Problem
When the $\mathbfit{\nabla}$ operator ($\mathbfit{\nabla}=\mathbfit{\hat{x}}\frac{\partial}{\partial x}+\mathbfit{\hat{y}}\frac{\partial}{\partial y}+\mathbfit{\hat{z}}\frac{\partial}{\partial z}$) acts on a scalar function, the result is called the gradient. $$\\$$
(a) What is the result called, when $\mathbfit{\nabla}$ acts upon a vector function $\mathbfit F$ via the dot product: $\mathbfit {\nabla} \cdot \mathbfit{F}$? $$\\$$
(b) Using the definition of the $\mathbfit{\nabla}$ operator ($\mathbfit{\nabla}=\mathbfit{\hat{x}}\frac{\partial}{\partial x}+\mathbfit{\hat{y}}\frac{\partial}{\partial y}+\mathbfit{\hat{z}}\frac{\partial}{\partial z}$), derive the mathematical equation for $\mathbfit{\nabla}\cdot \mathbfit {F}$, if $F_x$, $F_y$, and $F_z$ are the projections of $\mathbfit F$ along the $x$, $y$, and $z$ axes, respectively. $$\\$$
(c) Is the result a vector or a scalar?