let's start with topics I have quite good command of :
Hilbert spaces, Measure theory, functional analysis/operator theory
topics I totally don't : vector calculus
so I was studying PDE's using Strauss's book in conjunction with Evans book and it was going well until I dove into green's functions.
it seems that the divergence theorem is used to show some identities.
since I don't really have much free time I'd like to know what's the minimum necessary of vector calculus that I should review before proceeding away ?
also if you have any specific reference to recommend me it would be great. thanks !
Depending on which chapters you focus on, for most applications those identities are the important one. Let $u,v:\Omega \subset\mathbb{R}^n \to \mathbb{R}$ be sufficently smooth, integrable function s.t. all requirements for the divergence theorem are met and $\phi\in C_0^{\infty}(\Omega)$. Also, let $\Omega$ be a compact domain with smooth enough boundary.
1) Applying the divergence theorem to the function $f=u\phi e_i$, where $e_i$ denotes the $i$-th canonical basis vector. Then you have: $$ \int_{\Omega}div(f)dx=\int_{\Omega}u_i\phi+u\phi_idx=\int_{\partial \Omega}f \cdot n \;dS=0 $$ which implies: $$ \int_{\Omega}u_i\phi=-\int_{\Omega}u\phi_idx $$ This is an "integration by parts identity" used for defining weak solutions and studying sobolev spaces as well as acquiring integral indentities.
2)The second is applying the divergence theorem to $f=v\nabla u$ which gives you greens identity: $$ \int_{\Omega} div(f)dx=\int_{\Omega} v \Delta u + \nabla u\cdot\nabla v \; dx=\int_{\partial\Omega} f \cdot n \; dS=\int_{\partial \Omega} \frac{\partial u}{\partial n}v \; dS $$ In one line: $$ \int_{\Omega} v \Delta u + \nabla u\cdot\nabla v \; dx=\int_{\partial \Omega} \frac{\partial u}{\partial n}v \; dS $$ which is greens second identity. It is often used for studying equations involving the laplacian.
Those are the 2 most important applications. Apart from that, any calculus II/III books should cover you. I personally used Analysis II+III by Amann and Escher.