Полная производная по времени от рывка

Last modified by Alexey Popov on 2019/06/07 10:39

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

где

$\nabla_{\mathbf v} = \frac{\partial }{\partial v_x}+\frac{\partial }{\partial v_y}+\frac{\partial }{\partial v_z}$

$\nabla_{\mathbf a} = \frac{\partial }{\partial a_x}+\frac{\partial }{\partial a_y}+\frac{\partial }{\partial a_z}$

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

$\left(\mathbf {a} \cdot \nabla_{\mathbf v} \right)\mathbf {j} =\frac{1}{2} \left ( \nabla_{\mathbf v} \left(\mathbf {j} \cdot \mathbf {a} \right)- \mathbf {j} \left(\nabla_{\mathbf v} \cdot \mathbf {a} \right)\,+\,\mathbf {a} \left(\nabla_{\mathbf v} \cdot \mathbf {j} \right) \,+\,\nabla_{\mathbf v} \times \left(\mathbf {j} \times \mathbf {a} \right) - \mathbf {j} \times \left(\nabla_{\mathbf v} \times \mathbf {a} \right)-\mathbf {a} \times \left(\nabla_{\mathbf v} \times \mathbf {j} \right) \right)$

$(\mathbf j \cdot \nabla_{\mathbf a})\mathbf j = \frac{1}{2}\nabla_{\mathbf a} (\mathbf j \cdot \mathbf j) - \mathbf j \times (\nabla_{\mathbf a} \times \mathbf j)$

Ток текущий поперёк магнитных диполей
$\mathbf {u} \times \left(\nabla \times \mathbf {j} \right)$

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Created by Alexey Popov on 2019/02/28 11:00