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Vector Formulas
$\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b})$
$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$
$(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c}) (\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d}) (\vec{b} \cdot \vec{c})$
$\nabla \times \nabla Ψ = 0$
$\nabla \cdot (\nabla \times \vec{a}) = 0$
$\nabla \times (\nabla \times \vec{a}) = \nabla (\nabla \cdot \vec{a}) - \nabla^{2} \vec{a}$
$\nabla \cdot (Ψ \vec{a}) = \vec{a} \cdot \nabla Ψ + Ψ \nabla \cdot \vec{a}$
$\nabla \times (Ψ \vec{a}) = \nabla Ψ \times \vec{a} + Ψ \nabla \times \vec{a}$
$\nabla (\vec{a} \cdot \vec{b}) = (\vec{a} \cdot \nabla) \vec{b} + (\vec{b} \cdot \nabla) \vec{a} + \vec{a} \times (\nabla \times \vec{b}) + \vec{b} \times (\nabla \times \vec{a})$
$\nabla \cdot (\vec{a} \times \vec{b}) = \vec{b} \cdot (\nabla \times \vec{a}) - \vec{a} \cdot (\nabla \times \vec{b})$
$\nabla \times (\vec{a} \times \vec{b}) = \vec{a} (\nabla \cdot \vec{b}) - \vec{b} (\nabla \cdot \vec{a}) + (\vec{b} \cdot \nabla) \vec{a} - (\vec{a} \cdot \nabla) \vec{b}$

( $\vec{r}$ is the coordinate of a point with respect to some origin.)
$\nabla \cdot \vec{r} = 3$	$\nabla \times \vec{r} = 0$
$\nabla \cdot [\vec{n} f (r)] = \frac{2}{r} f + \frac{\partial f}{\partial r}$	$\nabla \times [\vec{n} f (r)] = 0$
$(\vec{a} \cdot \nabla) \vec{n} f (r) = \frac{f (r)}{r} [\vec{a} - \vec{n} (\vec{a} \cdot \vec{n})] + \vec{n} (\vec{a} \cdot \vec{n}) \frac{\partial f}{\partial r}$
$\nabla (\vec{r} \cdot \vec{a}) = \vec{a} + \vec{r} (\nabla \cdot \vec{a}) + i (\vec{L} \times \vec{a})$
where $\vec{L} = \frac{1}{i} (\vec{r} \times \nabla)$ is the angular-momentum operator.

\int_{V} \nabla \cdot \vec{A} d^{3} x = \int_{S} \vec{A} \cdot \vec{n} d a

\int_{V} \nabla Ψ d^{3} x = \int_{S} Ψ \vec{n} d a

\int_{V} \nabla \times \vec{A} d^{3} x = \int_{S} \vec{n} \times \vec{A} d a

\int_{V} (Φ \nabla^{2} Ψ + \nabla Φ \cdot \nabla Ψ) d^{3} x = \int_{S} Φ \vec{n} \cdot \nabla Ψ d a

\int_{V} (Φ \nabla^{2} Ψ - Ψ \nabla^{2} Φ) d^{3} x = \int_{S} (Φ \nabla Ψ - Ψ \nabla Φ) \cdot \vec{n} d a

\int_{S} (\nabla \times \vec{A}) \cdot \vec{n} d a = \oint_{C} \vec{A} \cdot d \vec{l}

\int_{S} \vec{n} \times \nabla Ψ d a = \oint_{C} Ψ d \vec{l}