Partial derivatives with dependent variables (fixed) question.

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In statistical mechanics, taking partial derivatives while holding certain variables constant can be confusing, especially when those variables are functions of the variable being differentiated. The distinction between partial and total derivatives is crucial in these cases, as treating dependent variables incorrectly can lead to contradictions. For example, when differentiating S with respect to T while holding P and V constant, one must consider the relationships between these variables, such as P*V being a function of T. The confusion often arises when substituting relationships into the function without properly accounting for the dependencies. Resources like videos or intuitive explanations can help clarify these concepts further.
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In statistical mechanics we express partial derivatives of functions, keeping some variables fixed. But these variables are functions of the other variables (which are not fixed).

I'm just confused by this, what is the convention for taking these derivatives? For example, if we have S as a function of T, V and P, or S=S(T,V,P)

we want to find partial of S with respect to T, holding P and V constant. Put P and V are functions of T, i.e. P*V=constant*T. How does this work?

If I have, say, S=log(T*V*P), then is the partial derivative mentioned above just equal to (ds/dt)V,P = 1/T ?

What if I write it as S=log(T*(constant*T)) using the relation P*V=constant*T. Shouldn't the derivative be the same thing? Why is there a contradiction?

Does anyone have any good way of explaining this or some good links? I tried to look through books/wikipedia/websites and found nothing.

Thanks.
 
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You need to distinguish between partial derivative and total derivative when the"constant" parameters are functions of the "variable" parameter. That's what getting you into trouble in your example.
 
Okay, I think I know what you mean. But can you give me an example or point me to some examples elsewhere?

My old calc book does not cover this, I already looked.
 
As mathman already mentioned you have to know the difference between partial and total derivative. Have a look at this excellent video by David Metzler in which he gives an intuitive explanation with a bug on a hot plate.

 
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