We know from simple Calculus that,
$$\int\_K f(x)f'(x)dx=\frac{f^2(x)}{2}|\_K$$
Suppose we have a function $u$ defined on $\Omega\in\mathbb{R}^d$ for $d=2,3$. Then, can we say
$$\int\_\Omega u\nabla u.\mathbb{n}dS=\frac{u^2}{2}|\_{\Omega}$$
where $\mathbb{n}$ is the outward unit normal and $\Omega$ is a simple domain such as a square or triangle.