What do you think?
Rate this book
About Vectors
No calculus needed, but this is not an elementary book. Introduces vectors, algebraic notation and basic ideas, vector algebra and scalars. Covers areas of parallelograms, triple products, moments, angular velocity, areas and vectorial addition, and more. Concludes with discussion of tensors. Includes 386 exercises.
144 pages, Paperback
First published June 1, 1975
6 people are currently reading
59 people want to read
About the author
Banesh Hoffmann
26 books10 followersBanesh Hoffmann studied mathematics and theoretical physics at the University of Oxford, where he earned his bachelor of arts and went on to earn his doctorate at Princeton University.
While at the Institute for Advanced Study in Princeton, Hoffmann collaborated with Einstein and Leopold Infeld on the classic paper Gravitational Equations and the Problem of Motion. Einstein’s original work on general relativity was based on two ideas. The first was the equation of motion: a particle would follow the shortest path in four-dimensional space-time. The second was how matter affects the geometry of space-time. What Einstein, Infeld, and Hoffmann showed was that the equation of motion followed directly from the field equation that defined the geometry (see main article).
In 1937 Hoffmann joined the mathematics department of Queens College, part of the City University of New York, where he remained till the late 1970s. He retired in the 1960s but continued to teach one course a semester — in the fall a course on quantum mechanics and in the spring one on the special and general theories of relativity.
He was a member of the Baker Street Irregulars and wrote the short story "Sherlock, Shakespeare, and the Bomb," published in Ellery Queen Mystery Magazine in February 1966.
While at the Institute for Advanced Study in Princeton, Hoffmann collaborated with Einstein and Leopold Infeld on the classic paper Gravitational Equations and the Problem of Motion. Einstein’s original work on general relativity was based on two ideas. The first was the equation of motion: a particle would follow the shortest path in four-dimensional space-time. The second was how matter affects the geometry of space-time. What Einstein, Infeld, and Hoffmann showed was that the equation of motion followed directly from the field equation that defined the geometry (see main article).
In 1937 Hoffmann joined the mathematics department of Queens College, part of the City University of New York, where he remained till the late 1970s. He retired in the 1960s but continued to teach one course a semester — in the fall a course on quantum mechanics and in the spring one on the special and general theories of relativity.
He was a member of the Baker Street Irregulars and wrote the short story "Sherlock, Shakespeare, and the Bomb," published in Ellery Queen Mystery Magazine in February 1966.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 3 of 3 reviews
December 1, 2022
I give a fair warning to readers of this review that I usually have a bit of a bias against Dover books. While they make some old texts available again, I find most of the titles overly dense in the grand scheme of things. There are certainly some gems out there, but, personally, I find it is a sea of confusion to find the one gem that is readable in modern standards. However, despite this, I was rather excited to check out "About Vectors," which is a recommendation I stumbled upon on The Math Sorceror's YouTube channel. Whom I find is a great resource for those interested in mathematics, especially if you feel you need some inspiration for doing math.
Now, I came at this book knowing a decent amount about vectors. However, one subject seems to have eluded me in all of my schooling and I've been looking for that hidden gem to teach myself Tensors for a while now. The school I attended seems rather phobic when it comes to breaching the topics of Differential Geometry and thus there was nothing offered, so I was very much on my own to read about this material if I wanted to learn it. However, most textbooks are not written with "self learning" in mind, which will often require more examples than a generic textbook, which relies on an instructor to fill in details or motivate students with more examples of ideas.
If your intent is to teach yourself about the nature of vectors steer clear of this book. This is not the authors intent at all and he says so within the first few pages. This book is written as a companion to learning the subject of vectors in a math class. It seems Hoffmann was rather dissatisfied with the presentation of vectors at his time writing and sought to rectify the situation with this book. He tries to motivate the reader asking questions about "what does magnitude really mean?" and other more vague descriptions. I think he does a good job presenting what types of objects can be combined versus which can't in a vector space, and he is correct, even in the modern day maths texts we don't do an exceptional job of talking about some of these nuances in the applied realms. However, I will say a good deal of his discussion centered around applied situations, which are things mathematics texts just don't care about when developing their theory. In maths we just say things like "the length of this vector is..." we don't even care what kind of vector, what forces, what displacements etc. are being discussed. We usually leave it up to the applied teachers/science teachers to hammer away at peoples notions of types and units.
