ベクトルの公式

$\displaystyle \mathop{\mathop{\emph ▽}\nolimits }\nolimits (fg)$	$\displaystyle =$	$\displaystyle f(\mathop{\mathop{\emph ▽}\nolimits }\nolimits g)+g(\mathop{\mathop{\emph ▽}\nolimits }\nolimits f)$	(B.1)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits (f\emph A)$	$\displaystyle =$	$\displaystyle A(\mathop{\mathop{\emph ▽}\nolimits }\nolimits f)+f(\mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits \emph A)$	(B.2)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \times}\nolimits (f\emph A)$	$\displaystyle =$	$\displaystyle f(\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph A)+(\mathop{\mathop{\emph ▽}\nolimits }\nolimits f)\times\emph A$	(B.3)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits }\nolimits (\emph A\cdot\emph B)$	$\displaystyle =$	$\displaystyle (\emph B\cdot\mathop{\emph ▽}\nolimits )\emph A +(\emph A\cdot\mathop{\emph ▽}\nolimits )\emph B$
		$\displaystyle +\emph A\times(\mathop{\mathop{\emph ▽}\nolimits \times}\nolimit... ...h B)+\emph B\times(\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph A)$	(B.4)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits (\emph A\times\emph B)$	$\displaystyle =$	$\displaystyle \emph B(\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph A)-\emph A(\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph B)$	(B.5)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \times}\nolimits (\emph A\times\emph B)$	$\displaystyle =$	$\displaystyle \emph A(\mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits \emph B)-\emph B(\mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits \emph A)$
		$\displaystyle +(\emph B\cdot\mathop{\emph ▽}\nolimits )\emph A-(\emph A\cdot\mathop{\emph ▽}\nolimits )\emph B$	(B.6)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits (\mathop{\mathop{\emph ▽}\nolimits }\nolimits f)$	$\displaystyle =$	$\displaystyle \mathop{\emph ▽}\nolimits ^2f$	(B.7)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \times}\nolimits (\mathop{\mathop{\emph ▽}\nolimits }\nolimits f)$	$\displaystyle =$	$\displaystyle \emph0$	(B.8)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits }\nolimits (\mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits \emph A)$	$\displaystyle =$	$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \times}\nolimits (\mathop{\mat... ...mph ▽}\nolimits \times}\nolimits \emph A)+\mathop{\emph ▽}\nolimits ^2\emph A$	(B.9)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits (\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph A)$	$\displaystyle =$	0	(B.10)
$\displaystyle \mathop{\mathop{\emph ▽}\nolimits \times}\nolimits (\mathop{\mathop{\emph ▽}\nolimits \times}\nolimits \emph A)$	$\displaystyle =$	$\displaystyle \mathop{\mathop{\emph ▽}\nolimits }\nolimits (\mathop{\mathop{\emph ▽}\nolimits \cdot}\nolimits \emph A)-\mathop{\emph ▽}\nolimits ^2\emph A$	(B.11)