Help with vector calculus integration identity proof I have limited knowlledge of how integration of a vector is correctly apply. In the following example, a vector is multiply to a vector dot product with an area. The integration is around the area of the sphere 4 Pi steradian.
I am interested to see how finding proof of this vector integral. Not sure how to get started. I have found this identity in textbooks but cant see how this integration results in 1/4 the surface area of a sphere.
$$\int _{4 \pi }\overset{\rightharpoonup }{s}   (\overset{\rightharpoonup }{s}\cdot A) d \omega=\frac{1}{3} (4 \pi ) A$$
Any good reference?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

Your question is NOT so clear but I guess that the answer is something like

\begin{align}
\int_{4\pi}\vec{s}\pars{\vec{s}\cdot\vec{A}}\dd\omega & =
\sum_{i,j\ \in\ \braces{x,y,z}}\hat{e}_{i}A_{j}\int_{4\pi}s_{i}s_{j}\,\dd\omega =
\sum_{i,j\ \in\ \braces{x,y,z}}
\hat{e}_{i}A_{j}\,\delta_{ij}\int_{4\pi}s_{i}^{2}\,\dd\omega
\\[5mm] & =
\pars{{1 \over 3}\int_{4\pi}\ \overbrace{\sum_{k\ \in\ \braces{x,y,z}}s_{k}^{2}}^{\ds{=\ 1}}\
\,\dd\omega}\
\overbrace{\sum_{i\ \in\ \braces{x,y,z}}\hat{e}_{i}A_{i}}^{\ds{=\ \vec{A}}}\
\\[5mm] & =
\pars{{1 \over 3}\ \overbrace{\int_{4\pi}\dd\omega}^{\ds{=\ 4\pi}}}\vec{A} =
\bbx{{1 \over 3}\pars{4\pi}\vec{A}}
\end{align}
