Полная производная по времени от ускорения

Last modified by Alexey Popov on 2020/02/26 10:53

$\mathbf j = \frac{D\mathbf a(t,\mathbf r(t))}{Dt} = \frac{\partial \mathbf a}{\partial t} + (\mathbf u \cdot \nabla)\mathbf a$

где

$\left(\mathbf {u} \cdot \nabla \right)\mathbf {a} =\frac{1}{2} \left ( \nabla \left(\mathbf {a} \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {a} \right) \,+\,\nabla \times \left(\mathbf {a} \times \mathbf {u} \right) - \mathbf {a} \times \boldsymbol \omega-\mathbf {u} \times \left(\nabla \times \mathbf {a} \right) \right)$

$\mathbf j = \frac{\partial \mathbf a}{\partial t} + \frac{1}{2} \left ( \nabla \left(\mathbf {a} \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {a} \right) \,+\,\nabla \times \left(\mathbf {a} \times \mathbf {u} \right) - \mathbf {a} \times \boldsymbol \omega-\mathbf {u} \times \left(\nabla \times \mathbf {a} \right) \right)$

$\mathbf j = \frac{\partial \mathbf a}{\partial t} + \frac{1}{2} \left ( \nabla \left(\mathbf {a} \cdot \mathbf {u} \right) - \mathbf {a} \times \boldsymbol \omega-\mathbf {u} \times \left(\nabla \times \mathbf {a} \right) \right)$

$\frac{\partial \mathbf a}{\partial t} = \frac{1}{2} \left ( - \nabla \left(\mathbf {a} \cdot \mathbf {u} \right) \,-\,\mathbf {u} \left(\nabla \cdot \mathbf {a} \right)+ \mathbf {a} \times \boldsymbol \omega+\mathbf {u} \times \left(\nabla \times \mathbf {a} \right) \right)$

$\nabla \cdot \mathbf a = \nabla^2 \varphi$

$\nabla^2 \mathbf u =- \frac{1}{c^2}(\nabla \cdot \mathbf a) \mathbf u$

Tags:

Created by Alexey Popov on 2019/02/28 10:43