Tools from the Calculus

Classical thermodynamics makes extensive use of the calculus in fact, thermodynamics employs calculus so extensively that it is worthwhile to have a summary of the most important concepts. That summary is provided here. Throughout this appendix, as in all thermodynamics, we presume that functions such as f x) and fix, y) satisfy the required conditions of continuity and differentiability. 

Basic relations among thermodynamic variables are routinely stated in terms of partial derivatives these relations include the fundamental equations from the first and second laws, as well as innumerable relations among properties. Here we define the partial derivative and give a graphical interpretation. Consider a variable z that depends on two independent variables, x and y, 

At a specified y-value the derivative of z with respect to (wrt) x, if it exists, is called the partial derivative of z wrt x it is defined by 

General forms for property changes and balance equations are usually posed as total differentials. Consider a quantity z that depends on two independent variables. 

We would like to know how z responds when we change either x or y or both. We have already found that the partial derivative dz/dx)y tells how z responds to a change in x at a fixed value of y likewise, dz/dy)x tells how z responds to a change in y at a fixed value of x. Then, if these partial derivatives are continuous, the total differential dz tells how z responds when we simultaneously change both x and y  

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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). 

Here we review the tools from multivariate calculus that we need to describe processes in which multiple degrees of freedom change together. We need these methods to solve two main problems to find the extrema of multivariate functions, and to integrate them. Here we introduce the mathematics. 

In fuzzy logic, the use of fuzzy if-then rules is governed by the calculus of fuzzy rules, CFR. A major part of CFR is the Fuzzy Dependency and Command Language, or FDCL for short. Basically, FDCL is a fuzzy programming language that provides a powerful tool for the representation and manipulation of imprecise or ill-defined dependencies. Two issues play pivotal roles in FDCL. The first is interpolation, and the second relates to the induction of rules from observations. 

Perhaps the most remarkable feature of modem chemical theory is the seamless transition it makes from a microscopic level (dealing directly with the properties of atoms) to describe the structure, reactivity and energetics of molecules as complicated as proteins and enzymes. The foundations of this theoretical structure are based on physics and mathematics at a somewhat higher level than is normally found in high school. In particular, calculus provides an indispensable tool for understanding how particles move and interact, except in somewhat artificial limits (such as perfectly constant velocity or acceleration). It also provides a direct connection between some observable quantities, such as force and energy. 

Next follows a detailed discussion of probability theory, stochastic simulation, statistics, and parameter estimation. As engineering becomes more focused upon the molecular level, stochastic simulation techniques gain in importance. Particular attention is paid to Brownian dynamics, stochastic calculus, and Monte Carlo simulation. Statistics and parameter estimation are addressed from a Bayesian viewpoint, in which Monte Carlo simulation proves a powerful and general tool for making inferences and testing hypotheses from experimental data. 

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