Example Find ${\rm div}{\bf F}$．

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

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

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

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

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

$$= 3y + x^{2} + y^{2}$$