Monday, October 29, 2018
How to Think about Analysis, Lara Alcock, Oxford University Press, Oxford, 2014
I have previously commented on Alcock's earlier work, How to study as a Mathematics Major. This book is even better. The first 50 or so pages are about learning analysis. The balance of the book reviews the central concepts of analysis unencumbered by the need to prove everything - which will, of course, be done in the actual course. So what we have here are research-informed strategies for learning analysis. I would recommend this to all students (Math, Physics, Chemistry, EE, etc.) who need to take advanced courses in mathematics. It would also help faculty who are preparing mathematically oriented courses that make use of the material covered in rigorous calculus and real analysis classes.
Sunday, October 28, 2018
Advanced Calculus for Applications, Francis B. Hildebrand, Prentice-Hall, Englewood Cliffs, NJ, 1962
In an earlier review, I promised that I would include a review of this book in a few days. So many months later here it is. The present book is a revision of Advanced Calculus for Engineers, first published by the same publishing house in 1948. This earlier edition is now available as a photographic reprint from Martin Fine Books at a modest price. A second edition was published in 1976, the paperback of which is still readily available - but used hardbacks are available at better prices, especially for the earlier editions.
This is one of the few books that lives up to its name - it really is advanced calculus for applications. It covers ordinary differential equations and their numerical solution, Laplace Transforms, series solutions, boundary value problems and characteristic-function representations, vector analysis, higher dimensional calculus, partial differential equations, with particular emphasis on those met in physics, and complex variable theory including contour integration and conformal mapping.
This is a very useful book. Don't hesitate to get a copy of any edition!
This is one of the few books that lives up to its name - it really is advanced calculus for applications. It covers ordinary differential equations and their numerical solution, Laplace Transforms, series solutions, boundary value problems and characteristic-function representations, vector analysis, higher dimensional calculus, partial differential equations, with particular emphasis on those met in physics, and complex variable theory including contour integration and conformal mapping.
This is a very useful book. Don't hesitate to get a copy of any edition!
A First Course in Calculus, 3rd Ed., Serge Lang, Addison-Wesley, Reading, Mass, 1973
This text would eventually go to 5 editions and a reissue of the first edition called Short Calculus. The edition in the title is the version that I used as a freshman at University College Cardiff in the fall of 1975. The course was taught by Dr. G.R.H. Greaves. At the time I thought that Dr. Greaves was in his 40s, but he was actually in his twenties - in fact, he was the last of my teachers still active in the classroom, only retiring a few years ago, and I will be 62 in a few days.
Dr. Greaves was a great teacher and the book was ideal for the course (I took 1A for mathematics majors rather than IB for science majors who didn't plan to pursue a degree in mathematics.) As a result, I was in a much better course, one that looked forward to real analysis. It remains a favorite, though since I have had to repair my International Student edition, I more often refer to Short Calculus. It isn't the book I would recommend to students today - though it is better than most current calculus. I'd recommend the books by Gilbert Strang or Peter Lax. Personally, if I were beginning calculus again, I'd like to be in a course that used Peter Lax's book. I'll write a review at some point. My all-time favorite remains A course in Pure Mathematics by G.H. Hardy.
Dr. Greaves was a great teacher and the book was ideal for the course (I took 1A for mathematics majors rather than IB for science majors who didn't plan to pursue a degree in mathematics.) As a result, I was in a much better course, one that looked forward to real analysis. It remains a favorite, though since I have had to repair my International Student edition, I more often refer to Short Calculus. It isn't the book I would recommend to students today - though it is better than most current calculus. I'd recommend the books by Gilbert Strang or Peter Lax. Personally, if I were beginning calculus again, I'd like to be in a course that used Peter Lax's book. I'll write a review at some point. My all-time favorite remains A course in Pure Mathematics by G.H. Hardy.
