Формулы векторного анализа для производной Лагранжа

Last modified by Alexey Popov on 2019/07/23 11:11

${\frac {D\phi }{Dt}}={\frac {\partial \phi }{\partial t}}+({\mathbf {u}}\cdot \nabla )\phi$

${\frac {D\phi }{Dt}}= {\frac {\partial \phi }{\partial t}}+\mathbf {u} \cdot \nabla \phi$

$\nabla \cdot \left(\phi \mathbf {A} \right)=\phi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla \phi$

${\frac {D\phi }{Dt}}= {\frac {\partial \phi }{\partial t}}+\nabla\cdot( \phi \mathbf {u} ) - \phi (\nabla \cdot \mathbf {u} )$

$\nabla \cdot \mathbf {u} = 0$

${\frac {D\phi }{Dt}}= {\frac {\partial \phi }{\partial t}}+\nabla\cdot( \phi \mathbf {u} )$

${\frac {D{\mathbf {A}}}{Dt}}={\frac {\partial {\mathbf {A}}}{\partial t}}+({\mathbf {u}}\cdot \nabla ){\mathbf {A}}$

$\nabla \left(\mathbf {A} \cdot \mathbf {u} \right)=\left(\mathbf {A} \cdot \nabla \right)\mathbf {u} +\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} +\mathbf {A} \times \left(\nabla \times \mathbf {u} \right)+\mathbf {u} \times \left(\nabla \times \mathbf {A} \right)$

$\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} = \nabla \left(\mathbf {A} \cdot \mathbf {u} \right)-\left(\mathbf {A} \cdot \nabla \right)\mathbf {u} - \mathbf {A} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {A} \right)$

$\nabla \times \left(\mathbf {A} \times \mathbf {u} \right)\ =\ \mathbf {A} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {A} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} \,-\,\left(\mathbf {A} \cdot \nabla \right)\mathbf {u}$

$\left(\mathbf {A} \cdot \nabla \right)\mathbf {u} \ =\ \mathbf {A} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {A} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} \,-\,\nabla \times \left(\mathbf {A} \times \mathbf {u} \right)$

$\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} = \nabla \left(\mathbf {A} \cdot \mathbf {u} \right)- \mathbf {A} \left(\nabla \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {A} \right)\,-\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {A} \,+\,\nabla \times \left(\mathbf {A} \times \mathbf {u} \right) - \mathbf {A} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {A} \right)$

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

$\nabla \cdot \mathbf {u} = 0$

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

${\frac {D{\mathbf {A}}}{Dt}}={\frac {\partial {\mathbf {A}}}{\partial t}}+\frac{1}{2} \left ( \nabla \left(\mathbf {A} \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {A} \right) \,+\,\nabla \times \left(\mathbf {A} \times \mathbf {u} \right) - \mathbf {A} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {A} \right) \right)$

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Created by Alexey Popov on 2019/07/23 10:55