Comments | sharpc: А ТЫ помнишь векторный анализ?

sharpc

А ТЫ помнишь векторный анализ?

Dec 22, 2009 22:34

Сегодня я наткнулся на забавную головоломку.

Согласно формуле в Википедии,

$\nabla \times \nabla \times {\bf{A}} = \nabla \left( {\nabla \cdot {\bf{A}}} \right) - \nabla ^2 {\bf{A}}$
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задачи, математика

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sharpc December 23 2009, 14:24:09 UTC

В этом и состоит хитрость: никакого умножения там нет. Просто лапласиан еще и скалярное поле принимает в качестве аргумента.
$\begin{array}{l} \nabla ^2 {\bf{A}} = \frac{{\partial ^2 {\bf{A}}}}{{\partial x^2 }} + \frac{{\partial ^2 {\bf{A}}}}{{\partial y^2 }} + \frac{{\partial ^2 {\bf{A}}}}{{\partial y^2 }} = \\ = \left( {\frac{{\partial ^2 {\bf{A}}_x }}{{\partial x^2 }};\frac{{\partial ^2 {\bf{A}}_y }}{{\partial x^2 }};\frac{{\partial ^2 {\bf{A}}_z }}{{\partial x^2 }}} \right) + \left( {\frac{{\partial ^2 {\bf{A}}_x }}{{\partial y^2 }};\frac{{\partial ^2 {\bf{A}}_y }}{{\partial y^2 }};\frac{{\partial ^2 {\bf{A}}_z }}{{\partial y^2 }}} \right) + \left( {\frac{{\partial ^2 {\bf{A}}_x }}{{\partial z^2 }};\frac{{\partial ^2 {\bf{A}}_y }}{{\partial z^2 }};\frac{{\partial ^2 {\bf{A}}_z }}{{\partial z^2 }}} \right) = \\ = \left( {\frac{{\partial ^2 {\bf{A}}_x }}{{\partial x^2 }} + \frac{{\partial ^2 {\bf{A}}_x }}{{\partial y^2 }} + \frac{{\partial ^2 {\bf{A}}_x }}{{\partial z^2 }};\frac{{\partial ^2 {\bf{A}}_y }}{{\partial x^2 }} + \frac{{\partial ^2 {\bf{A}}_y }}{{\partial y^2 }} + \frac{{\partial ^2 {\bf{A}}_y }}{{\partial z^2 }};\frac{{\partial ^2 {\bf{A}}_z }}{{\partial x^2 }} + \frac{{\partial ^2 {\bf{A}}_z }}{{\partial y^2 }} + \frac{{\partial ^2 {\bf{A}}_z }}{{\partial z^2 }}} \right) = \\ = \left( {\nabla ^2 {\bf{A}}_x ;\nabla ^2 {\bf{A}}_y ;\nabla ^2 {\bf{A}}_z } \right) \\ \end{array}$

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