I am trying to solve problem 1.15 in Jackson's Classical Electrodynamics (3rd Edition). I found a bunch of solutions (proofs) to the problem, one of which shows: $$ \begin{align*} \delta W &=\epsilon_0\int_V \nabla \Phi\cdot\nabla\delta\Phi \ d^3x\\=&\epsilon_0\nabla\Phi\delta\Phi\bigg|_{\partial V}-\epsilon_0\int_V \delta\Phi\nabla\Phi\ d^3x \end{align*} $$ where, $\delta\Phi$ is a small incremental of potential $\Phi$. The last step was derived by integration by parts. This most puzzled me. To me, the correct integration by parts would be $$ \epsilon_0\int_V \nabla \Phi\cdot\nabla\delta\Phi \ d^3x=\epsilon_0\oint_S\delta\Phi\frac{\partial \Phi}{\partial n}\ d^2S-\epsilon_0\int_V \delta\Phi\Delta\Phi\ d^3x $$
Question:
How $\oint_S\delta\Phi\frac{\partial \Phi}{\partial n}\ d^2S$ can be written as $\nabla\Phi\ \delta\Phi\bigg|_{\partial V}$ is beyond me. It seems to me that $\nabla\Phi\ \delta\Phi\bigg|_{\partial V}$ is a vector while $\oint_S\delta\Phi\frac{\partial \Phi}{\partial n}\ d^2S$ is a scalar.
Thanks