I have some problems when I reading mathematical notations. For example:
$$\mathrm{E}_{0}\left\{\sum_{\mathrm{t}=0}^{\infty} \beta^{t}\left[\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)-\int_{0}^{\mathrm{1}} \mathrm{v}\left(\mathrm{h}_{\mathrm{t}}(\mathrm{i}) ; \xi_{\mathrm{t}}\right) \mathrm{di}\right]\right\}$$ This is a optimization equation for representative agent in an economy. This problem here is maximizing this formula. where
\begin{aligned}
&\mathrm{E}_{0}= \text{expected value in period 0} \\
&\mathrm{C}_{\mathrm{t}}= \text{ index of household consumption of goods supplied in period t}\\ &\mathrm{P}_{\mathrm{t}}= \text{index of prices of goods supplied in period t }\\
&\mathrm{M}_{\mathrm{t}}= \text{nominal money balances in period t }\\
&\mathrm{h}_{\mathrm{t}}(\mathrm{i})= \text{quantity of labour of type i supplied in period t}\\
&\xi_{t}=\text{ vector of aggregate shocks in period t}\\
&\mathrm{u}(.)=\text { utility in period } t\\
&\mathrm{v}(.)=\text { disutility in period } \mathrm{t}\\
&\beta^{\mathrm{t}}=\text { discount factor for period } \mathrm{t}
\end{aligned}
1- First of all, I don't understand the $\xi_{t}$ notation in $\mathrm{u}\left(\mathrm{C}_{\mathrm{t}}, \mathrm{M}_{\mathrm{t}} / \mathrm{P}_{\mathrm{t}} ; \xi_{\mathrm{t}}\right)$. Does semi colon here mean that this vector notation can affect the equation's behavior exceptionally, but the other variables affect it all the time?. If there would be a positive or negative shock already, then u and v will be effected through the prices and wages. Why should we use this vector here?
2- Why do we need to take the integral of the disutility function, but don't need to take the integral of the utility function?
3- There are no enough explanations for newbies in articles in mathematical economics, and I can't find a textbook for this topic. I studied Varian's Intermediate Microeconomics, but there is no chapter for this. I looked up books that recommended here but again there is no chapter at all. Is there a book that covers all classical mathematical solutions and explains them in detail for economics?