This book has one purpose: to help you understand and apply the mathematics used in college-level astronomy. The authors have instructed several thousand students in introductory astronomy courses at large and small universities, and in our experience a common response to the question “How's the course going for you?” is “I'm doing fine with the concepts, but I'm struggling with the math.” If you're a student in that situation, or if you're a life-long learner who'd like to be able to delve more deeply into the many wonderful astronomy books and articles in bookstores and on-line, this book is here to help.

We want to be clear that this book is not intended to be your first exposure to astronomy, and it is not a comprehensive treatment of the many topics you can find in traditional astronomy textbooks. Instead, it provides a detailed treatment of selected topics that our students have found to be mathematically challenging. We have endeavored to provide just enough context for those topics to help foster deeper understanding, to explain the meaning of important mathematical relationships, and most of all to provide lots of illustrative examples.

We've also tried to design this book in a way that supports its use as a supplemental text. You'll notice that the format is modular, so you can go right to the topic of interest. If you're solid on gravity but uncertain of how to use the radiation laws, you can skip Chapter 2 and dive right into Section 3.2 of Chapter 3.

While it remains true that “entropy is not a localized, microscale phenomenon at which we can point, even in our imaginations, and say, ‘Look! There is entropy’” and that, “if we insist on trying to understand a subject in ways inconsistent with its nature, we will be disappointed,” the eight chapters of this guide have prepared us to give a constructive answer to the question “What is entropy?”

Any short description of entropy will necessarily be figurative. After all, one task of a figure of speech is to transfer a complex meaning from an extended description to a word or short phrase. In fact, we have already considered several figurative descriptions of entropy that are appropriate in special contexts: transformation content, disorder, uncertainty, spread in phase space, and missing information. Transformation content was Clausius’s way of referring to how the entropy function indicates the direction in which an isolated system may evolve. Spread in phase space, while appropriate for statistical systems, depends upon familiarity with the technical concept of phase space.

Disorder has long been a popular synonym for entropy. But recently order and disorder as describing low and high entropy systems have fallen into disfavor. This is because scientists have become fascinated with isolated systems that generate apparent order from apparent disorder. For instance, consider a thoroughly shaken bottle of water and olive oil. When left undisturbed, the water and olive oil begin to separate into distinct layers with the less dense olive oil on top. Yet even in this process the entropy of the oil–water system increases. Thus, while order and disorder are suggestive, they can mislead.