Divergence

例題ベクトル場 ${\bf F}$ の発散を求めよ．

$\displaystyle {\bf F} = 3xy{\bf i} + x^{2}y{\bf j} + y^{2}z{\bf k}$

解答ベクトル場の発散は次のように定義します．

$\displaystyle {\rm div} {\bf F} = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$

また， $\nabla = \frac{\partial }{\partial x}{\bf i} + \frac{\partial }{\partial y}{\bf j} + \frac{\partial }{\partial z}{\bf k}$ 演算子を用いて ${\rm div} {\bf F} = \nabla \cdot {\bf F}$ と表します．

$\displaystyle {\rm div} \boldsymbol{F}$	$\displaystyle =$	$\displaystyle \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$
	$\displaystyle =$	$\displaystyle 3y + x^{2} + y^{2}$

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