[Twickel]

Hi
This has been doing my head can anyone explain how they get nidlnni+lnnidi - dni

Cansomeone explain it to  me mathematically.

[Jorriss]

Stirlings approximation of N!.

lnN!  ~ NlnN-N

You can find many, many sources online that explain it in various detail.

[Twickel]

I understand that, so then where does dlnni*!=nidlnni+lnnidni-dni come from should it only be  dlnni*!=nidlnn-dni

[Jorriss]

This notation is very difficult to read, but I think the problem is that you are forgetting to use the product rule when finding a differential.

lnN! ~ NlnN-N

d(NlnN-N) = (dN)lnN + N(dlnN) - dN, where the first time two terms come from d(NlnN).

[Twickel]

Doh!

Thanks, ok so how do they cancel the next line, where does the ln dissapear to?

[Jorriss]

Doh!

Thanks, ok so how do they cancel the next line, where does the ln dissapear to?

dlnx/dx = 1/x so it cancels with the third term.

NdlnN = dN as dlnN = (1/N)dN

[Twickel]

Thank you, so what I am puzzled about is if I have taken the derivative to simplify the expression, why then do I keep the d in the simplified term. E.g

dinx= 1/x I dont write dlnx=d1/x

[Jorriss]

Thank you, so what I am puzzled about is if I have taken the derivative to simplify the expression, why then do I keep the d in the simplified term. E.g

dinx= 1/x I dont write dlnx=d1/x

It's neither of those. dlnN = (1/N)dN. Think chain rule.

You're not taking the derivative - the derivative showed up as part of the chain rule. You're finding the total differential. \

That's what we're actually trying to do here. Find the total differential of W (the multinomial coefficient). If you don't know what total differentials are, it's worth reviewing. They are pervasive throughout all of thermodynamics.