Формулы векторного анализа
Обозначения
- оператор набла: [math]\displaystyle{ \nabla }[/math]
- градиент скалярного поля: [math]\displaystyle{ \nabla \psi = \operatorname{grad}\ \psi }[/math]
- дивергенция векторного поля: [math]\displaystyle{ \nabla \cdot \mathbf{A} = \operatorname{div}\ \mathbf{A} }[/math]
- ротор векторного поля: [math]\displaystyle{ \nabla \times \mathbf{A} = \operatorname{rot}\ \mathbf{A} }[/math]
- лапласиан: [math]\displaystyle{ \Delta = \nabla^2 = \nabla \cdot \nabla }[/math]
Линейность
Для любого числа [math]\displaystyle{ \alpha }[/math]:
| [math]\displaystyle{ \ \nabla ( \alpha \phi + \psi ) = \alpha \nabla \phi + \nabla \psi }[/math] | [math]\displaystyle{ \ \mathbf{grad} ( \alpha \phi + \psi ) = \alpha\ \mathbf{grad}\ \phi + \mathbf{grad}\ \psi }[/math] |
| [math]\displaystyle{ \ \nabla \cdot ( \alpha \mathbf{A} + \mathbf{B} ) = \alpha \nabla \cdot \mathbf{A} + \nabla \cdot \mathbf{B} }[/math] | [math]\displaystyle{ \ \mathbf{div}\ ( \alpha \mathbf{A} + \mathbf{B} ) = \alpha\ \mathbf{div}\ \mathbf{A} + \mathbf{div}\ \mathbf{B} }[/math] |
| [math]\displaystyle{ \ \nabla \times ( \alpha \mathbf{A} + \mathbf{B} ) = \alpha \nabla \times \mathbf{A} + \nabla \times \mathbf{B} }[/math] | [math]\displaystyle{ \ \mathbf{rot} ( \alpha \mathbf{A} + \mathbf{B} ) = \alpha\ \mathbf{rot}\ \mathbf{A} + \mathbf{rot}\ \mathbf{B} }[/math] |
Операторы второго порядка
| [math]\displaystyle{ \ \nabla \times ( \nabla \psi ) = 0 }[/math] | [math]\displaystyle{ \ \mathbf{rot}(\mathbf{grad}\ \psi) = 0 }[/math] |
| [math]\displaystyle{ \ \nabla \cdot ( \nabla \times \mathbf{A} ) = 0 }[/math] | [math]\displaystyle{ \ \mathbf{div}\ (\mathbf{rot}\ \mathbf{A} ) = 0 }[/math] |
| [math]\displaystyle{ \ \Delta\ \psi = \nabla \cdot (\nabla \psi) = \nabla^2 \psi }[/math] | [math]\displaystyle{ \ \Delta\ \psi = \mathbf{div}\ (\mathbf{grad}\ \psi) }[/math] |
| [math]\displaystyle{ \ \nabla \times \nabla \times \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A} }[/math] | [math]\displaystyle{ \ \mathbf{rot}\ (\mathbf{rot}\ \mathbf{A}) = \mathbf{grad}\ (\mathbf{div}\ \mathbf{A}) - \Delta\mathbf{A} }[/math] |
Дифференцирование произведений полей
| [math]\displaystyle{ \nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A} }[/math] | [math]\displaystyle{ \mathbf{div}(\psi\mathbf{A}) = \mathbf{A} \cdot \mathbf{grad}\psi + \psi\ \mathbf{div}\mathbf{A} }[/math] |
| [math]\displaystyle{ \nabla \times (\psi\mathbf{A}) = \nabla\psi \times \mathbf{A} + \psi\nabla \times \mathbf{A} }[/math] | [math]\displaystyle{ \mathbf{rot} (\psi\mathbf{A}) = \mathbf{grad}\psi \times \mathbf{A} + \psi\ \mathbf{rot}\mathbf{A} }[/math] |
| [math]\displaystyle{ \nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + }[/math]
[math]\displaystyle{ + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) }[/math] |
[math]\displaystyle{ \ \mathbf{grad}(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + }[/math]
[math]\displaystyle{ + \mathbf{A} \times \mathbf{rot} \mathbf{B} + \mathbf{B} \times \mathbf{rot} \mathbf{A} }[/math] |
| [math]\displaystyle{ \frac{1}{2} \nabla A^2 = \mathbf{A} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{A} }[/math] | [math]\displaystyle{ \frac{1}{2}\ \mathbf{grad} A^2 = \mathbf{A} \times (\mathbf{rot} \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{A} }[/math] |
| [math]\displaystyle{ \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot \nabla \times \mathbf{A} - \mathbf{A} \cdot \nabla \times \mathbf{B} }[/math] | [math]\displaystyle{ \mathbf{div}\ (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot \mathbf{rot}\ \mathbf{A} - \mathbf{A} \cdot \mathbf{rot}\ \mathbf{B} }[/math] |
| [math]\displaystyle{ \ \nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + }[/math]
[math]\displaystyle{ \;+ (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} }[/math] |
[math]\displaystyle{ \ \mathbf{rot} (\mathbf{A} \times \mathbf{B}) = \mathbf{A}\ (\mathbf{div}\ \mathbf{B}) - \mathbf{B}\ (\mathbf{div}\ \mathbf{A}) + }[/math]
[math]\displaystyle{ \;+ (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} }[/math] |