statistical mechanics - 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:

${Fj|j=1,..,N}$

$S= \sum_{j=1}^{N} F_{j}$

$=< \sum_{j=1}^{N} F_{j}> = \sum_{j=1}^{N} $

$\sigma^{2}_{S} =-^{2}$

line a:

$=\sum_{j=1}^{N}\sum_{k=1}^{N} - \sum_{j=1}^{N} \sum_{k=1}^{N}$

line b:

$=\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} +\sum_{j=1}^{N} - \sum_{j=1}^{N} \sum_{k=1}^{N}$

line c:

$=\sum_{j=1}^{N} (-^{2})$ $=\sum_{j=1}^{N} \sigma_{j}^{2}$

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

Answer

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

$\sum_{j=1}^{N}\sum_{k=1(k\neq j))}^{N} - \sum_{j=1}^{N} \sum_{k=1}^{N}$ you get

$\sum_{j=1}^{N} ^2$

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.