The Space of Thermodynamic Response Vectors

In what sense might a thermodynamic variable be considered as a vector, or the associated equilibrium state as a geometrical space of vectors  

In characterizing a thermodynamic state variable (e.g., temperature, pressure, or energy), we may choose to focus on either of two distinctive aspects of the variable (i) its numerical value, which may serve as a scalar descriptor or identifier of the state, and (ii) its variability or responsiveness to some stimulus (i.e., an attempted change of a chosen control variable of the state). These distinct aspects may require distinct mathematical objects and symbols for proper description. 

In the calculus-based description of thermodynamic phenomena, these two aspects of a variable are distinguished as the value (F) versus the variability (dF) of an underlying function [F(...)], for example, 

The calculus-based symbolism (10.1), (10.2) provides dependable, if unwieldy, descriptions of thermodynamic responses. The differential aspect (10.2) often transcends the functional aspect (10.1) for thermodynamic purposes, and, indeed, special criteria may be invoked to test whether an underlying function (10.1) exists for measured differential 

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

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