Green’s Three Identities are the fundamental results for vector calculus, which is widely used in basic PDE theories and fluid mechanics. Here I present the results.
Assume that $U \subset \mathbb{R}^n$ be open and has $C^1$ boundary, then:
Integration by Part
Result 1 (Integration by Part): for $u,v \in C^1(\bar{U})$, we have:
$$\int_U u_{x_i} v dx + \int_U v_{x_i} u dx = \int_{\partial U} uv \hat{n}^i dS, \quad (i = 1,\cdots, n)$$
where $\hat{n}^i$ are the outward normal’s i-th component.
Divergence Theorem
Result 2 (Divergence Theorem): for $u \in C^1(\bar{U})$, we have:
$$\int_U u_{x_i} dV = \int_{\partial U} u \hat{n}^i dS $$
The divergence theorem turns volume integral into surface integral, by turning gradient operator into dot product with the outward normal:
$$\int_U \nabla u dV = \int_{\partial U} u \hat{n} dS$$
In 3D this equation has a special form with divergence:
$$\int_V \nabla \cdot F dV = \int_{partial V} F \cdot \hat{n} dS$$
Green’s 1st Identity
Result 3 (Green’s 1st Identity): Applying the divergence theorem to the vector field $F = u\nabla v$, then:
$$\int_U (u \Delta v + \nabla u \cdot \nabla v) dV = \int_{\partial U} u (\nabla v \cdot \hat{n})dS = \int_{\partial U} \frac{\partial v}{\partial \hat{n}}u$$
where $\Delta = \nabla \cdot \nabla$ is the Laplace operator, and the notation of $\frac{\partial v}{\partial \hat{n}}u$ is equivalent to $u (\nabla v\cdot \hat{n})$. This identity is useful for energy-based proofs used in PDE theory, since $\nabla u \cdot \nabla u$ is $||u||_2^2$.
Green’s 2nd Identity
Result 4 (Green’s 2nd Identity): Use IBP, with $u_{x_i}$ in place of $u$ and $v=1$, we have:
$$\begin{aligned} & \int_U u_{x_i x_i} dx = \int_{\partial U} u_{x_i} \hat{n}^i dS\\ & \int_U \Delta u dx = \int_{\partial U} \frac{\partial u}{\partial \hat{n}}dS \end{aligned}$$
Green’s 3rd Identity
Result 5 (Green’s 3rd Identity): obtained from 1st identity:
$$\int_U u\nabla v - v\nabla u dx = \int_{\partial U} u \frac{\partial v}{\partial \hat{n}} - v \frac{\partial u}{\partial \hat{n}}dS$$