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A Student's Guide #2
A Student's Guide to Vectors and Tensors
Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.
206 pages, Paperback
First published September 22, 2011
84 people are currently reading
263 people want to read
About the author
Daniel Fleisch
8books39followersProf. Dan Fleisch short biography
Dan Fleisch is a Professor in the Department of Physics
at Wittenberg University, where he specializes in
electromagnetics and space physics. He is the author of
the internationally best-selling book A Student’s Guide to
Maxwell’s Equations, published by Cambridge
University Press in January 2008 and already in its 12th printing. This book has been
translated into Japanese, Korean, and Chinese. Dr. Fleisch is also the author of A
Student’s Guide to Vectors and Tensors, published by Cambridge Press in 2011, and A
Student’s Guide to the Mathematics of Astronomy, to be published in September of 2013.
He is currently under contract with Cambridge Press for A Student’s Guide to Waves,
which will be published in 2014. Fleisch is also the co-author with the late Prof. John
Kraus of The Ohio State University of the McGraw-Hill textbook Electromagnetics with
Applications.
Prof. Fleisch has published technical articles in the IEEE Transactions, The Journal of
Atmospheric and Terrestrial Physics, and Microwave Journal, and has presented more
than a dozen professional papers on topics related to high-speed microwave
instrumentation and radar cross-section measurement. He has been a regular contributor
of science commentary to PBS station WYSO of Yellow Springs, and in 2006 he
appeared in the documentary "The Dayton Codebreakers" shown on Public
Television. In 2009 he was the first U.S. citizen to receive an Arthur Award from Stuart
McLean of the Canadian Broadcasting Corporation.
Prof. Fleisch was named Outstanding Faculty Member at the Wittenberg Greek
scholarship awards in 2000, and in 2002 he won the Omicron Delta Kappa award for
Excellence in Teaching. In 2003 and 2005 he was recognized for Faculty Excellence and
Innovation by the Southwestern Ohio Council for Higher Education (SOCHE), and in
2004 he received Wittenberg’s Distinguished Teaching Award, the university’s highest
faculty award.
In November of 2010 Prof. Fleisch was named the Ohio Professor of the Year by the
Carnegie Foundation and the Council for the Advancement and Support of Education.
In August of 2013 Prof. Fleisch was named one of the Top 25 Science, Technology,
Engineering, and Mathematics (STEM) Professors in Ohio.
Fleisch received his B.S. in Physics from Georgetown University in 1974 and his M.S.
and Ph.D. in Space Physics and Astronomy from Rice University in 1976 and 1980,
respectively.
Dan Fleisch is a Professor in the Department of Physics
at Wittenberg University, where he specializes in
electromagnetics and space physics. He is the author of
the internationally best-selling book A Student’s Guide to
Maxwell’s Equations, published by Cambridge
University Press in January 2008 and already in its 12th printing. This book has been
translated into Japanese, Korean, and Chinese. Dr. Fleisch is also the author of A
Student’s Guide to Vectors and Tensors, published by Cambridge Press in 2011, and A
Student’s Guide to the Mathematics of Astronomy, to be published in September of 2013.
He is currently under contract with Cambridge Press for A Student’s Guide to Waves,
which will be published in 2014. Fleisch is also the co-author with the late Prof. John
Kraus of The Ohio State University of the McGraw-Hill textbook Electromagnetics with
Applications.
Prof. Fleisch has published technical articles in the IEEE Transactions, The Journal of
Atmospheric and Terrestrial Physics, and Microwave Journal, and has presented more
than a dozen professional papers on topics related to high-speed microwave
instrumentation and radar cross-section measurement. He has been a regular contributor
of science commentary to PBS station WYSO of Yellow Springs, and in 2006 he
appeared in the documentary "The Dayton Codebreakers" shown on Public
Television. In 2009 he was the first U.S. citizen to receive an Arthur Award from Stuart
McLean of the Canadian Broadcasting Corporation.
