Preface: A Gift to My Former Self, or a Line of Defense in the Age of Social Media

I wrote this book to answer the questions I had as an undergraduate. It assumes, roughly, the background I had at the time.

The book also needs a few lines of defense. Once you publish something mathematical-looking on the internet, you cannot avoid them. In several places, a more advanced textbook would immediately abstract everything away. I deliberately begin instead with matrix representations and computations on finite cells.

That is not a rejection of the more advanced point of view. Quite the opposite. My aim is to give the reader concrete tools, intuition, and a feel for “rigor” that will remain useful when they eventually face the world beyond this book.

The detailed problem setting comes in the next section, “Introduction.”

Now, to the matter at hand.

Can you explain what $dx$ is?

Note (on notes)

This book contains many notes. Some readers may find them unkind, since they interrupt the flow of the main text.

Here, however, the notes have three jobs.

First, they are lines of defense for me. They leave a small record, beside the main text, of where something is being omitted, where two things are being identified, and what kind of rigorous formulation lies further ahead.

Second, they help train the reader’s nose. Even if you do not understand everything yet, it helps to sense that “something is going on here” or that “this way of speaking has its limits.” That sense will help later.

Third, they are meant to work a little like flashbacks in a mystery novel. At first, a note may look like a fragment. As you read on, it gradually acquires meaning, until later you realize, “Ah, so that was what this was for.” I am aiming for a little of that reading experience as well.

The notes are not there so that you understand everything on the spot. Pick up the parts you can read, and feel free to pass over the parts you cannot.

Note (historical order and understandable order)

Modern readers can learn matrices, linear algebra, differential forms, and vector analysis as already-organized tools. At least in the education I received, matrices first appeared in high school as algebraic tools one could compute by hand. Historically, however, these tools seem to have developed in a much less linear way, influencing one another along the way. I suspect that some confusion born of that history still remains in education. This book is not an attempt to reconstruct the history of mathematical physics. It is an attempt to unwind multivariable calculus and vector analysis in a different order, using the matrix algebra that modern readers already have.

Remark (this book contains anger)

This is not anger at any particular person. It is anger at the fact that a problem that should have been visible was not sufficiently solved when I was a student, and has still not been sufficiently solved even now. Vector analysis implicitly contains the metric, orientation, duality, and the exterior derivative, yet it is taught as a collection of formulas while hiding them. Meanwhile, differential forms have been placed up in the heights of manifold theory, instead of being brought down as tools for beginners trying to understand vector analysis. This book is an attack on that gap, and at the same time, a proposed alternative.