General Transformation Theory in Thermodynamic Metric Space

As shown by these values, the pressure vector dominates the thermodynamic responsiveness of the ideal gas, conferring near one-dimensional character on its thermodynamic geometry. 

4 GENERAL TRANSFORMATION THEORY IN THERMODYNAMIC METRIC SPACE 

Thus far the discussion has been largely confined to a specific set of reference intensities and extensities. However, equations such as (11.10) make clear the possibility of treating more general types of thermodynamic variations. Transformations among thermodynamic variables will correspond to ordinary Euclidean vector transformations, which can therefore be treated simply and systematically in Ms- 

An application of (11.10) was already seen in (11.16), where a specified linear combination of intensive variables was found to be associated with variations of an extensive coordinate. Such linear combinations are also necessary to represent variations along a coexistence curve, or along other paths in a phase diagram that are not parallel to one of the axes. Additional incentives to describe more general variations may arise from purely experimental considerations, where the variables under practical experimental control may involve simultaneous changes of two or more reference variables. It is therefore desirable that general expressions be available to allow easy transformation from one thermodynamic coordinate system to another. 

To develop such transformation properties of Ms we now consider the simultaneous transformations of the/reference intensities R, that are generated by an arbitrary (but nonsingular) real / x / matrix A  

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