$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$
Let $D$ be a bounded domain in $\mathbb{R}^2$ and $L=\partial D$ be its boundary. Then \begin{equation} -\int_{L} \mathbf{A}\cdot \mathbf{n} \,ds= \iint_D (\nabla \cdot \mathbf{A})\,dS \label{eq-A.1.1} \end{equation} where the left-hand side expression is a linear integral, the right-hand side expression is an area integral and $\mathbf{n}$ is a unit inner normal to $L$. This is Green formula.
Let $V$ be a bounded domain in $\mathbb{R}^3$ and $\Sigma=\partial V$ be its boundary. Then \begin{equation} -\iint_{\Sigma} \mathbf{A}\cdot \mathbf{n} \,dS= \iiint_D (\nabla \cdot \mathbf{A})\,dV \label{eq-A.1.2} \end{equation} where the left-hand side expression is a surface integral, the right-hand side expression is a volume integral and $\mathbf{n}$ is a unit inner normal to $\Sigma$. This is Gauss formula.
Remark 1.
Let $D$ be a bounded domain in $\mathbb{R}^2$ and $L=\partial D$ be its boundary, counter–clockwise oriented (if $L$ has several components then inner components should be clockwise oriented). Then \begin{equation} \oint_{L} \mathbf{A}\cdot \, d\, \mathbf{r}= \iint_D (\nabla \times \mathbf{A})\cdot\mathbf{n}\,dS \label{eq-A.1.4} \end{equation} where the left-hand side expression is a line integral, the right-hand side expression is an area integral and $\mathbf{n}=\mathbf{k}$. This is Green formula again.
Let $\Sigma$ be a bounded piece of the surface in $\mathbb{R}^3$ and $L=\partial \Sigma$ be its boundary. Then \begin{equation} \oint_{L} \mathbf{A}\cdot \, d\, \mathbf{l}= \iint_\Sigma (\nabla \times \mathbf{A})\cdot\mathbf{n}\,dS \label{eq-A.1.5} \end{equation} where the left-hand side expression is a line integral, the right-hand side expression is a surface integral and $\mathbf{n}$ is a unit normal to $\Sigma$; orientation of $L$ should match to direction of $\mathbf{n}$. This is Stokes formula.
Remark 2.
Definition 1.
Definition 2. \begin{equation} \Delta= \nabla^2 = \nabla\cdot \nabla= \partial_x^2 + \partial_y^2+ \partial_z^2. \label{eq-A.1.7} \end{equation} is Laplace operator or simply Laplacian.
Four formulae to remember: \begin{gather} \nabla (\nabla \phi)= \Delta \phi,\label{eq-A.1.8}\\[3pt] \nabla \times (\nabla \phi)= 0,\label{eq-A.1.9}\\[3pt] \nabla \cdot (\nabla \times \mathbf{A})= 0,\label{eq-A.1.10}\\[3pt] \nabla \times (\nabla \times \mathbf{A})= -\Delta \mathbf{A} + \nabla (\nabla \cdot \mathbf{A}) \label{eq-A.1.11} \end{gather} where all but the last one are obvious and the last one follows from \begin{equation} \mathbf{a}\times (\mathbf{a} \times \mathbf{b})= - \mathbf{a}^2 \mathbf{b}+ (\mathbf{a}\cdot \mathbf{b}) \mathbf{a} \label{eq-A.1.12} \end{equation} which is the special case of \begin{equation} \mathbf{a}\times (\mathbf{b} \times \mathbf{c})= \mathbf{b}(\mathbf{a}\cdot\mathbf{c})- \mathbf{c}(\mathbf{a}\cdot\mathbf{b}). \label{eq-A.1.13} \end{equation}
Recall Leibniz rule how to apply the first derivative to the product which can be symbolically written as \begin{equation*} \partial (uv)= (\partial_u + \partial_v)(uv)= \partial_u (uv)+\partial_v (uv)= v\partial_u (u) +u\partial_v (v)=v\partial u +u\partial v \end{equation*} where subscripts "$u$" or "$v$" mean that it should be applied to $u$ or $v$ only.
Since $\nabla$ is a linear combination of the first derivatives, it inherits the same rule. Three formulae are easy \begin{gather} \nabla ( \phi\psi)= \phi\nabla \psi +\psi \nabla \phi,\label{eq-A.1.14}\\[3pt] \nabla \cdot ( \phi\mathbf{A})= \phi\nabla \cdot \mathbf{A} +\nabla \phi\cdot \mathbf{A} , \label{eq-A.1.15}\\[3pt] \nabla \times ( \phi\mathbf{A})= \phi\nabla \times \mathbf{A} +\nabla \phi\times \mathbf{A} , \label{eq-A.1.16}\end{gather} and the fourth follows from the Leibniz rule and (\ref{eq-A.1.13}) \begin{equation} \nabla \times ( \mathbf{A}\times \mathbf{B})= (\mathbf{B}\cdot\nabla)A-\mathbf{B}(\nabla\cdot \mathbf{A}) - (\mathbf{A}\cdot\nabla)B+\mathbf{A}(\nabla\cdot \mathbf{B}). \label{eq-A.1.17} \end{equation}