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**1. Homework Statement**

Given the entropy of a system :

$$ S = AU^αV^βN^{1-α-β} $$

The problem requires me to write

$$ (\frac{∂T}{∂U})_{V,N} > 0, (\frac{∂P}{∂V})_{U,N} < 0, (\frac{∂μ}{∂N})_{U,V} > 0$$

to find the mathematical constraint of α and β

**2. Homework Equations**

dU = TdS - PdV + μdN

**3. The Attempt at a Solution**

Actually, I don't quite understand the rationale behind Maxwell relations.

I tried to write

dF = SdT - PdV + μdN

I can then get

$$ (\frac {∂T} {∂F})_{V,N}$$

but I just don't know how to transform the variable. Can anyone give me an idea of these kind of problems. Thanks!