Полная производная по времени от скорости

Last modified by Alexey Popov on 2019/06/26 18:20

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

где
\boldsymbol \omega = \nabla \times \mathbf v

(\mathbf v \cdot \nabla)\mathbf v = \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) - \mathbf v \times (\nabla \times \mathbf v) =  \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) - \mathbf v \times \boldsymbol \omega

Получим

\mathbf a = \frac{\partial \mathbf v}{\partial t} + \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) - \mathbf v \times \boldsymbol \omega

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Created by Alexey Popov on 2019/06/17 18:20