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1 {{formula}}
2 \mathbf u = \frac{D\mathbf r}{Dt} = \frac{\partial \mathbf r}{\partial t} + \left (\mathbf u \cdot \nabla)\mathbf r
3 {{/formula}}
4
5 {{formula}}
6 \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)=\left(\mathbf {r} \cdot \nabla \right)\mathbf {u} +\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} +\mathbf {r} \times \left(\nabla \times \mathbf {u} \right)+\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)
7 {{/formula}}
8
9 {{formula}}
10 \left(\mathbf {u} \cdot \nabla \right)\mathbf {r} = \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)-\left(\mathbf {r} \cdot \nabla \right)\mathbf {u} - \mathbf {r} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)
11 {{/formula}}
12
13 {{formula}}
14 \nabla \times \left(\mathbf {r} \times \mathbf {u} \right)\ =\ \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,-\,\left(\mathbf {r} \cdot \nabla \right)\mathbf {u}
15 {{/formula}}
16
17 {{formula}}
18 \left(\mathbf {r} \cdot \nabla \right)\mathbf {u} \ =\ \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right)
19 {{/formula}}
20
21 {{formula}}
22 \left(\mathbf {u} \cdot \nabla \right)\mathbf {r} = \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,-\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)
23 {{/formula}}
24
25 {{formula}}
26 \left(\mathbf {u} \cdot \nabla \right)\mathbf {r} =\frac{1}{2} \left ( \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right) \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {r} \right) \right)
27 {{/formula}}
28
29 {{formula}}
30 \left(\mathbf {u} \cdot \nabla \right)\mathbf {r} =\frac{1}{2} \left ( \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega \right)
31 {{/formula}}
32
33 {{formula}}
34 \mathbf u = \frac{D\mathbf r}{Dt} = \frac{\partial \mathbf r}{\partial t} + \frac{1}{2} \left ( \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega \right)
35 {{/formula}}
36
37 {{formula}}
38 2\mathbf u = 2\frac{\partial \mathbf r}{\partial t} + \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega
39 {{/formula}}
40
41 {{formula}}
42 2\mathbf u \,-\,3\mathbf {u}= 2\frac{\partial \mathbf r}{\partial t} + \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right) \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega
43 {{/formula}}
44
45 {{formula}}
46 \mathbf {u}= -2\frac{\partial \mathbf r}{\partial t} - \nabla \left(\mathbf {r} \cdot \mathbf {u} \right) \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) + \mathbf {r} \times \boldsymbol \omega
47 {{/formula}}
48
49
50 == Дивергенция вектора скорости ==
51
52
53 {{formula}}
54 2\nabla \cdot \mathbf u =\nabla^2(\mathbf {r} \cdot \mathbf {u}) - \mathbf r \cdot \nabla^2 \mathbf u = 2\nabla \cdot ((\mathbf u \cdot\nabla )\mathbf r)
55 {{/formula}}