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

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thegirl
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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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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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