The indispensable guide to Undergraduate Research, Anne H. Charity Hudley, Cheryl L. Dickter, and Hannah A. Franz, Teachers College Press, New York, 2017
This is my second review of books about getting involved in student research of the day. This is the better of the two books. It is informed by the experiences of the authors and the many researchers with whom they have interacted. It still isn't the book that I was looking for. I'm looking for a short book that is equally valuable for students and for their faculty members, particularly new prospective mentors. I may have commented before that I have found books like the Cambridge Monograph Series of old to be the most useful, and the older ones tended to be between 80 and 120 pages. At 180 pages, this book is too long.
The book begins with a section on what is research and why do we do it, it then has the following chapters:
1. Finding the tools to become a scholar
2. Getting started with undergraduate research: What, Why, and How.
3. How to fit research in with everything else: Time and Energy Mangement
4. Research with Professors and Mentors
5. Writing and Presenting Research
6. Underrepresented Scholars in the Academy: Making A Way
7. In Conclusion: Research in Action
This book contains much that is useful - but it is aimed at all disciplines. If I want a book that focuses on the sciences and mathematics and mentions the latest on funding and the appropriate tools, I think that I will have to write it myself. I should note that I would have bought a copy of this one - even if I had first borrowed a copy from the library.
The book begins with a section on what is research and why do we do it, it then has the following chapters:
1. Finding the tools to become a scholar
2. Getting started with undergraduate research: What, Why, and How.
3. How to fit research in with everything else: Time and Energy Mangement
4. Research with Professors and Mentors
5. Writing and Presenting Research
6. Underrepresented Scholars in the Academy: Making A Way
7. In Conclusion: Research in Action
This book contains much that is useful - but it is aimed at all disciplines. If I want a book that focuses on the sciences and mathematics and mentions the latest on funding and the appropriate tools, I think that I will have to write it myself. I should note that I would have bought a copy of this one - even if I had first borrowed a copy from the library.
Getting In, David G. Oppenheimer and Paris H. Grey, Secret Handshake Press, Gainesville, Fl, 2015
I have long considered writing a book on getting started in student research. I likely still will. In the meantime, I'm going to review some books that are already on the market. This one has the sub-title The Insider's Guide to Finding the Perfect Undergraduate Research Experience. This one was written from a life sciences perspective. It has some useful information, but it isn't perfect. The book has a very annoying feature, it prints the things that it thinks the reader should highlight in bold. This is also to support skimming. The book is worth reading - this feature detracts from its usefulness.
The book begins with a section about why students should become involved in research and describes the benefits, both short-term and long-term, that can be derived from such involvement. It follows this with a chapter on what research is - this likely would have made a better first chapter. It's tough to get motivated if you don't know what you are doing! The next chapter is about whether students will like research - some will, some won't. It does point out that students don't always know at first that they are learning things that will help them. This gets to one of my major points about this book. Is it for students, or is it for faculty - the answer is both, but it would have been better to make this clear. Many new faculty would be helped by a book like this, but it would help readers if the sections for each audience were more clearly identified. Both audiences need to be familiar with the other's viewpoint, but it would help readers if the information for these roles was better separated. The authors address the question as to when a student should begin undergraduate research - for me the only acceptable answer is: Now. What are you waiting for? The authors describe different scenarios at length, but, in my experience, the earlier the better.
The book then has a search strategy for finding the ideal position. This will be helpful for some, but my philosophy on this is that in the beginning topics don't matter - as long as you are going to learn research skills, if you have an offer, and if the environment is welcoming and supportive, then you have found a good place to start. Over my academic career, I have seen prospective faculty turning down tenure-track offers without realizing that one such offer is all that most people are going to get, and I have seen faculty keep trying to publish in top-tier journals, resulting in insufficient publications for tenure or promotion. So my advice, if the opportunity is a good opportunity, take it. You will have the rest of your life to follow your own research directions. The search and interview process described in this book are also too complicated. If you are in an environment that is supportive of student research, you can probably get started by talking to your teachers, attending seminars and asking if you can attend some research group meetings. The CV and interview process described in this book is probably not going to be used for the sorts of experiences that most students are looking for.
