Формулы векторного анализа

Обозначения

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

Линейность

Для любого числа ${\displaystyle \alpha }$:

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

Операторы второго порядка

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

Дифференцирование произведений полей

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

См. также

• Векторный анализ
• Теорема Стокса
• Оператор набла в различных системах координат