Whew! For those not following my Twitter account, let me declare that I finished the last major section of my textbook today! Now I'm going to do a bit of touching up of various sections that I worked on recently, and I should get rid of the whole damn thing first thing next week.
Today was a real brain-buster for me -- I finally sat down to face the section on tensor analysis. For those not familiar, tensor analysis is a very abstract and hard-to-digest field of mathematics, and the hardest one in my book. To give you an idea of how painful it is, Einstein formulated his general theory of relativity using tensor mathematics.
I've been reading about tensors for months now, trying to get a good enough understanding to write a clear, concise section for the book. The experience really drove home for me the difference between understanding something and understanding it well enough to teach it! I was determined to get through it today, no matter what, and fortunately it finally clicked. (Though I'm pretty sure I'll need to revise the text significantly later this week.)
The amusing thing is that the section on tensors is one of the less important ones in the book but one that may be absolutely essential in the future. The relatively recent introduction of "cloaking devices" has spawned a new field of optics referred to as transformation optics, in which the ideas of general relativity and the tools of tensor analysis are used to design materials that manipulate light in novel and even "unnatural" ways. Though the field isn't a standard topic in optics education, I didn't want to leave out such a possible glaring omission.
Fortunately, I'm done! Just a few revisions, and the addition of a few additional exercises, and I can get rid of it! At least until the editor's comments start rolling in...