Multivariate Calculus

Many of the functional relationships needed in thermodynamics are direct applications of the rules of multivariable calculus. This section reviews those rules in the context of the needs of themodynamics. These ideas were expounded in one of the classic books on chemical engineering thermodynamics [see Hougen, O. A., et al., Part II, Thermodynamics, in Chemical Process Principles, 2d ed., Wiley, New York (1959)]. 

An understanding of optimization techniques does not require complex mathematics. We require as background only basic tools from multivariable calculus and linear algebra to explain the theory and computational techniques and provide you with an understanding of how optimization techniques work (or, in some cases, fail to work). 

G. Thomas, M. D. Weir, J. Hass, and F. R. Giordano Thomas Calculus, 11th ed., Pearson Education, Boston, 2004 J. Stewart, B. Pirtle, and K. Sandberg, Multivariate Calculus Early Transcendentals, Brooks/Cole, Belmont, CA, 2003 R. Courant, Differential and Integral Calculus, 2nd ed., Blackie, Glasgow, Scotland, 1937. 

MTW] Marsden, J.E., A.J. Tromha and A. Weinstein, Basic Multivariable Calculus, Springer Verlag, New York, 1993. 

Kinetic properties (rates of chemical reactions) and thermodynamic properties (equilibrium constants, energy, entropy) are described by a large number of different mathematical relations, which are usually just presented for the student to memorize. Part of the reason for this is the complexity associated with a full treatment of these properties these subjects are taught in graduate chemistry and physics courses at every major university, and multivariate calculus is needed to formulate a rigorous treatment. Unfortunately, simple memorization does not provide much intuition. 

Schrodinger s description, called wave mechanics, is the easier one to present at the level of this book. A general description requires multivariate calculus, but some useful special cases (such as motion of a particle in one dimension) can be described by a single position x, and we will restrict our quantitative discussion to these cases. 

Schrodinger s equation for a single electron and a nucleus with Z protons is an extension into three dimensions (x,y,z) of Equation 6.8, with the potential replaced by the Coulomb potential U(r) = Ze2/Ajt8or. This problem is exactly solvable, but it requires multivariate calculus and some very subtle mathematical manipulations which are beyond the scope of this book. 

Chapter 2 presents the basics of differential and integral calculus. I use derivatives of one variable extensively in the rest of the book. I also use the concept of integration as a way to determine the area under a curve, but the students are only asked to gain a qualitative understanding (at a level which allows them to look up integrals in a table), particularly in the first five chapters. Multivariate calculus is never used. 

This book is mainly intended as a supplement for the mathematically sophisticated topics in an advanced freshman chemistry course. My intent is not to force-feed math and physics into the chemistry curriculum. It is to reintroduce just enough to make important results understandable (or, in the case of quantum mechanics, surprising). We have tried to produce a high-quality yet affordable volume, which can be used in conjunction with any general chemistry book. This lets the instructor choose whichever general chemistry book covers basic concepts and descriptive chemistry in a way which seems most appropriate for the students. The book might also be used for the introductory portions of a junior-level course for students who have not taken multivariate calculus, or who do not need the level of rigor associated with the common one-year junior level physical chemistry sequence for example, an introduction to biophysical chemistry or materials science should build on a foundation which is essentially at this level. 

Since mathematics is the language of thermodynamics, there are many equations in this book. However, the mathematics used is no more complicated than necessary. Facility with differentiation and integration at the level of a first-year course in calculus is assumed and a few relationships from multivariable calculus are used repeatedly. All the reader has to know about this subject, however, is presented in Appendix A. Although the mathematically advanced reader can skim over this, it remains as a handy reference for any question that arises on multivariable calculus. 

A proof of this can be found in most textbooks on multivariable calculus. Alternatively, the reader can just pick a few functions of two variables and compare the cross-second-derivatives, as is done in Example 4. The latter procedure has the advantage of providing some practice in taking partial derivatives. 

It is an interesting exercise in multivariable calculus to rewrite the Laplacian in terms of these new variables. If you do this you will get ... 

In this Chapter, we quickly review some basic definitions and concepts from thermodynamics. We then provide a brief description of the first and second laws of thermodynamics. Next, we discuss the mathematical consequences of these laws and cover some relevant theorems in multivariate calculus. Finally, free energies and their importance are introduced. 

Ideos behind the proof The argument is a familiar one, and comes up in multivariable calculus, complex variables, electrostatics, and various other subjects. We think of C as a balloon and suck most of the air out it, being careful not to hit any of the fixed points. The result of this deformation is a new closed curve r, consisting of n small circles, ..., about the fixed points, and two-way bridges connecting these circles (Figure 6.8.8). Note that 7 = 1, by property (1), since we didn t cross... 

This rule follows from a fact of multivariable calculus if J is the Jacobian of a... 

Richly illustrated, and with many exercises and worked examples, this book is ideal for an introductory course at the junior/senior or first-year graduate level. It is also ideal for the scientist who has not had formal instruction in nonlinear dynamics, but who now desires to begin informal study. The prerequisites are multivariable calculus and introductory physics. 

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