Thermodynamics, like other sciences, has a theoretical side, expressed in mathematical language, and a practical side, in which experiments are performed to produce the physical data required and interpreted by the theoretical side. The mathematical side of thermodynamics is simple and elegant and is easily derived from first principles. This might lead to the conclusion that thermodynamics is a simple subject, one that can be easily absorbed early in one's education before going on to more challenging and interesting topics. This is true, if by learning thermodynamics one means learning to manipulate its equations and variables and showing their interrelationships. But for most students the subject is actually far from simple, and for professors it is a considerable challenge to present the necessary material intelligibly. The equations and the variables are somehow related to the real world of beakers and solutions, fuels and engines, rocks and minerals, and it is this interface that provides most of the difficulties. What do variables such as entropy and free energy really mean, and what physical processes do the equations describe? The difficulty in understanding and using thermodynamics is conceptual, not mathematical. We will attempt to explain the relationship between the mathematical and the physical sides of thermodynamics, but it is advisable first to review the mathematics involved and subsequently to define the terms used in thermodynamics. The mathematics required for thermodynamics consists for the most part of nothing more complex than differential and integral calculus. However, several aspects of the subject can be presented in various ways that are either more or less mathematically based, and the "best" way for various individuals depends on their mathematical background. The more mathematical treatments are elegant, concise, and satisfying to some people, and too abstract and divorced from reality for others. In this book we attempt to steer a middle-of-the-road course. We review in the first part of this chapter those aspects of mathematics that are absolutely essential to an understanding of thermodynamics. The chapter closes with mathematical topics that, although not essential, do help in understanding certain aspects of thermodynamics.
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