Vector Geometry and Metric Spaces

Although geometrical representations of propositions in the thermodynamics of fluids are in general use, and have done good service in disseminating clear notions in this science, yet they have by no means received the extension in respect to variety and generality of which they are capable. 

A thermodynamic example may be illustrative. Consider Maxwell s model of the Gibbs USV surface for water (Fig. 1.1), as depicted schematically in Fig. 9.1. In this model, the physical (77, S, V) coordinates are associated with mutually perpendicular axes, and three chosen points on this surface form a triangle whose edges, angles, and area are as shown in Fig. 9.1a. However, the model might have been constructed (with equal thermodynamic justification) in a skewed /io/ orthogonal axis system (Fig. 9.1b) in which the 

Classical and Geometrical Theory of Chemical and Phase Thermodynamics. By Frank Weinhold Copyright 2009 John Wiley Sons, Inc. 

As mentioned in the Preface, our goal in Part III is not merely to re-generate the material of Parts I and II (as summarized in Section 8.9) in new mathematical dress. We re-derive (rather trivially) many earlier thermodynamic identities and stability conditions to illustrate the geometrical techniques, but our primary emphasis is on thermodynamic extensions (particularly, to saturation properties, critical phenomena, multicomponent Gibbs-Konowalow-type relationships, higher-derivative properties, and general reversible changes 

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Having made this long detour into vector geometry and metric spaces, the student of thermodynamics will naturally be impatient to learn the missing link that connects these disparate domains, i.e., that associates the scalar products of the geometry domain... 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

We now wish to introduce a still deeper form of geometry as first suggested by Bernhard Riemann (Sidebar 13.2). Riemann s formalism makes possible a distinction between the space of vectors whose metrical relationships are specified by the metric M and an associated linear manifold by which the vectors and metric are parametrized. Let be an element of a linear manifold (in general, having no metric character) that can uniquely identify the state of a collection of metrical objects X). The Riemannian geometry permits the associated metric M to itself be a function of the state,... 

The general least-squares treatment requires that the generalized sum of squares of the residuals, the variance a2, be minimized. This is, by the geometry of error space, tantamount to the requirement that the residual vector be orthogonal with respect to fit space, and this is guaranteed when the scalar products of all fit vectors (the rows of XT) with the residual vector vanish, XTM 1 = 0, where M 1 is the metric of error space. The successful least-squares treatment [34] yields the following minimum-variance linear unbiased estimators (A) for the variables, their covariance matrix, the variance of the fit, the residuals, and their covariance matrix ... 

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