Prof. Fleisch was named Outstanding Faculty Member at the Wittenberg Greek
scholarship awards in 2000, and in 2002 he won the Omicron Delta Kappa award for
Excellence in Teaching. In 2003 and 2005 he was recognized for Faculty Excellence and
Innovation by the Southwestern Ohio Council for Higher Education (SOCHE), and in
2004 he received Wittenberg’s Distinguished Teaching Award, the university’s highest
faculty award.
In November of 2010 Prof. Fleisch was named the Ohio Professor of the Year by the
Carnegie Foundation and the Council for the Advancement and Support of Education.
In August of 2013 Prof. Fleisch was named one of the Top 25 Science, Technology,
Engineering, and Mathematics (STEM) Professors in Ohio.
Fleisch received his B.S. in Physics from Georgetown University in 1974 and his M.S.
and Ph.D. in Space Physics and Astronomy from Rice University in 1976 and 1980,
respectively.
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Displaying 1 - 17 of 17 reviews
August 15, 2018
This is an outstanding introduction to tensors, going over the conceptual side without the (in my view senseless) rigor you find in purely mathematical textbooks. Fleisch provides very clearly the ideas behind such things as the metric tensor, christoffel symbols, covariant derivatives and the Riemann tensor (for those interested in general relativity). There is also a section on covariant and contravariant tensors and their geometrical interpretation but I found this of rather little use. The ordinary definitions given for these in most textbooks really do suffice, even though it often sounds like tensors are "things that transform like tensors". But that's just what they are! Defined by their transformation property.
The first three chapters on vectors are also very helpful for really seeing how the curl or laplacian works, together with applications in mechanics and electromagnetism.
All in all, I would greatly advise self-learners in physics this book, as well as other books by Fleisch, particularly his book on waves.
The first three chapters on vectors are also very helpful for really seeing how the curl or laplacian works, together with applications in mechanics and electromagnetism.
All in all, I would greatly advise self-learners in physics this book, as well as other books by Fleisch, particularly his book on waves.
February 5, 2014
I finally understood tensors with this very well written and explained book.
January 21, 2025
Yet another excellent guide from Professor Fleisch. Having embarked on the General Relativity and trying to absorb some of its concepts from a few books I had collected, I quickly realised I needed to understand the subject of tensors before getting any traction in this challenge. This book helped me to clear some of the complexity of the subject.
The subject is presented in a number of chapters of increasing complexity. However, at the end of each I could feel confident to embark on the next one. There are a lot of examples discussed in the text and each chapter concludes with a series of exercises, with solutions available on the author's website.
The first chapters on vectors, vector operations and applications were pretty straightforward. The fun started with the fourth chapter on covariant and contra-variant vector components. Having explained the issue of coordinate-system transformations I could proceed to basis-vector transformations. Following the introduction of non-orthogonal coordinate systems I continued with the dual basis vectors and learnt how to find covariant and contra-variant components. Starting with Einstein's index notations we were shown which quantities transform contravariantly, and which ones are subject of covariant transformation.
The chapter five deals with the Higher-rank tensors and goes into definitions of covariant, contra-variant and mixed tensors, followed by the operations on them. Finally, we got to a Metric tensor and how it fits into the four-dimension space-time of Einstein. Tensor derivatives, Christoffel symbols and Covariant differentiation were introduced next. This concluded the traditional approach to General Relativity and we were told that the traditional approach tends to treat contra-variant and covariant components as representations of the same object,whereas in the modern approach objects are classified either as “vectors� or as“one-forms� (also called “covectors�). In the modern terminology, vectors transform as contra-variant quantities, and one-forms transform as covariant quantities.
The final chapter explains tensor applications. I focused on General Relativity using the Riemann curvature tensor, the Ricci tensor and scalar and the Einstein tensor. The other sections in this chapter cover the Inertia tensor, which is used to summarise all moments of inertia of an object. The separate section is devoted to the electromagnetic field tensor in connection with Maxwell's equations.