Thus, in summary, this book contains much that is useful, but it probably pertains better to students attending major universities rather than the bulk of students who are attending colleges and comprehensive institutions where most students will gain their first research experiences. It will remain on my bookshelf, but I wouldn't have bought a copy if I had been able to borrow it from the library. It has some good points, but it has glaring omissions like funding, the availability of NSF REUs and how to progress.
The book begins with a section about why students should become involved in research and describes the benefits, both short-term and long-term, that can be derived from such involvement. It follows this with a chapter on what research is - this likely would have made a better first chapter. It's tough to get motivated if you don't know what you are doing! The next chapter is about whether students will like research - some will, some won't. It does point out that students don't always know at first that they are learning things that will help them. This gets to one of my major points about this book. Is it for students, or is it for faculty - the answer is both, but it would have been better to make this clear. Many new faculty would be helped by a book like this, but it would help readers if the sections for each audience were more clearly identified. Both audiences need to be familiar with the other's viewpoint, but it would help readers if the information for these roles was better separated. The authors address the question as to when a student should begin undergraduate research - for me the only acceptable answer is: Now. What are you waiting for? The authors describe different scenarios at length, but, in my experience, the earlier the better.
The book then has a search strategy for finding the ideal position. This will be helpful for some, but my philosophy on this is that in the beginning topics don't matter - as long as you are going to learn research skills, if you have an offer, and if the environment is welcoming and supportive, then you have found a good place to start. Over my academic career, I have seen prospective faculty turning down tenure-track offers without realizing that one such offer is all that most people are going to get, and I have seen faculty keep trying to publish in top-tier journals, resulting in insufficient publications for tenure or promotion. So my advice, if the opportunity is a good opportunity, take it. You will have the rest of your life to follow your own research directions. The search and interview process described in this book are also too complicated. If you are in an environment that is supportive of student research, you can probably get started by talking to your teachers, attending seminars and asking if you can attend some research group meetings. The CV and interview process described in this book is probably not going to be used for the sorts of experiences that most students are looking for.
Thus, in summary, this book contains much that is useful, but it probably pertains better to students attending major universities rather than the bulk of students who are attending colleges and comprehensive institutions where most students will gain their first research experiences. It will remain on my bookshelf, but I wouldn't have bought a copy if I had been able to borrow it from the library. It has some good points, but it has glaring omissions like funding, the availability of NSF REUs and how to progress.
Wednesday, October 24, 2018
The Code Book, Simon Singh, Anchor Books, New York, 1999
If you have never read a book by Simon Singh you are missing out; who could resist the Simpsons and their Mathamatical Secrets? Singh is a a PhD (Cantab) in physics. I discovered the present book when I was looking for a text for an Honors course in cybersecurity. If you want an overview of cryptography from Herodetus to qunatum computing, there is no better place to start. Writing a summary of this book would be tough as it is itself a summary. If you have any interest in cybersecurity or cryptography you will enjoy this book.
How to study as a Mathematics Major, Lara Alcock, Oxford University Press, Oxford, 2013
Laura Alcock works in the Mathematics Education Center at the University of Loughborough in the United Kingdom. In recent months I have been compiling my various notes on mathematics to put together all the tools I use in mathematical physics so I don't have to go back to original sources. In the process, I have been using more modern notation than I was taught and am writing the notes formally so that my students can use them as a starting point in reading more mathematically formal papers, particularly those that are being published on entropy, weak value quantum mechanics, and quantum computing. Alcock's book has a lot to offer and specifically talks about making the transition from calculational mathematics to advanced formal mathematics. As well as talking about this transition, Alcok reviews major definitions and provides a useful entry to formal mathematics.
The items that I mentioned in the paragraph above would make the book useful. But, the book doesn't end there, it has additional material on proofs, on reading mathematics, on good writing in mathematics, study skills, time management, and it describes what mathematicians and mathematics professors actually do. While some information applies to British universities almost everything in the book is universal. Having attended universities in both the US and the UK, I would say that Ms. Alcock has made an effort to address students at US universities.
