Integral and Wick rotation (Srednicki ch75) I was reading chapter 75 of Srednicki's QFT book and I   ran into this statement.
To determine the value of its integral, we
make a Wick rotation to euclidean space, which yields a factor of i as
usual; then we have
\begin{equation}
\int \frac{d^{4}l}{(2\pi)^4}\frac{\partial}{\partial l^{\beta}}f_{\alpha}(l)=i\lim_{l\rightarrow\infty}\int\frac{dS_{\beta}}{(2\pi)^{4}}f_{\alpha}(l)
\end{equation}
where $dS_{\beta}=l^{2}l_{\beta}d\Omega$ is a surface-area element, and $d\Omega$ is the differential
solid angle in four dimensions.
I don't understand how the LHS of the equation can be written as the RHS. Especially, how did the derivative $\frac{\partial}{\partial l^{\beta}}$ disappear? Can someone give me a detailed explanation?
 A: I believe Prof. Srednicki first performed the Wick rotation and then used a version of the 4-dimensional divergence theorem. 
Given the comments below the question, people are unaware of the "component-wise" divergence theorem, so I'll give a derivation below starting from the typical expression of the divergence theorem.
We have $$\int dV\:  \nabla \cdot \vec A  = \int dS\: \hat n \cdot \vec A $$ from the typical divergence theorem in d-dimensions. Now consider an $\vec A = \vec c f$ where $\vec c $ is a constant vector. This yields, using the usual product rule for derivatives, 
$$\vec c \cdot \int dV\:  \nabla f  = \vec c \cdot \int dS\: \hat n f $$
This is true for any $\vec c$, including the unit basis vectors, so we have
$$\int dV\:  \nabla f  = \int dS\: \hat n f. $$
Note that the above is a vector equation that holds componentwise:
$$\int dV\:  \partial_\mu f  = \int dS\: n_\mu f. $$
This is true for any $f$, so it is true for e.g. functions $g_0, g_1, ..., g_d$, which we can write as 
$$\int dV\:  \partial_\mu g_\nu  = \int dS\: n_\mu g_\nu. $$
Please let me know if you need any clarifications or if that addressed the heart of your question. I beleive that should answer where the derivative goes on the right hand side of your equation. All the functions above are assumed to be well-enough behaved.