That being said, I do quite prefer the modern treatment of vectors. We must remember this book was originally written in 1966, so while we refer to things like "vector u" or "vector v" he will use notation like "vector AB" or "vector BC" and the like. I find our newer notation to be far more condensed in that regard and easier to follow/read. Aside from that, his abstract overview of how vectors work is pretty good, but it truly does act as a supplement to other texts on vectors. Such that, I don't think I would have been able to read this without first knowing something about vectors at least a little bit.
One of the things I really did not like about the book is that Hoffmann spends a rather inordinate amount of time going over physics problems and the types of physics problems you can do. While I suppose this is all somewhat useful, it just seems like you could leave this to the physicists to discuss in detail. It's been nearly a decade since I've seen some of this stuff and I don't remember a good deal of it, and Hoffmann's ultra brief overview of these vector situations did not clarify anything really. He also avoids bringing up any calculus in this book, which is rather commendable, until we get to the physics parts of the book. Where, frankly, talking about the motion of these objects is really hard to understand without calculus. So, on the one hand, it is nice that you only need to know some vector algebra to do this book, but on the other... the physics can be hard to follow since there is no calculus to help make more sense of the situations.
The other part I found disappointing is really the sections on why I bought this book. In one part he discusses the possibility of creating a division of vectors. He brings up the fact that Hamilton's quaternions might do this, but then hand waves it away as being beyond the book. He shows a few examples where we can't divide, but does not tell us much about when we can or what it means. Why not just say "division of vectors is beyond the scope of this book"? The major reason I picked this up was the last chapter on Tensors. Now in this regard, I definitely learned more than I walked in with. However, I cannot say I truly understood it. A huge portion of this was spent on just trying to explain the notation as briefly as possible, which makes sense in a small book. The real problem is that Tensors are not really taught as their own stand alone thing, so I'm not sure what he's supplementing, unless there is a vectors course that was offered in the 60's when he was actively teaching that just doesn't exist anywhere anymore? I imagine at one time courses on Vector Analysis were rather common and these are still taught to a certain degree, but I notice most curriculums tend to sift the vector analysis throughout a bunch of courses rather than having a single dedicated course to the topic... and in that they may not even get to Tensors! Alas, my search for Tensors will remain in effect. I am not very interested in a brief overview on the topic. I would like to know and be able to do challenging problems, in that regard online resources are usually quite scant.
In any event, if you are interested in this book, just be aware that this book is not written as a "first book on vectors" it's written more as a "second book on vectors". It has some interesting material, but I just feel that more modern titles will cover this and do a better job of it. So, in that regard the book feels rather dated, which, of course, it is. That was my take away at the end of the day.
Now, I came at this book knowing a decent amount about vectors. However, one subject seems to have eluded me in all of my schooling and I've been looking for that hidden gem to teach myself Tensors for a while now. The school I attended seems rather phobic when it comes to breaching the topics of Differential Geometry and thus there was nothing offered, so I was very much on my own to read about this material if I wanted to learn it. However, most textbooks are not written with "self learning" in mind, which will often require more examples than a generic textbook, which relies on an instructor to fill in details or motivate students with more examples of ideas.
If your intent is to teach yourself about the nature of vectors steer clear of this book. This is not the authors intent at all and he says so within the first few pages. This book is written as a companion to learning the subject of vectors in a math class. It seems Hoffmann was rather dissatisfied with the presentation of vectors at his time writing and sought to rectify the situation with this book. He tries to motivate the reader asking questions about "what does magnitude really mean?" and other more vague descriptions. I think he does a good job presenting what types of objects can be combined versus which can't in a vector space, and he is correct, even in the modern day maths texts we don't do an exceptional job of talking about some of these nuances in the applied realms. However, I will say a good deal of his discussion centered around applied situations, which are things mathematics texts just don't care about when developing their theory. In maths we just say things like "the length of this vector is..." we don't even care what kind of vector, what forces, what displacements etc. are being discussed. We usually leave it up to the applied teachers/science teachers to hammer away at peoples notions of types and units.
