimension" are both missing from the index.) This entire quirky approach reminds me of that TV game where the contestants are required to respond with a question instead of an answer. It requires a pointlessly awkward presentation just to cram the material into their ideological system. The definitions and formulas are not designated separately, as in other books, but imbedded in the problem "solutions".

Most important, the book is full of mathematical errors. Page 22 refers to "vectors of the same magnitude" when they mean "vectors of the same dimension". Page 23 has a problem involving vector addition and subtraction, in which they present a1, b1, c1, a2, b2, and c2 as components of vectors (using subscripts), and proceed to use a1c1 as the DIFFERENCE between those two components (or the distance between the terminal points of the horizontal projections of vectors a and c, which is the same thing). Of course, they should have used c1 - a1. This is not just non-standard notation; it is simply mathematically wrong. And I believe they repeat that same error in other problems later on.

In proving an important theorem (the equivalence of the algebraic and geometric/trigonometric definitions of dot products), they suddenly pull out a formula from analytic geometry that very few calculus students in the US will ever have seen. Since the use of that formula renders the rest of the proof entirely trivial, the student will learn nothing at all about how to prove theorems from reading that proof. The same sort of thing happens in Problem 3-6, where they use a formula for finding the components of a projection - a formula which has never been mentioned before, and which they neither derive nor explain.

The book constantly rounds off intermediate results. This should never be done in these days of 12-digit calculators, and ESPECIALLY when the calculated number is to be used as input into an inverse trigonometric or exponential function. In fact, they don't even really round those numbers off - they just truncate them! This causes their final answers to be wrong in many cases, such as in Problem 3-11, where the answer should be the vector (0.279, 0.885, -0.373). This is amateurish. If the authors don't know better than that then they shouldn't be writing math books.

After reading "Problem" 3-16 on pages 81-82, I finally gave up entirely on this book. The problem statement asks for the cross product of two vectors, while sort of half-defining the cross product within the problem statement itself. Then, in the "solution", they don't find the cross-product at all! They find only the magnitude! They simply forgot to finish the problem!

This book was never competently proofread, peer-reviewed, or edited. Presumably that would have been too expensive. But this amateurish production is unusable for learning Vector Calculus. Not recommended.