For me it was an eye opener, thoroughly enjoyed. Highly recommended for those who are trying to crack this enigma of one hundred years old theory.
The subject is presented in a number of chapters of increasing complexity. However, at the end of each I could feel confident to embark on the next one. There are a lot of examples discussed in the text and each chapter concludes with a series of exercises, with solutions available on the author's website.
The first chapters on vectors, vector operations and applications were pretty straightforward. The fun started with the fourth chapter on covariant and contra-variant vector components. Having explained the issue of coordinate-system transformations I could proceed to basis-vector transformations. Following the introduction of non-orthogonal coordinate systems I continued with the dual basis vectors and learnt how to find covariant and contra-variant components. Starting with Einstein's index notations we were shown which quantities transform contravariantly, and which ones are subject of covariant transformation.
The chapter five deals with the Higher-rank tensors and goes into definitions of covariant, contra-variant and mixed tensors, followed by the operations on them. Finally, we got to a Metric tensor and how it fits into the four-dimension space-time of Einstein. Tensor derivatives, Christoffel symbols and Covariant differentiation were introduced next. This concluded the traditional approach to General Relativity and we were told that the traditional approach tends to treat contra-variant and covariant components as representations of the same object,whereas in the modern approach objects are classified either as “vectors� or as“one-forms� (also called “covectors�). In the modern terminology, vectors transform as contra-variant quantities, and one-forms transform as covariant quantities.
The final chapter explains tensor applications. I focused on General Relativity using the Riemann curvature tensor, the Ricci tensor and scalar and the Einstein tensor. The other sections in this chapter cover the Inertia tensor, which is used to summarise all moments of inertia of an object. The separate section is devoted to the electromagnetic field tensor in connection with Maxwell's equations.
For me it was an eye opener, thoroughly enjoyed. Highly recommended for those who are trying to crack this enigma of one hundred years old theory.
August 22, 2018
I didn't expect it to be this good. It's a short book, but it tackles things in a great way! Don't expect rigor or really high-level stuff, but what is here is full of intuition.The material that is here, is presented in the clearest way possible. It also offers some discussions that shed light on material that might be a bit confusing in other books.
I love this treatment. This book is amazing as a supplement. No serious physics/mathematics student should learn vectors or tensors from this book alone, but having one more thorough book with more details and this one as an aside is the best thing to do.
All in all, I was surprised by the quality of this small gem.
I love this treatment. This book is amazing as a supplement. No serious physics/mathematics student should learn vectors or tensors from this book alone, but having one more thorough book with more details and this one as an aside is the best thing to do.
All in all, I was surprised by the quality of this small gem.
December 20, 2018
Not bad text but it is useless without proofs and examples.
So, take Borisenko A., Tarapov I. "Vector and Tensor Analysis" if you really want to understand what the author wrote about.
So, take Borisenko A., Tarapov I. "Vector and Tensor Analysis" if you really want to understand what the author wrote about.
March 13, 2024
The introduction covers vectors from roughly high-school level, before going on to give a great description of covariant and contravariant vector components.
Then some introductory differential (Riemannian) geometry is discussed, with applications soon to follow, some of which I felt were unnecessarily advanced.
In the derivation of the Christoffel symbol I did find that I had to work out why de_i/dx_j = de_j/dx_i, which actually is not necessarily true unless the connection is torsionless. It is stated as trivial, and I did not feel this was so, so a pointer wouldn't have hurt. To be fair, if the student follows the derivation of e_i closely, they will not run into this problem.
I think the book gets a bit too advanced too quickly as there are other tools worth teaching which are easy to grasp but give high reward for the student, such as the Levi-Civita symbol for deriving product rules etc.
I think a further area which is missing is the description of axial and polar vectors, and how these are used to make vector analysis a useful tool for physics, but can lead to confusion where the transformation properties are concerned. It would have been nice to see some discussions of angular momentum being better represented as a tensor as a simple example.
Also, the cross product is a special trick which really involves mapping a bivector to its hodge dual which happens to be a vector. This mapping of the bivector to vector only works in three dimensions (and also seven so I've heard?), so some discussion of this might have been good.
