Math for Thermodynamics Basics I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present.
There is a section that i can not understand:
${Fj|j=1,..,N}$
$S= \sum_{j=1}^{N} F_{j}$
$<S>=< \sum_{j=1}^{N} F_{j}> = \sum_{j=1}^{N} <F_{j}>$
$\sigma^{2}_{S} =<S^{2}>-<S>^{2}$
line a:
$=\sum_{j=1}^{N}\sum_{k=1}^{N} <F_{j}F_{k}> - \sum_{j=1}^{N} <F_{j}>\sum_{k=1}^{N}<F_{k}>$
line b:
$=\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} <F_{j}><F_{k}> +\sum_{j=1}^{N} <F_{j}^{2}> - \sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$
line c:
$=\sum_{j=1}^{N} (<F_{j}^{2}>-<F_{j}>^{2})$
$=\sum_{j=1}^{N} \sigma_{j}^{2}$
My question is what happened after line a to line b and after that to line c?
My other question is, i have a little math, what should i study to understand such thermodynamics root math studies, calculus 1 or 2 or what else, can you specify a math topic?
Thanks
 A: I'll use a much simpler notation for starters, going to drop $\langle$ and $\rangle$. So the first term in line a is 
$\sum_{i}\sum_{j}A_iA_j$
and if you write it explicitly you have
$\sum_{i}\sum_{j}A_iA_j=(A_1A_1+A_2A_2+\dots+A_nA_n)+(A_1A_2+A_1A_3+\dots+A_1A_n)+\dots+(A_nA_1+A_nA_2+\dots+A_nA_{n-1})=\sum_iA_{i}^2+A_1\sum_{i\ne1}A_i+A_2\sum_{i\ne2}A_i+\dots+A_n\sum_{i\ne n}A_i=\sum_i A_{i}^2+\sum_i\sum_{j\ne i}A_iA_j$
So this gives you term one and two in line b. The third term in line b stays the same. Now for the last line when you take the following difference
$\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} <F_{j}><F_{k}> - \sum_{j=1}^{N} <F_ {j}>\sum_{k=1}^{N}<F_{k}>$
 you get 
$\sum_{j=1}^{N} <F_{j}>^2$
this is because the first double sum contains only terms like $F_iF_j$ and the second sum cointains terms like $F_iF_i$ and $F_iF_j$. So when you take the difference all the terms  $F_iF_j$ will cancel out and your left with $\langle F_i\rangle\langle F_i\rangle=\langle F_i\rangle^2$. Thus you find your final result.