In summary, I recommend this book to everyone who teaches mathematical topics and to all mathematics and physics students.
The items that I mentioned in the paragraph above would make the book useful. But, the book doesn't end there, it has additional material on proofs, on reading mathematics, on good writing in mathematics, study skills, time management, and it describes what mathematicians and mathematics professors actually do. While some information applies to British universities almost everything in the book is universal. Having attended universities in both the US and the UK, I would say that Ms. Alcock has made an effort to address students at US universities.
In summary, I recommend this book to everyone who teaches mathematical topics and to all mathematics and physics students.
Friday, July 6, 2018
Vectorial Mechanics, E. A. Milne, Interscience New York, Methuen, London (1948)
I have used index notation to manipulate vectors for almost 40 years. Recently, my principal research collaborator John Gray and I were discussing the method and its origins and why it was not more widely known. This conversation led to a review of past articles and books that used the index method. I have referred to Milne's text many times, it contains much that is useful - but it isn't easy reading. Milne's book (which he began writing in the 1920s) was one of the first books that provided detailed examples of vector and tensor methods in applied mathematics (in the British sense) or theoretical physics - and as a result of changing notation, and the fact that almost every result is demonstrated, it is difficult to read.
The work was inspired by Milne's teacher Sydney Chapman, F.R.S. In the preface, Milne states,
It was he who first expounded to me the view that vectors were not merely a pretty toy, suitable only for elegant proofs of general theorems, but were a powerful weapon of workaday mathematical investigation, both in research and in solving problems of the types set in English examinations. I did not at first believe him; I had been brought up in the idea that, in the words of a distinguished applied mathematician, vectors were like a pocket-rule, which needs to be unfolded before it can be applied and used; similarly I thought (with most others at that time) that vector expressions were a mere shorthand for sets of Cartesian expressions, and that before they could be interpreted they always needed to be translated into Cartesians. Professor Chapman’s reply was; ‘Try for
yourself ’ I had faith in him to do so, and rapidly convinced myself, in spite of my previous beliefs, that he was right. I found further, again under his inspiration, that one could soon learn to think and work that problems, dull or difficult by conventional methods, vectorially gained when treated by vector methods an interest, an ease and a delight which they previously lacked; and that, when a problem was formulated and solved vectorially, the vector solution provided a kinematic picture of the motion in question that gave far more insight into the phenomenon than the corresponding Cartesian solution.
He went on to say,
I began in consequence to use vector methods directly in my own researches, and I began to follow Chapman in lecturing systematically to students through the medium of vectors. The credit for the discovery of the practical advantages of vectors is, however, entirely due to Chapman.
Of course, others would correctly argue that the discovery of the practical advantages of the use of vectors was due to many others including Gibbs, Heaviside, and many others. If you are interested in the history of vector analysis, I recommend reading A History of Vector Analysis by Micahel J. Crowe.
Milne's books reards careful reading. If you want to learn more about vectors and tensors, you will find much that would not be found elsewhere. There is much useful information about the epsilon-delta relationship and a note that about what Milne calls suffix notation. Since it was index notation that took me back to this book, I'll collect some of Milne's comments about suffix (index) notation.
Milne preferred his vector notation to suffix notation (what we call today index notation), though his comments are often at odds with each other. Early on he states
The above notation for the vector calculus is the result of much thought and discussion, and it is hoped that the book may help towards standardizing vector notation. I have at times been criticized as 'clumsy' in my employment of vectors. I will not defend myself against this accusation, but I must defend the notation, which I believe to be concise and convenient. It is incomparably less clumsy and more insight-giving, for the purposes of dynamics, than the suffix notation, which for all its supposed terseness cannot readily describe vector products or lend itself to vector multiplication as an operation. In certain contexts(in Chapter IV), I have deliberately used the suffix notation to secure increased generality, but the results expressed in this notation are not attractive.
