I Why can U be expanded in terms of T and V?

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The discussion centers on the mathematical expansion of internal energy (dU) in terms of temperature (dT) and volume (dV) when expressing entropy (dS). Participants explore whether a specific mathematical rule permits this expansion and if it alters the meaning of the expression. The conversation also touches on the number of intensive properties needed to define the state of a single-phase material with constant composition. The differential expression dU/dT(V) + dU/dV(T) is highlighted as significant in this context. Understanding these relationships is crucial for thermodynamic analysis.
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when expressing dS as a function of dV and dT, dU was expanded out as you can see in the screenshot below, is there a mathematical rule which allows this? does the fact that the internal energy is expanded out change the meaning of the expression?
Screen Shot 2016-03-24 at 17.45.17.png
 
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How many intensive properties are required to specify the state of a single phase material of constant composition?
 
thegirl said:
when expressing dS as a function of dV and dT, dU was expanded out as you can see in the screenshot below, is there a mathematical rule which allows this? does the fact that the internal energy is expanded out change the meaning of the expression?View attachment 97881
dU/dT(V)+dU/dV(T) is the differential of U with the coordinated (T,V)
 
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