However the fact is that this book has managed to introduce fairly well tensor analysis from a grounding in vectors, so I will give four stars.
Then some introductory differential (Riemannian) geometry is discussed, with applications soon to follow, some of which I felt were unnecessarily advanced.
In the derivation of the Christoffel symbol I did find that I had to work out why de_i/dx_j = de_j/dx_i, which actually is not necessarily true unless the connection is torsionless. It is stated as trivial, and I did not feel this was so, so a pointer wouldn't have hurt. To be fair, if the student follows the derivation of e_i closely, they will not run into this problem.
I think the book gets a bit too advanced too quickly as there are other tools worth teaching which are easy to grasp but give high reward for the student, such as the Levi-Civita symbol for deriving product rules etc.
I think a further area which is missing is the description of axial and polar vectors, and how these are used to make vector analysis a useful tool for physics, but can lead to confusion where the transformation properties are concerned. It would have been nice to see some discussions of angular momentum being better represented as a tensor as a simple example.
Also, the cross product is a special trick which really involves mapping a bivector to its hodge dual which happens to be a vector. This mapping of the bivector to vector only works in three dimensions (and also seven so I've heard?), so some discussion of this might have been good.
However the fact is that this book has managed to introduce fairly well tensor analysis from a grounding in vectors, so I will give four stars.
July 21, 2022
Excellent depiction of a difficult subject
The author Daniel Fleisch presents both derivations and basic rationales for tensor and vector equations. He has a rare gift for explaining concepts simply and coherently. Use it to understand these equations when seeing them in other texts. For those who wish to mathematically work with them, there are problems at the end of the chapters. My only issue with this book is the lack of specific examples showing the basic vector and tensor manipulations. A more general issue is with kindle. The equations are hardly readable, they are in such a small font. When purchasing texts from kindle you have to weigh this against the advantages of highlighting, searching, note taking, and lookup. I highly recommend this book (paperback or ebook) to those who want to read and understand advanced texts on general relativity.
The author Daniel Fleisch presents both derivations and basic rationales for tensor and vector equations. He has a rare gift for explaining concepts simply and coherently. Use it to understand these equations when seeing them in other texts. For those who wish to mathematically work with them, there are problems at the end of the chapters. My only issue with this book is the lack of specific examples showing the basic vector and tensor manipulations. A more general issue is with kindle. The equations are hardly readable, they are in such a small font. When purchasing texts from kindle you have to weigh this against the advantages of highlighting, searching, note taking, and lookup. I highly recommend this book (paperback or ebook) to those who want to read and understand advanced texts on general relativity.
June 29, 2018
BEST BOOK EVER!!!
I struggled alot while learning tensors, but this book has been great help, even I learned what is covarient and contravariant components of tensor, from this book. Prof. Daniel is a great teacher his way of explaining things is quite well, and how he discussed cristoffel symbol earlier than talking about covarient derivatives, it's just quite awesome
I struggled alot while learning tensors, but this book has been great help, even I learned what is covarient and contravariant components of tensor, from this book. Prof. Daniel is a great teacher his way of explaining things is quite well, and how he discussed cristoffel symbol earlier than talking about covarient derivatives, it's just quite awesome
November 6, 2024
If you don’t know anything about scalars, vectors and tensors get a hold of this book and stick with it. It has saved me from so many downhills and yet I have sleepless nights of crying my eyes out from analyzing the metric tensor. But what can I say this is my life now. I have huge respect for anyone who has studied and done General Relativity. Oh Gravity, thou heartless b!tch.
April 11, 2021
It was an OK book.
The first half of the book was too easy, just a reveiw of highschool and first year undergrad and the last two was too short to understand anything about tensors.
I wouldn't say it was a guide but it did help giving some perspective and direction to read more on the topic.
The first half of the book was too easy, just a reveiw of highschool and first year undergrad and the last two was too short to understand anything about tensors.