However, later he goes on to say
But in more complicated cases the manipulation by use of general suffixes is by far the simplest and safest, and the student is recommended to familiarize himself with this procedure.
Still later, he states
It is clear that the suffix notation is the more convenient analytical expression of the facts of differential and integral calculus contained in the theorem, whilst the vector notation gives the greater physical meaning in each particular case.
It is clear that Milne was ambivalent about index notation - but he provides information about it that is hard to find elsewhere.
So, in summary, this is a difficult book, the fact that notation wasn't standardized when it was written doesn't help. While the book is exhaustive, it is not well organized. I have borrowed this book from the library many times, I've decided that it is time that I owned a copy.
The work was inspired by Milne's teacher Sydney Chapman, F.R.S. In the preface, Milne states,
It was he who first expounded to me the view that vectors were not merely a pretty toy, suitable only for elegant proofs of general theorems, but were a powerful weapon of workaday mathematical investigation, both in research and in solving problems of the types set in English examinations. I did not at first believe him; I had been brought up in the idea that, in the words of a distinguished applied mathematician, vectors were like a pocket-rule, which needs to be unfolded before it can be applied and used; similarly I thought (with most others at that time) that vector expressions were a mere shorthand for sets of Cartesian expressions, and that before they could be interpreted they always needed to be translated into Cartesians. Professor Chapman’s reply was; ‘Try for
yourself ’ I had faith in him to do so, and rapidly convinced myself, in spite of my previous beliefs, that he was right. I found further, again under his inspiration, that one could soon learn to think and work that problems, dull or difficult by conventional methods, vectorially gained when treated by vector methods an interest, an ease and a delight which they previously lacked; and that, when a problem was formulated and solved vectorially, the vector solution provided a kinematic picture of the motion in question that gave far more insight into the phenomenon than the corresponding Cartesian solution.
He went on to say,
I began in consequence to use vector methods directly in my own researches, and I began to follow Chapman in lecturing systematically to students through the medium of vectors. The credit for the discovery of the practical advantages of vectors is, however, entirely due to Chapman.
Of course, others would correctly argue that the discovery of the practical advantages of the use of vectors was due to many others including Gibbs, Heaviside, and many others. If you are interested in the history of vector analysis, I recommend reading A History of Vector Analysis by Micahel J. Crowe.
Milne's books reards careful reading. If you want to learn more about vectors and tensors, you will find much that would not be found elsewhere. There is much useful information about the epsilon-delta relationship and a note that about what Milne calls suffix notation. Since it was index notation that took me back to this book, I'll collect some of Milne's comments about suffix (index) notation.
Milne preferred his vector notation to suffix notation (what we call today index notation), though his comments are often at odds with each other. Early on he states
The above notation for the vector calculus is the result of much thought and discussion, and it is hoped that the book may help towards standardizing vector notation. I have at times been criticized as 'clumsy' in my employment of vectors. I will not defend myself against this accusation, but I must defend the notation, which I believe to be concise and convenient. It is incomparably less clumsy and more insight-giving, for the purposes of dynamics, than the suffix notation, which for all its supposed terseness cannot readily describe vector products or lend itself to vector multiplication as an operation. In certain contexts(in Chapter IV), I have deliberately used the suffix notation to secure increased generality, but the results expressed in this notation are not attractive.
However, later he goes on to say
But in more complicated cases the manipulation by use of general suffixes is by far the simplest and safest, and the student is recommended to familiarize himself with this procedure.
Still later, he states
It is clear that the suffix notation is the more convenient analytical expression of the facts of differential and integral calculus contained in the theorem, whilst the vector notation gives the greater physical meaning in each particular case.
It is clear that Milne was ambivalent about index notation - but he provides information about it that is hard to find elsewhere.
So, in summary, this is a difficult book, the fact that notation wasn't standardized when it was written doesn't help. While the book is exhaustive, it is not well organized. I have borrowed this book from the library many times, I've decided that it is time that I owned a copy.
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