That being said, I do quite prefer the modern treatment of vectors. We must remember this book was originally written in 1966, so while we refer to things like "vector u" or "vector v" he will use notation like "vector AB" or "vector BC" and the like. I find our newer notation to be far more condensed in that regard and easier to follow/read. Aside from that, his abstract overview of how vectors work is pretty good, but it truly does act as a supplement to other texts on vectors. Such that, I don't think I would have been able to read this without first knowing something about vectors at least a little bit.
One of the things I really did not like about the book is that Hoffmann spends a rather inordinate amount of time going over physics problems and the types of physics problems you can do. While I suppose this is all somewhat useful, it just seems like you could leave this to the physicists to discuss in detail. It's been nearly a decade since I've seen some of this stuff and I don't remember a good deal of it, and Hoffmann's ultra brief overview of these vector situations did not clarify anything really. He also avoids bringing up any calculus in this book, which is rather commendable, until we get to the physics parts of the book. Where, frankly, talking about the motion of these objects is really hard to understand without calculus. So, on the one hand, it is nice that you only need to know some vector algebra to do this book, but on the other... the physics can be hard to follow since there is no calculus to help make more sense of the situations.
The other part I found disappointing is really the sections on why I bought this book. In one part he discusses the possibility of creating a division of vectors. He brings up the fact that Hamilton's quaternions might do this, but then hand waves it away as being beyond the book. He shows a few examples where we can't divide, but does not tell us much about when we can or what it means. Why not just say "division of vectors is beyond the scope of this book"? The major reason I picked this up was the last chapter on Tensors. Now in this regard, I definitely learned more than I walked in with. However, I cannot say I truly understood it. A huge portion of this was spent on just trying to explain the notation as briefly as possible, which makes sense in a small book. The real problem is that Tensors are not really taught as their own stand alone thing, so I'm not sure what he's supplementing, unless there is a vectors course that was offered in the 60's when he was actively teaching that just doesn't exist anywhere anymore? I imagine at one time courses on Vector Analysis were rather common and these are still taught to a certain degree, but I notice most curriculums tend to sift the vector analysis throughout a bunch of courses rather than having a single dedicated course to the topic... and in that they may not even get to Tensors! Alas, my search for Tensors will remain in effect. I am not very interested in a brief overview on the topic. I would like to know and be able to do challenging problems, in that regard online resources are usually quite scant.
In any event, if you are interested in this book, just be aware that this book is not written as a "first book on vectors" it's written more as a "second book on vectors". It has some interesting material, but I just feel that more modern titles will cover this and do a better job of it. So, in that regard the book feels rather dated, which, of course, it is. That was my take away at the end of the day.
October 28, 2011
Here is how this book ends. Quite a thrilling conclusion if you ask me!
10. WHAT THEN IS A VECTOR?
This being a book about vectors, we have presented only the sketchiest account of tensors-- barely enough to illustrate the advantages of thinking of vectors in terms of the way their components transform.
We have one final point to make. Notice that we defined contravariant vectors and covariant vectors-- indeed, tensors of all ranks--before we introduced the metrical tensor. Suppose there were no metrical tensor. What could we then say about the magnitudes of vectors? Or about the cosines of the angles between them?
You may be tempted to argue that such questions prove that there has to be a metrical tensor. But actually there does not. Mathematicians often work with spaces that do not possess one; they call them nonmetrical spaces.
Thus vectors do not have to have magnitudes. And this is as good a place as any to stop.
10. WHAT THEN IS A VECTOR?
This being a book about vectors, we have presented only the sketchiest account of tensors-- barely enough to illustrate the advantages of thinking of vectors in terms of the way their components transform.
We have one final point to make. Notice that we defined contravariant vectors and covariant vectors-- indeed, tensors of all ranks--before we introduced the metrical tensor. Suppose there were no metrical tensor. What could we then say about the magnitudes of vectors? Or about the cosines of the angles between them?
You may be tempted to argue that such questions prove that there has to be a metrical tensor. But actually there does not. Mathematicians often work with spaces that do not possess one; they call them nonmetrical spaces.
Thus vectors do not have to have magnitudes. And this is as good a place as any to stop.
This entire review has been hidden because of spoilers.
Displaying 1 - 3 of 3 reviews