Формулы векторного анализа

Обозначения

оператор набла: $\nabla$
градиент скалярного поля: $\nabla \psi =\operatorname {grad} \ \psi$
дивергенция векторного поля: $\nabla \cdot \mathbf {A} =\operatorname {div} \ \mathbf {A}$
ротор векторного поля: $\nabla \times \mathbf {A} =\operatorname {rot} \ \mathbf {A}$
лапласиан: $\Delta =\nabla ^{2}=\nabla \cdot \nabla$

Линейность

Для любого числа $\alpha$ :

$\ \nabla (\alpha \phi +\psi )=\alpha \nabla \phi +\nabla \psi$	$\ \mathbf {grad} (\alpha \phi +\psi )=\alpha \ \mathbf {grad} \ \phi +\mathbf {grad} \ \psi$
$\ \nabla \cdot (\alpha \mathbf {A} +\mathbf {B} )=\alpha \nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B}$	$\ \mathbf {div} \ (\alpha \mathbf {A} +\mathbf {B} )=\alpha \ \mathbf {div} \ \mathbf {A} +\mathbf {div} \ \mathbf {B}$
$\ \nabla \times (\alpha \mathbf {A} +\mathbf {B} )=\alpha \nabla \times \mathbf {A} +\nabla \times \mathbf {B}$	$\ \mathbf {rot} (\alpha \mathbf {A} +\mathbf {B} )=\alpha \ \mathbf {rot} \ \mathbf {A} +\mathbf {rot} \ \mathbf {B}$

Операторы второго порядка

$\ \nabla \times (\nabla \psi )=0$	$\ \mathbf {rot} (\mathbf {grad} \ \psi )=0$
$\ \nabla \cdot (\nabla \times \mathbf {A} )=0$	$\ \mathbf {div} \ (\mathbf {rot} \ \mathbf {A} )=0$
$\ \Delta \ \psi =\nabla \cdot (\nabla \psi )=\nabla ^{2}\psi$	$\ \Delta \ \psi =\mathbf {div} \ (\mathbf {grad} \ \psi )$
$\ \nabla \times \nabla \times \mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$	$\ \mathbf {rot} \ (\mathbf {rot} \ \mathbf {A} )=\mathbf {grad} \ (\mathbf {div} \ \mathbf {A} )-\Delta \mathbf {A}$

Дифференцирование произведений полей

$\nabla \cdot (\psi \mathbf {A} )=\mathbf {A} \cdot \nabla \psi +\psi \nabla \cdot \mathbf {A}$	$\mathbf {div} (\psi \mathbf {A} )=\mathbf {A} \cdot \mathbf {grad} \psi +\psi \ \mathbf {div} \mathbf {A}$
$\nabla \times (\psi \mathbf {A} )=\nabla \psi \times \mathbf {A} +\psi \nabla \times \mathbf {A}$	$\mathbf {rot} (\psi \mathbf {A} )=\mathbf {grad} \psi \times \mathbf {A} +\psi \ \mathbf {rot} \mathbf {A}$
$\nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +$ $+\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} )$	$\ \mathbf {grad} (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +$ $+\mathbf {A} \times \mathbf {rot} \mathbf {B} +\mathbf {B} \times \mathbf {rot} \mathbf {A}$
${\frac {1}{2}}\nabla A^{2}=\mathbf {A} \times (\nabla \times \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A}$	${\frac {1}{2}}\ \mathbf {grad} A^{2}=\mathbf {A} \times (\mathbf {rot} \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A}$
$\nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot \nabla \times \mathbf {A} -\mathbf {A} \cdot \nabla \times \mathbf {B}$	$\mathbf {div} \ (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot \mathbf {rot} \ \mathbf {A} -\mathbf {A} \cdot \mathbf {rot} \ \mathbf {B}$
$\ \nabla \times (\mathbf {A} \times \mathbf {B} )=\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+$ $\;+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B}$	$\ \mathbf {rot} (\mathbf {A} \times \mathbf {B} )=\mathbf {A} \ (\mathbf {div} \ \mathbf {B} )-\mathbf {B} \ (\mathbf {div} \ \mathbf {A} )+$ $\;+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B}$

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