$$
\int_{a V} v_{i} n_{j} \mathrm{~d} A=\int_{V} \frac{\partial v_{i}}{\partial x_{j}} \mathrm{~d} V .
$$
GrEEN’s identities also follow from adequate application of the Divergence Theorem. The first GREEN’s identity is obtained by choosing in $f=\phi v$. This yields
$$
\begin{aligned}
\int_{\partial V} \phi v \cdot n \mathrm{~d} A &=\int_{V} \operatorname{div}(\phi v) \mathrm{d} V \
&=\int_{V}{\phi \operatorname{div} v+v \operatorname{grad} \phi} \mathrm{d} V .
\end{aligned}
$$
If we substitute here $v=\operatorname{grad} \psi$ and observe that div $v=\operatorname{div} \operatorname{grad} \psi=\Delta \psi$, where $\Delta$ is the LAPLACE operator, then
$$
\int_{V}{\operatorname{grad} \phi \cdot \operatorname{grad} \psi} \mathrm{d} V=-\int_{V} \phi \Delta \psi \mathrm{d} V+\int_{\partial V} \phi \frac{\partial \psi}{\partial n} \mathrm{~d} A
$$
which is GREEN’s first identity.
If in the roles of $\phi$ and $\psi$ are interchanged, we obtain
$$
\int_{V}{g r a d \psi \cdot \operatorname{grad} \phi} \mathrm{d} V=-\int_{V} \psi \Delta \phi \mathrm{d} V+\int_{\partial V} \psi \frac{\partial \phi}{\partial n} \mathrm{~d} A
$$
PHS1001 COURSE NOTES ：
which, when combined, yields
$$
\int_{\mathcal{C}} \phi \mathrm{d} x=-\int_{A_{C}}(\operatorname{grad} \phi) \times n \mathrm{~d} A
$$
If $\phi$ in is the ith component of a vector field, then it also implies
$$
\int_{\mathcal{C}} v \otimes \mathrm{d} x=-\int_{A_{\mathcal{C}}}(\operatorname{grad} v) \times \boldsymbol{n} \mathrm{d} A
$$
A further interesting formula is obtained by selecting
$$
e_{i} \times \int_{\mathcal{C}} v_{i} \mathrm{~d} x=\oint_{\mathcal{C}} v \times \mathrm{d} x=-\int_{A_{C}} \underbrace{e_{i} \times \operatorname{grad} v_{i}}{-\text {curl } v} \times n \mathrm{~d} A $$ so that $$ \oint{\mathcal{C}} v \times \mathrm{d} x=\int_{A_{\mathcal{C}}}(\operatorname{curl} v) \times \boldsymbol{d} A
$$