A Student's Guide to Maxwell's Equations by D. Fleisch- focuses attention on the four Maxwell's equations in 130 pages. This book consists of five chapters, and the first four chapters deal with four Maxwell's equations, respectively. The final chapter is about inducing electric and magnetic wave equations from Maxwell's equations. In my opinion, the most appropriate audience of the book is undergraduate students who are studying or studied the freshmen-level of physics and calculus. To understand the book, readers should know, in advance, the Coulomb's law, electric field, capacitor, what a current is, etc. (Although they are explained, if you have not encountered them before, you may have difficulty in understanding the book).<br><br>The Maxwell's equations are expressed in the language of vector calculus such as the vector fields, the divergence of a vector field, the curl of a vector field, the line integral of a vector field on a curve, and the surface integral of a vector field on a surface. So, a firm understanding of vector calculus is very helpful to understand Maxwell's equations. The virtue of the book is that it puts all its efforts into understanding the language of vector calculus in detail. Although there had been a great need for such books, only few were available. In the way of doing it, the author shows the physical meaning of divergence, curl, integrals of vector fields, not just giving mathematical explanations. That was also very helpful to me. I titled my book review as "Learning vector calculus via Maxwell's equations" in the hope that people who learned vector calculus but failed to have a unified picture would benefit from the book.<br><br>In mathematics curriculum, even in physics, some professors spend much time in giving detailed proofs of theorems of vector calculus, for example, the Green's theorem, the divergence theorem, and the Stokes' theorem. But I think that this is not a wise choice. Like the book, it would be better that professors explain vector calculus using physical meaning, show how to apply them to concrete examples, and leave their proof to students. For students, it is sufficient to know that there are mathematical proofs for them.<br><br>In the book, the differential version of Maxwell's equations appear. The Maxwell's equations that we generally know are in fact the integral version. Freshmen-level physics textbooks don't deal with the differential version. In the final chapter, it is shown that the two versions are equivalent by using the divergence theorem and the Stokes' theorem. Here, the readers should not think that there are mathematical proofs for them in the book. Proofs are unavailable there. To show that the two versions of Maxwell's equations are equivalent, it uses two equivalent definitions of the divergence (and the curl) without proof. But that's the point that we have to prove mathematically, that is, we have to show that the different-looking definitions are actually equivalent. But as I said, I think that we don't need to know the detailed proof for that. Just have a glance at the proof in a mathematical textbook. In the case of vector calculus, having a whole picture and easy understanding is more important than following a detailed proof. I pointed that, since we must know for sure what the logical structure is in a book even though we don't need the proofs right away.<br><br>Here are some detailed points.<br><br>1. Suppose that there are some electric charges in space. Consider an imaginary sphere that does not contain any of the charges. By Gauss's law for electric fields, the flux of the electric field of the charges through the sphere is zero. Some teachers say that Gauss's law holds "because" any field line coming in must come out. But as we see in the definition of the flux of a field through a surface, we must consider the angles at which the field lines and the surface meet. So such an argument is wrong. It is the Gauss's law that makes the argument hold, and Gauss's law is an experimental fact or a physical principle that we should accept. I realized this while reading the book.<br><br>Such kind of inaccuracy can also happen when people say about Gauss's law for magnetic fields. Some teachers might say that Gauss's law for magnetic fields holds because the magnetic charge always occur in pairs and so any field line coming out a surface must come in. By the same reason, this argument is wrong and we should accept Gauss's law for magnetic fields as an experimental fact.<br><br>2. Most electromagnetic theory textbooks, including the book, say that the four Maxwell's equations explain all electromagnetic phenomena. In my opinion, some precaution is needed. To say that, we should know what electric and magnetic fields are. And to know them, we should know what electric and magnetic forces are, in other words, how to measure them in a laboratory. I think that this point should be explained in more detail when authors say that Maxwell's equations explains all.<br><br>3. The book is excellent as a supplement, not as a main textbook, I think. Considering the amount of 130 pages, this is not a bad point.<br><br>4. The book is sloppy in some places. For example, in explaining the meaning of the integral of a vector field on a curve, the book says that a finite sum of some quantities is equal to what we want. But the finite sum cannot be equal to it! Only the limit of a sequence of finite sums is what we want.<br><br>5. The Appendix seems to be just a summary rather than something from which we can learn.<br><br>6. In the book, two Faraday's laws are introduced. One is the generally known, and the other is an alternative form. They are equivalent to each other. But the difference of the two in the physical sense is not so clear in the book. At least, not explained in an easy way. I had to make my own effort to understand it. In my opinion, the key to understand it lies in the question of what is the source of a current in a circuit with a battery. Does a current flow because the battery forms an electric field in the circuit or because the battery does work for that by EMF (electromotive force)? This confuses me. If you actually consider the charge distribution in a circuit, you will find the electric field cannot be confined in the wire. Then is it by the EMF? But EMF is also ultimately due to an electric force (ultimately, there are only four forces in Nature, gravitational, electromagnetic, weak and strong nuclear forces), so in the case also, we have to say that the battery forms an electric field in the circuit, and the electric field is the source of a current. I hope it comes clear to me someday.