eDivergence

Example Find ${\rm div}{\bf F}$ ．

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

Answer Divergence of vector field ${\bf F}$ is defined as follows:

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

Using $\nabla = \frac{\partial }{\partial x}{\bf i} + \frac{\partial }{\partial y}{\bf j} + \frac{\partial }{\partial z}{\bf k}$ , we can define ${\rm div} {\bf F} = \nabla \cdot {\bf F}$ ．

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