I wouldn't say it was a guide but it did help giving some perspective and direction to read more on the topic.
February 6, 2024
Good book
I read the book and was able to understand nearly everything, although a summary page on each chapter with a formula page would have been very helpful.
Besides the General Relativity section, this book was understandable.
I read the book and was able to understand nearly everything, although a summary page on each chapter with a formula page would have been very helpful.
Besides the General Relativity section, this book was understandable.
February 3, 2019
gives you a good intuition and naive introduction to tensors, but not as a reference for working with tensors
November 10, 2024
#nsreads 56
I can't believe how many times I've tried and failed to finish this since y2 but yay at least I understand it now!
I can't believe how many times I've tried and failed to finish this since y2 but yay at least I understand it now!
February 23, 2014
The book is excellent in introducing vectors, vector fields (with operations like div, grad), transformations (covariant/contravariant), and also in introducing tensors with some of their basic properties.
One problem is that concepts from one chapter to another are not very well linked, for instance you learn some facts like "contravariant transformations are for vector components and covariant transformations are for vector themselves", but this knowledge is not put to proper use in the chapters that follow.
Also, when it comes to exemplifying tensors, the part with inertia tensor was OK, but the electromagnetic field strength tensor chapter is very weak. I did not even dare to go further to the last chapter (space metric and general relativity) because I felt sure I would not understand it as it happened with the electromagnetic field tensor. Therefore, I started reading another book on this subject.
All in all, it is a great place to start and a good value, but it does not serve its fundamental purpose to introduce you to tensors, in that you get the theory but exemplification is really problematic.
If you really want to learn tensors in context of physics, I would highly recommend directly a general relativity text such as Lambourne's or Peter Collier's (the first if you are a advanced physics undergrad / the second if you are weaker on math). As a physicist, learning about tensors in the context of relativity is much more rewarding, it gives back more than simply understanding some terminology (like covariant or contravariant). You understand why they are used by physicists, and you learn how to use them in real life examples.
One problem is that concepts from one chapter to another are not very well linked, for instance you learn some facts like "contravariant transformations are for vector components and covariant transformations are for vector themselves", but this knowledge is not put to proper use in the chapters that follow.
Also, when it comes to exemplifying tensors, the part with inertia tensor was OK, but the electromagnetic field strength tensor chapter is very weak. I did not even dare to go further to the last chapter (space metric and general relativity) because I felt sure I would not understand it as it happened with the electromagnetic field tensor. Therefore, I started reading another book on this subject.
All in all, it is a great place to start and a good value, but it does not serve its fundamental purpose to introduce you to tensors, in that you get the theory but exemplification is really problematic.
If you really want to learn tensors in context of physics, I would highly recommend directly a general relativity text such as Lambourne's or Peter Collier's (the first if you are a advanced physics undergrad / the second if you are weaker on math). As a physicist, learning about tensors in the context of relativity is much more rewarding, it gives back more than simply understanding some terminology (like covariant or contravariant). You understand why they are used by physicists, and you learn how to use them in real life examples.
May 14, 2024
An excellent book! It is an excellent book for engineers, mathematicians, physicists, etc. If you have a fair to a good foundation of algebra, vectors, vector calculus, calculus, and differential equations, then you'll do fine (as I have the math background, but was very stale with it, I did some review and it helped). If not, the book will be good to get a conceptual-level understanding of tensors, but the math will be hard to follow. Let me also recommend Professor Fleisch's companion book, ASG to Maxwell's equations. Written in the same easy-to-read style. If you want to read that book, I'd recommend buying this book too, as it goes into better background on the underlying math and that helped me a lot reading ASG to Maxwell's equations. Hope this helps. Happy reading!
August 4, 2014
He explains the theory behind vectors and tensors well, but don't expect to get any practice solving problems. I was really dismayed to find only ten measly problems per chapter and no solutions included in the book. As my Russian math professors would say, "You don't learn math by reading books."
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July 1, 2015I just passed over it quickly
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